where p is the differential operator.

The stator winding is directly connected to the power system. Therefore, stator voltage becomes constant in the normal state, leading to the conditions UM1δ ≈ 0 and UT1δ ≈ constant. And L2σ is negligible because it is generally small [25].

Thus, Equations (26) and (27) become as follows:

$$I\_{M2} = \mathcal{U}\_{M2} \;/\; R\_2 \tag{28}$$

$$I\_{T2} = \left(\mathcal{U}\_{T2} - \mathcal{U}\_{T1\delta} + \alpha\_2 \mathcal{\boldsymbol{\nu}}\_{\delta}\right) / \mathcal{R}\_2 \tag{29}$$

Equations (28) and (29) show that rotor voltages along the M and T axes, respectively depend only on the rotor currents along the M and T axes. In other words, the relationships between the currents and voltages along the M and T axes are linear. Consequently, the rotor currents IM2 and IT2 can be controlled independently by regulating the rotor voltages UM2 and UT2.

### **2.4. Composition of the control system**

These equations are transformed by using the relationships ψT1δ = ψT2δ, ψM1δ = ψM2δ, and ωS =

( ) ( ) 2 2 1 2 22 2 2 2 *M M M M S T S T*

( ) ( ) 2 2 1 2 22 2 2 2 *T T T T S M S M*

 d

 d w wy

w wy

 d

 d w y

w y

s

2 22 / *M M I UR* = (28)

(29)

s

y

y

d

d

 d

 d  d

 d

(22)

(23)

(24)

(25)

*dt dt*

*dt dt*

*M M S TM T*

*T T S MT M*

*I U T T S MT* 2 2 ( ( ) *LI U* 22 1 2 2 2 ) /( ) *R pL*

2 2 12 2 ( ) / *T TT I UU R* =-+d

Equations (28) and (29) show that rotor voltages along the M and T axes, respectively depend only on the rotor currents along the M and T axes. In other words, the relationships between the currents and voltages along the M and T axes are linear. Consequently, the rotor currents IM2 and IT2 can be controlled independently by regulating the rotor voltages UM2 and UT2.

 d w y

 d

The stator winding is directly connected to the power system. Therefore, stator voltage becomes constant in the normal state, leading to the conditions UM1δ ≈ 0 and UT1δ ≈ constant.

When the Equations (22), (23), (24), and (25) are transformed by the rotor current IM2 and IT2,

 d*L I U R pL* 22 1 2 2 ) /( )

= ++ - + (26)

 d w y

= -+ - + + (27)

 = + ++ + w wds

 = + -+ + + w wds

d s

 = + ++ + ++ w w

( ) 2 2 2 22 2 2 1 22

( ) 2 2 2 22 2 2 1 22

*M*

*T*

*I U M M S TM* 2 2 ( ( ) w wds

> w wds

And L2σ is negligible because it is generally small [25].

Equations (26) and (27) are given:

where p is the differential operator.

Figure 4 illustrates the control system diagram. Considering the analysis of control system, it can be interpreted that M-T frame can be used to describe the composition of the active and the reactive power. Mentioned control system has six parts: (1) power control loop (regulation of the rotor current references from the deviation between detection and reference values for both active and reactive power); (2) current PI regulator for rotor currents IM2 and IT2 (same application with the first part for regulation of the voltage); (3) air-gap flux calculator (using the stator currents, voltages, and the signals of the position sensor); (4) P1 and Q1 detector (calculation of the stator active and reactive power); (5) detector of the rotor current (vector values of M-T axis, Equation (30)); (6) coordinate transformer (three-phase voltage references, Equation (31)) [25].

$$
\begin{bmatrix} I\_{M2} \\ I\_{I2} \end{bmatrix} = \begin{bmatrix} \cos(\lambda + \gamma) & \sin(\lambda + \gamma) \\ \sin(\lambda + \gamma) & \cos(\lambda + \gamma) \end{bmatrix} \begin{bmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \end{bmatrix} \begin{bmatrix} I\_{s2} \\ I\_{b2} \\ I\_{c2} \end{bmatrix} \tag{30}
$$

$$
\begin{bmatrix} \mathcal{U}\_{a2} \\ \mathcal{U}\_{b2} \\ \mathcal{U}\_{c2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -1/2 & \sqrt{3}/2 \\ -1/2 & -\sqrt{3}/2 \end{bmatrix} \begin{bmatrix} \cos(\lambda + \gamma) & -\sin(\lambda + \gamma) \\ \sin(\lambda + \gamma) & \cos(\lambda + \gamma) \end{bmatrix} \begin{bmatrix} \mathcal{U}\_{M2} \\ \mathcal{U}\_{T2} \end{bmatrix} \tag{31}
$$

**Figure 4.** Block diagram of the control system

### **2.5. Power reference generation**

Power Reference Generation (PRG) is one of the key modules of the algorithm. Strangely, it is the only module that depends on the parameter values. Due to direct connection to the torque, active power reference is the simplest way to find out the torque value desired to produce. On the other hand, there are some difficulties related with the active power reference.

Normally, the following equation is enough to calculate active power reference [10]:

$$P\_{ref} = T\_{ref} \alpha\_{m2} = T\_{ref} \left(2\pi m\_2\right) \tag{32}$$

where Tref is supplied from some outer loop such as a speed control loop, and n2 is the actual mechanical shaft speed at a particular instant of time.

Equation (32) is reliable under some limits so that calculated value of active power is not adequate to generate the torque Tref. Main reason is the losses (losses in the stator and rotor resistance and iron losses) where all input power can not be transmitted to output. Loss in rotor resistance is influenced by the slip of the machine. Schematic demonstration of the power flows under motoring and generating for an induction machine are shown in Fig. 5.

**Figure 5.** Schematic demonstration of the power flows in an induction machine under (a) motoring and (b) generating (Pi = input power (taken from supply in motor mode, given from shaft in generator mode), Pm = mechanical power, Pδ = air-gap power (transferred from stator to rotor under motoring and from rotor to stator under generating), Pl = pow‐ er loss, Pfe = iron loss, Pcu = copper loss, Pa = additional losses produced by harmonics, Pf,w = friction and windage loss‐ es, Pel = effective electrical power taken from rotor circuit, Po = output power (shaft power in motor mode, electrical power in generation mode) (1 subscript means stator, 2 subscript means rotor) [28]

As seen in literature, whilst some researchers can assume that the stator resistance power and the iron losses can be ignored (under many practical situations), the power in the rotor resistance cannot be ignored, especially under heavy load conditions when the slip of the machine can be large [10].

As the diagram shows, under the motoring condition the input power separates into two parts: one part is for the losses (stator, iron, and harmonics) and the other part is related with the rotating field power (air-gap power).

Also, under generating the input shaft power separates into two parts: the losses (friction, windage) and mechanical power. All these parts should be balanced to reach the goal power value. To manage this aim, we should be able to calculate the slip and afterward we could calculate required power expressions as follows for the motoring/generating situation (if the friction and windage losses are neglected):

$$P\_{ref} = \frac{P\_{shaft}}{\left(1 - s\right)} = \frac{T\_{ref} \alpha\_{m2}}{\left(1 - s\right)}\tag{33}$$

where Pshaft is the desired shaft power, and Pref is the terminal reference power as defined previously.

The reactive power reference generation is linked with the slip. So, first we should deal with it. Figure 2 gives an approximate vector diagram of the voltages, currents, and fluxes of an induction machine. Reactive power can be calculated as the multiplication of the emf voltage E in one axis (e.g. qr) of the machine by the current in the other axis (e.g. dr):

$$\left| \mathbf{Q} \right| = I\_{nr} E \tag{34}$$

$$\left| \mathbf{Q} \right| = I\_{mr} \alpha\_1 \nu\_m \tag{35}$$

where ψm is the flux magnitude, Imr is the magnetising current, and ω1 is the stator angular speed related with electrical frequency.

Rearranging (35) one can write:

$$\alpha\_{sref} = -\left(\frac{\left|Q\_{rf}\right|}{\left(I\_{mrrf}\nu\_{mrf}\right)}\right) - \alpha\_2\tag{36}$$

realizing:

= pow‐

active power reference is the simplest way to find out the torque value desired to produce. On

where Tref is supplied from some outer loop such as a speed control loop, and n2 is the actual

Equation (32) is reliable under some limits so that calculated value of active power is not adequate to generate the torque Tref. Main reason is the losses (losses in the stator and rotor resistance and iron losses) where all input power can not be transmitted to output. Loss in rotor resistance is influenced by the slip of the machine. Schematic demonstration of the power

**Figure 5.** Schematic demonstration of the power flows in an induction machine under (a) motoring and (b) generating

er loss, Pfe = iron loss, Pcu = copper loss, Pa = additional losses produced by harmonics, Pf,w = friction and windage loss‐ es, Pel = effective electrical power taken from rotor circuit, Po = output power (shaft power in motor mode, electrical

As seen in literature, whilst some researchers can assume that the stator resistance power and the iron losses can be ignored (under many practical situations), the power in the rotor

power in generation mode) (1 subscript means stator, 2 subscript means rotor) [28]

 = input power (taken from supply in motor mode, given from shaft in generator mode), Pm = mechanical power, Pδ = air-gap power (transferred from stator to rotor under motoring and from rotor to stator under generating), Pl

 p

(32)

the other hand, there are some difficulties related with the active power reference.

Normally, the following equation is enough to calculate active power reference [10]:

2 2 ( ) 2 *ref ref m ref PT T n* = = w

flows under motoring and generating for an induction machine are shown in Fig. 5.

mechanical shaft speed at a particular instant of time.

58 Induction Motors - Applications, Control and Fault Diagnostics

(Pi

f2 = pp.n2

ω2 = pp.ωm2

ω1 = ω2 + ωs

ωs = slip angular speed related with slip frequency

ωs ref = the desired slip angular speed related with slip frequency

Imr ref = the desired magnetizing current

pp = the pole pairs

│Qref│ = the desired reactive power

The negative sign in Equation (36) results from the sign of Qref, the reference reactive power. Given Equation (36), we can now write the expression for the slip:

$$\mathbf{s} = \frac{\alpha \mathbf{o}\_s}{\left(\alpha \mathbf{o}\_2 + \alpha \mathbf{o}\_s\right)} = \mathbf{1} + \left(\alpha \mathbf{o}\_2 \mathbf{I}\_{mnrq} \mathbf{y} \mathbf{v}\_{mrq} \,/\, \mathbf{Q}\_{nq}\right) = \mathbf{1} + \left(\alpha \mathbf{o}\_2 \mathbf{L}\_n \mathbf{I}\_{mnrq}^2 \,/\, \mathbf{Q}\_{nq}\right) \tag{37}$$

This expression can be substituted into the active power slip compensation term 1/(1-s). The Imr ref and ψm ref terms in this expression are reference values. Clearly, Imr and ψm for an induction machine are related, that is, ψm = Lm.Imr. Hence, we have written the numerator of Equation (37) as Lm.Imr ref2. This requires acquiring the value of magnetizing inductance of the machine.

The reactive power expression can be written as:

$$\left| \bigotimes\_{ref} \right| = I\_{mr} \alpha \rho\_1 \nu\_m = I\_{mr} \nu\_m \left( \alpha \rho\_2 + o \rho\_s \right) \tag{38}$$

So, it can be concluded that the reactive power is dependent on the slip frequency. The slip frequency is well-related with the torque. In case a rapid change in torque is needed, slip frequency changes as a step and consequently the reactive power value changes. The torque T and ωs expressions of an induction machine can be written as (using the standard expression from Field Oriented Control):

$$T = \mathbf{1.5p}\_p \mathbf{L}\_m^2 \left| I\_{mr} \right| I\_p \text{ / } \mathbf{L}\_2 \tag{39}$$

$$
\alpha \alpha\_s = I\_p \nmid \left( \pi\_2 \left| I\_{mr} \right| \right) \tag{40}
$$

where τ2 = L2 /R2, L2 = the rotor inductance, and R2 = the rotor resistance. Therefore, τ2 is the rotor time constant.

Rearranging Equation (39), one can write:

Active and Reactive Power Control of Wound Rotor Induction Generators by Using the Computer and Driver http://dx.doi.org/10.5772/61130 61

$$I\_p = \left(TL\_2\right) / \left(1.5 p\_p L\_m^2 \left| I\_{mr} \right|\right) \tag{41}$$

Substituting Equation (41) into Equation (40) gives:

ω1 = ω2 + ωs

pp = the pole pairs

ωs = slip angular speed related with slip frequency

60 Induction Motors - Applications, Control and Fault Diagnostics

Imr ref = the desired magnetizing current

│Qref│ = the desired reactive power

2

w w

from Field Oriented Control):

rotor time constant.

Rearranging Equation (39), one can write:

w

*s*

The reactive power expression can be written as:

ωs ref = the desired slip angular speed related with slip frequency

Given Equation (36), we can now write the expression for the slip:

The negative sign in Equation (36) results from the sign of Qref, the reference reactive power.

( ) ( ) ( ) 2 2 2

This expression can be substituted into the active power slip compensation term 1/(1-s). The Imr ref and ψm ref terms in this expression are reference values. Clearly, Imr and ψm for an induction machine are related, that is, ψm = Lm.Imr. Hence, we have written the numerator of Equation (37) as Lm.Imr ref2. This requires acquiring the value of magnetizing inductance of the machine.

> y w w

So, it can be concluded that the reactive power is dependent on the slip frequency. The slip frequency is well-related with the torque. In case a rapid change in torque is needed, slip frequency changes as a step and consequently the reactive power value changes. The torque T and ωs expressions of an induction machine can be written as (using the standard expression

2

( ) 2 / *s p mr*

 t

where τ2 = L2 /R2, L2 = the rotor inductance, and R2 = the rotor resistance. Therefore, τ2 is the

w

*mrref mref ref m mrref ref*

 w

+ (37)

1 2 ( )*s* == + (38)

2 1.5 / *p m mr p T pL I I L* = (39)

= *I I* (40)

1 /1 / *s*

*s I Q LI Q*

wy

*QI I ref mr m mr m* wy

= =+ = +

$$
\rho \alpha\_s = \left( TL\_2 \right) / \left( 1.5 \, p\_p \, L\_m^2 \left| I\_{mr} \right|^2 \pi\_2 \right) \tag{42}
$$

The denominator in Equation (42) can be simplified by assuming that the leakage inductance of the machine is very small in relation to L2, and hence L2 ≈ Lm. Therefore, Equation (42) can be written as:

$$\rho o\_s \approx \frac{2T}{\left(3p\_p L\_m \left| I\_{mr} \right|^2 \tau\_2\right)} = \frac{2T}{\left(3p\_p \nu\_m \left| I\_{mr} \right| \tau\_2\right)}\tag{43}$$

We are now in a position to write an expression for the reference reactive power. Substituting Equation (43) into Equation (38) and simplifying, we get Equation (44):

$$\mathbf{Q}\_{n\circ f} = -\left[ \left( \left| I\_{n\circ} \right|\_{n\circ f} \alpha\_2 \mathbf{y}\_{n\circ n\circ f} \right) + \left( \frac{2T\_{n\circ f}}{3\mathcal{p}\_p \pi\_2} \right) \right] = -\left[ L\_m \left( \left| I\_{n\circ r} \right|\_{n\circ f} \right)^2 \alpha\_2 + \left( \frac{2T\_{n\circ f}}{3\mathcal{p}\_p \pi\_2} \right) \right] \tag{44}$$

### **3. Experimental studies**

An experimental setup using the control system shown in Fig. 4 was established by Yamamoto and Motoyoshi and the characteristics of the control system along the M-T frame have been experimentally examined. Experimental data are taken from their study [25]. The schematic diagram of the experimental system is shown in Fig. 6. Specifications of the generator used in experimental system are shown in Table 1. Equations (16) and (17) show that active and reactive power is proportional to the rotor currents IT2 and IM2.

**Table 1.** Specifications of the generator used in experimental system [25]

**Figure 6.** Experimental system [25]

### **3.1. Experimental results**

Figures 7 and 8 show the relationships between stator power and rotor current reference and Figs. 9 and 10 show the relationships between stator power and stator power reference.

Figure 10 shows the step response of the stator reactive power Q1 and the step response of the rotor current IM2 which is in proportion to the stator reactive power (Q1). Figure 8 shows the step response of the rotor current IT2 which is in proportion to the stator active power (P1). Figure 9 shows the step response of the stator active power P1. IM2 and IT2 respond to stepping of I\*M2 and I\*T2 in 20 ms without any effect on IT2 and IM2. P1 and Q1 respond to stepping of P\*1 and Q\*1 in 80 ms without any effect on Q1 and P1. The effects of this active and reactive power control method have been proved by these experimental results.

**Figure 7.** Step response of the M-axis current [25]

Active and Reactive Power Control of Wound Rotor Induction Generators by Using the Computer and Driver http://dx.doi.org/10.5772/61130 63

**Figure 8.** Step response of the T-axis current [25]

**3.1. Experimental results**

**Figure 6.** Experimental system [25]

62 Induction Motors - Applications, Control and Fault Diagnostics

**Figure 7.** Step response of the M-axis current [25]

Figures 7 and 8 show the relationships between stator power and rotor current reference and Figs. 9 and 10 show the relationships between stator power and stator power reference.

Figure 10 shows the step response of the stator reactive power Q1 and the step response of the rotor current IM2 which is in proportion to the stator reactive power (Q1). Figure 8 shows the step response of the rotor current IT2 which is in proportion to the stator active power (P1). Figure 9 shows the step response of the stator active power P1. IM2 and IT2 respond to stepping of I\*M2 and I\*T2 in 20 ms without any effect on IT2 and IM2. P1 and Q1 respond to stepping of P\*1 and Q\*1 in 80 ms without any effect on Q1 and P1. The effects of this active and reactive

power control method have been proved by these experimental results.

**Figure 9.** Step response of the active power [25]

**Figure 10.** Step response of the reactive power [25]

### **3.2. Harmonic analysis**

For the power converter of this system, the cycloconverter which is suitable for a large capacity system is often used. Generally, when the large-scale converter is applied to the electric power network system, it is a very important item to analyze the harmonic currents of the power converter. As known, the harmonic currents of the power converter are transmitted to the electric power network system through the rotor and stator windings. In this chapter, the characteristics of the transmission of the harmonic currents caused by the cycloconverter are analyzed theoretically.

Before beginning analysis, let us remember the basic principles of cycloconverter and mathe‐ matical background (taken from [30]) for the understanding of the subject.

### *3.2.1. Principle of the cycloconverter*

[30]

As known, cycloconversion is concerned mostly with direct conversion of energy to a different frequency by synthesizing a low-frequency wave from appropriate sections of a higherfrequency source. As shown in Fig. 11a, a cycloconverter can be considered to be composed of two converters connected back-to-back. The load waveforms of Fig. 11b show that in the general case, the instantaneous power flows in the load fall into one of four periods. The two periods, when the product of load voltage and current is positive, require power flow into the load, dictating a situation where the converter groups rectify, the positive and negative groups conducting respectively during the appropriate positive and negative load-current periods. periods. The two periods, when the product of load voltage and current is positive, require power flow into the load, dictating a situation where the converter groups rectify, the positive and negative groups conducting respectively during the appropriate positive and negative load‐current periods.

Figure 11. General cycloconverter layout (a) Block diagram representation, (b) Ideal load waveforms **Figure 11.** General cycloconverter layout (a) Block diagram representation, (b) Ideal load waveforms [30]

group labeled P and the negative group for reverse current labeled N.

Figure 12. Single‐phase load fed from a three‐pulse cycloconverter [30]

The other two periods represent times when the product of load voltage and current is negative; hence, the power flow is out of the load, demanding that the converters operate in the inverting mode. As shown in Fig. 11a, the principle of the cycloconverter can be demonstrated by using the simplest possible single‐phase input to single‐phase output with a pure resistance as load. Each converter is a bi‐phase half‐wave connection, the positive The other two periods represent times when the product of load voltage and current is negative; hence, the power flow is out of the load, demanding that the converters operate in the inverting mode. As shown in Fig. 11a, the principle of the cycloconverter can be demon‐ strated by using the simplest possible single-phase input to single-phase output with a pure resistance as load. Each converter is a bi-phase half-wave connection, the positive group labeled P and the negative group for reverse current labeled N.

[Placeholder for fig. 1Please, do not alter]

The operation of the blocked group cycloconverter with various loads can be readily explained by reference to the three‐pulse connection shown in Fig. 12, with the associated waveforms for inductive load in Figs. 13 and 14. As known, the wound rotor induction motor is an inductive load. In this chapter, waveforms are drawn for inductive load.

Active and Reactive Power Control of Wound Rotor Induction Generators by Using the Computer and Driver http://dx.doi.org/10.5772/61130 65

**Figure 12.** Single-phase load fed from a three-pulse cycloconverter [30]

**3.2. Harmonic analysis**

64 Induction Motors - Applications, Control and Fault Diagnostics

analyzed theoretically.

*3.2.1. Principle of the cycloconverter*

negative load‐current periods.

[30]

For the power converter of this system, the cycloconverter which is suitable for a large capacity system is often used. Generally, when the large-scale converter is applied to the electric power network system, it is a very important item to analyze the harmonic currents of the power converter. As known, the harmonic currents of the power converter are transmitted to the electric power network system through the rotor and stator windings. In this chapter, the characteristics of the transmission of the harmonic currents caused by the cycloconverter are

Before beginning analysis, let us remember the basic principles of cycloconverter and mathe‐

As known, cycloconversion is concerned mostly with direct conversion of energy to a different frequency by synthesizing a low-frequency wave from appropriate sections of a higherfrequency source. As shown in Fig. 11a, a cycloconverter can be considered to be composed of two converters connected back-to-back. The load waveforms of Fig. 11b show that in the general case, the instantaneous power flows in the load fall into one of four periods. The two periods, when the product of load voltage and current is positive, require power flow into the load, dictating a situation where the converter groups rectify, the positive and negative groups conducting respectively during the appropriate positive and negative load-current periods.

periods. The two periods, when the product of load voltage and current is positive, require power flow into the load, dictating a situation where the converter groups rectify, the positive and negative groups conducting respectively during the appropriate positive and

matical background (taken from [30]) for the understanding of the subject.

(a) (b)

**Figure 11.** General cycloconverter layout (a) Block diagram representation, (b) Ideal load waveforms [30]

Figure 11. General cycloconverter layout (a) Block diagram representation, (b) Ideal load waveforms

The other two periods represent times when the product of load voltage and current is negative; hence, the power flow is out of the load, demanding that the converters operate in the inverting mode. As shown in Fig. 11a, the principle of the cycloconverter can be demon‐ strated by using the simplest possible single-phase input to single-phase output with a pure resistance as load. Each converter is a bi-phase half-wave connection, the positive group

demonstrated by using the simplest possible single‐phase input to single‐phase output with a pure resistance as load. Each converter is a bi‐phase half‐wave connection, the positive

[Placeholder for fig. 1Please, do not alter]

The operation of the blocked group cycloconverter with various loads can be readily explained by reference to the three‐pulse connection shown in Fig. 12, with the associated waveforms for inductive load in Figs. 13 and 14. As known, the wound rotor induction motor is an inductive load. In this chapter, waveforms are drawn for inductive load.

The other two periods represent times when the product of load voltage and current is negative; hence, the power flow is out of the load, demanding that the converters operate in

the inverting mode. As shown in Fig. 11a, the principle of the cycloconverter can be

group labeled P and the negative group for reverse current labeled N.

labeled P and the negative group for reverse current labeled N.

Figure 12. Single‐phase load fed from a three‐pulse cycloconverter [30]

The operation of the blocked group cycloconverter with various loads can be readily explained by reference to the three-pulse connection shown in Fig. 12, with the associated waveforms for inductive load in Figs. 13 and 14. As known, the wound rotor induction motor is an inductive load. In this chapter, waveforms are drawn for inductive load.

**Figure 13.** Waveforms with maximum voltage to an inductive load [30]

**Figure 14.** Waveforms when the load voltage is at half maximum (inductive load current continuous) [30]

bridge [30] The waveforms are as shown in Fig. 13 when the load is inductive, these being at a condition **Figure 15.** Cycloconverter connections with three-phase output. (a) Three-pulse bridge, (b) Six-pulse bridge [30]

Figure 15. Cycloconverter connections with three‐phase output. (a) Three‐pulse bridge, (b) Six‐pulse

of maximum voltage. The load current will lag the voltage and, as the load‐current direction determines which group is conducting, the group on‐periods are delayed relative to the desired output voltage. The group thyristors are fired at such angles to achieve an output as close as possible to a sinewave, but now the lagging load current takes each group The waveforms are as shown in Fig. 13 when the load is inductive, these being at a condition of maximum voltage. The load current will lag the voltage and, as the load-current direction determines which group is conducting, the group on-periods are delayed relative to the desired output voltage. The group thyristors are fired at such angles to achieve an output as close as possible to a sinewave, but now the lagging load current takes each group into the inverting mode. The group will cease conducting when the load current reverses. The loadvoltage waveform is shown as a smooth transfer between groups, but in practice, a short gap would be present to ensure cessation of current in, and the regaining of the blocking state in, the outgoing group, before the incoming group is fired. The waveforms drawn assume the current is continuous within each load half-cycle. The effects of overlap will in practice be present in the waveforms.

In practice, the load-current waveform in Fig. 13 is assumed to be sinusoidal, although it would contain a ripple somewhat smaller than that in the voltage waveform. A light load inductance would result in discontinuous current, giving short zero-voltage periods. Each thyristor will conduct the appropriate block of load current, having the branch currents shown. The current ia - ib would represent the transformer input line current if one assumed the supply to be via a delta primary transformer. The input-current waveform shows changes in shape from cycle to cycle but where the input and output frequencies are an exact multiple, the waveform will repeat over each period of output frequency.

**Figure 14.** Waveforms when the load voltage is at half maximum (inductive load current continuous) [30]

(a) (b)

bridge [30]

66 Induction Motors - Applications, Control and Fault Diagnostics

Figure 15. Cycloconverter connections with three‐phase output. (a) Three‐pulse bridge, (b) Six‐pulse

**Figure 15.** Cycloconverter connections with three-phase output. (a) Three-pulse bridge, (b) Six-pulse bridge [30]

The waveforms are as shown in Fig. 13 when the load is inductive, these being at a condition of maximum voltage. The load current will lag the voltage and, as the load‐current direction determines which group is conducting, the group on‐periods are delayed relative to the desired output voltage. The group thyristors are fired at such angles to achieve an output as close as possible to a sinewave, but now the lagging load current takes each group

The waveforms are as shown in Fig. 13 when the load is inductive, these being at a condition of maximum voltage. The load current will lag the voltage and, as the load-current direction determines which group is conducting, the group on-periods are delayed relative to the desired output voltage. The group thyristors are fired at such angles to achieve an output as close as possible to a sinewave, but now the lagging load current takes each group into the inverting mode. The group will cease conducting when the load current reverses. The loadvoltage waveform is shown as a smooth transfer between groups, but in practice, a short gap would be present to ensure cessation of current in, and the regaining of the blocking state in, the outgoing group, before the incoming group is fired. The waveforms drawn assume the

As shown in Fig. 14, a reduction in the output voltage can be obtained by firing angle delay. Here firing is delayed, even at the peak of the output voltage, so that control is possible over the magnitude of the output voltage. Comparison of Fig. 14 with Fig. 13 indicates a higher ripple content when the output voltage is reduced.

As shown in Fig. 15a, the three-pulse cycloconverter when feeding a three-phase load can be connected with a total of 18 thyristors. As shown in Fig. 15b, a six-pulse cycloconverter can be based on either six-phase half-wave blocks or the bridge connection when 36 thyristors are required.

**Figure 16.** Cycloconverter load-voltage waveform with a lagging power factor load (six-pulse connection) [30]

An example of the cycloconverter output waveforms for the higher-pulse connections is given in Fig. 16, with an output frequency of one-third of the input frequency. It is clear from these waveforms that the higher the pulse-number, the closer is the output waveform to the desired sinusoidal waveform. In general, the output frequency is in general limited to about one-half to one-third of the input frequency, the higher-pulse connections permitting a higher limit.

As in Fig. 15, when the three-pulse cycloconverter feeds a three-phase balanced load, the current loading on the supply is much more evenly balanced. The waveforms to illustrate this are given in Fig. 17 for a frequency ratio of 4/1 with a load of 0.707 power factor lagging. It has been assumed that the load current is sinusoidal, although in practice it must contain ripple. The total load current is not identical from one cycle to the next, obviously contains harmonics, and its fundamental component lags the supply voltage by a larger amount than the load power factor angle.

**Figure 17.** Development of total input current to three-pulse cycloconverter with three-phase lagging power factor load [30]

The thyristors of a cycloconverter are commutated naturally, and whether the load is resistive, inductive, or capacitive, the firing of the thyristors must be delayed to shape the output. The net result is that the AC supply input current will always lag its associated voltage.

### *3.2.2. Circulating current mode*

The previous section specified cycloconverter operation where either the positive or negative groups were conducting, but never together. As shown in Fig. 18, if a center-tapped reactor is connected between the positive group P and negative group N, then both groups can be permitted to conduct. The reactor will limit the circulating current, that is, the value of its inductance to the flow of load current from either group being one quarter of its value to the circulating current, because inductance is proportional to the square of the number of turns.

In Fig. 19, typical waveforms are shown for the three-pulse cycloconverter shown in Fig. 18. Each group conducts continuously, with rectifying and inverting modes as shown. The mean between the two groups will be fed to the load, some of the ripple being cancelled in the combination of the two groups. Both groups synthesize the same fundamental sinewave. The reactor voltage is the instantaneous difference between the two group voltages. The circulating current shown in Fig. 19 can only flow in one direction, the thyristors preventing reverse flow. Hence, the current will build up during the reactor voltage positive periods until in the steady state it is continuous, rising and falling as shown.

**Figure 18.** Three-pulse cycloconverter with intergroup reactor [30]

### *3.2.3. Mathematical analysis*

been assumed that the load current is sinusoidal, although in practice it must contain ripple. The total load current is not identical from one cycle to the next, obviously contains harmonics, and its fundamental component lags the supply voltage by a larger amount than the load

**Figure 17.** Development of total input current to three-pulse cycloconverter with three-phase lagging power factor

The thyristors of a cycloconverter are commutated naturally, and whether the load is resistive, inductive, or capacitive, the firing of the thyristors must be delayed to shape the output. The

The previous section specified cycloconverter operation where either the positive or negative groups were conducting, but never together. As shown in Fig. 18, if a center-tapped reactor is connected between the positive group P and negative group N, then both groups can be

net result is that the AC supply input current will always lag its associated voltage.

power factor angle.

68 Induction Motors - Applications, Control and Fault Diagnostics

load [30]

*3.2.2. Circulating current mode*

The above subjects have demonstrated that almost all the waveforms associated with power electronic equipment are non-sinusoidal, which contains harmonic components. The purpose of this subject is to analyze the harmonic content of the various waveforms and discuss their effects as regards both supply and load.

Any periodic waveform may be shown to be composed of the superposition of a direct component with a fundamental pure sinewave component, together with pure sinewaves known as harmonics at frequencies which are integral multiples of the fundamental. A nonsinusoidal wave is often referred to as a complex wave [30].

Mathematically, it is more convenient to express the independent variable as x and the dependent variable as y. Then, the series may be expressed as:

$$y = f\left(\mathbf{x}\right) = a\_0 + a\_1 \cos\left(\mathbf{x}\right) + a\_2 \cos\left(2\mathbf{x}\right) + \dots + a\_n \cos(n\mathbf{x}) + b\_1 \sin(\mathbf{x}) + b\_2 \sin(2\mathbf{x}) + \dots + b\_n \sin\left(n\mathbf{x}\right) \tag{45}$$

**Figure 19.** Waveform of a three-pulse cycloconverter with circulating current but without load [30]

Equation (45) is known as a Fourier series, and where f(x) can be expressed mathematically, a Fourier analysis yields that the coefficients are [31]:

$$a\_0 = \frac{1}{2\pi} \int\_{-\pi}^{\pi} f(\mathbf{x}) d\mathbf{x} \tag{46}$$

$$a\_n = \frac{1}{\pi} \int\_{-\pi}^{\pi} f(\mathbf{x}) \cos nx \mathbf{x} d\mathbf{x} \tag{47}$$

$$b\_n = \frac{1}{\pi} \int\_{-\pi}^{\pi} f(\mathbf{x}) \sin nx dx \tag{48}$$

Alternatively, the series may be expressed as:

$$y = f\left(\mathbf{x}\right) = \mathbf{R}\_0 + \mathbf{R}\_1 \sin\left(\mathbf{x} - \phi\_1\right) + \mathbf{R}\_2 \sin\left(2\mathbf{x} - \phi\_2\right) + \mathbf{R}\_3 \sin\left(3\mathbf{x} - \phi\_3\right) + \dots + \mathbf{R}\_n \sin\left(n\mathbf{x} - \phi\_n\right)\tag{49}$$

Equations (49) and (45) are equivalent, with:

Active and Reactive Power Control of Wound Rotor Induction Generators by Using the Computer and Driver http://dx.doi.org/10.5772/61130 71

$$a\_n \cos n\mathbf{x} + b\_n \sin n\mathbf{x} = R\_n \sin \left(n\mathbf{x} - \phi\_n\right) \tag{50}$$

From which the resultant and phase angle:

$$R\_u = \left(a\_u^2 + b\_u^2\right)^{1/2} \tag{51}$$

$$\phi\_n = \arctan \frac{a\_n}{b\_n} \tag{52}$$

Electrically, it expresses the independent variable as ωt instead of x and the dependent variable as volts or amperes instead of y. Then, the series may be expressed as:

$$\nabla \cdot \mathbf{v} = V\_0 + V\_1 \sin \left( \alpha t - \phi\_1 \right) + V\_2 \sin \left( 2 \alpha t - \phi\_2 \right) + V\_3 \sin \left( 3 \alpha t - \phi\_3 \right) + \dots + V\_n \sin \left( n \alpha t - \phi\_n \right) \tag{53}$$

where

Equation (45) is known as a Fourier series, and where f(x) can be expressed mathematically, a

= ò (46)

= ò (47)

= ò (48)

sin 2 sin 3 ... sin *n n* ( ) (49)

 f

0 ( ) 1 2 *a f x dx* p

**Figure 19.** Waveform of a three-pulse cycloconverter with circulating current but without load [30]

( ) 1 cos *n a f x nxdx* p

( ) 1 sin *n b f x nxdx* p

p p-

p p-

*y f x R R x R x R x R nx* = = + - + - + - ++ - ( ) 01 1 2 2 3 3 sin() ( ) ( ) fff

p p-

Fourier analysis yields that the coefficients are [31]:

70 Induction Motors - Applications, Control and Fault Diagnostics

Alternatively, the series may be expressed as:

Equations (49) and (45) are equivalent, with:

v is the instantaneous value of the voltage at any time t;

Vo is the direct (or mean) value of the voltage;

V1 is the maximum value of the fundamental component of the voltage;

V2 is the maximum value of the second harmonic component of the voltage;

V3 is the maximum value of the third harmonic component of the voltage;

Vn is the maximum value of the nth harmonic component of the voltage;

ϕ defines the relative angular reference; and

ω = 2πf, where f is the frequency of the fundamental component, 1/f

defining the time over which the complex wave repeats itself.

The constant term of Equation (46) is the mean value of the function, and is the value found in, for example, the calculation of the direct (mean) voltage output of a rectifier. In the analysis of a complex wave, certain statements and simplifications are possible by inspection of any given waveform. If the areas of the positive and negative half-cycles are equal, then a0 is zero. If f(x + π) = - f(x), then there are no even harmonics, that is, no second, fourth, etc. In plain terms, this means the negative half-cycle is a reflection of the positive half-cycle. If f(- x) = f(x), then an= 0; that is, there are no sine terms. If f(- x) = f(x), then bn= 0; that is, there are no cosine terms. Symmetry of the waveform can result in Equations (47) and (48) being taken as twice the value of the integral from 0 to π, or four times the value of the integral from 0 to π / 2, hence simplifying the analysis.

Where it is difficult to put a mathematical expression to f(x), or where an analysis of an experimental or practical waveform obtained from a piece of equipment is required, graphical analysis can be performed [30].

The conditions of analysis are as follows.

Using these conditions, the three-phase rotor current can be defined as follows:

$$I\_{a2} = \sum\_{m} \sum\_{n} \frac{A\_{\left(\delta n - 1\right)n} \operatorname{Sim}\left\{6m\theta\_{1} - \left(6n - 1\right)\theta\_{s}\right\} + A\_{\left(\delta n - 5\right)n} \operatorname{Sim}\left\{6m\theta\_{1} - \left(6n - 5\right)\theta\_{s}\right\}}{\left\{6m\theta\_{1} - \left(6n - 1\right)\theta\_{s}\right\} + B\_{\left(\delta n - 5\right)n} \operatorname{Sim}\left\{6m\theta\_{1} - \left(6n - 5\right)\theta\_{s}\right\}}\tag{54}$$

$$\begin{aligned} A\_{(\kappa n - 1)w} \operatorname{Sim} \left\{ \left. \left\{ \mathfrak{Im} \theta\_1 - \left( \mathfrak{Im} - 1 \right) \left( \theta\_s - \frac{2 \pi}{3} \right) \right\} \right| \\\ I\_{b2} = \sum\_m \sum\_n & + B\_{(\kappa n - 3)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \mathfrak{Im} - 5 \right) \left( \theta\_s - \frac{2 \pi}{3} \right) \right\} \right| \\\ I\_{b2} = \sum\_m \sum\_n & + B\_{(\kappa n - 1)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \mathfrak{Im} - 1 \right) \left( \theta\_s - \frac{2 \pi}{3} \right) \right\} \right| \\\ & + B\_{(\kappa n - 3)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \mathfrak{Im} - 5 \right) \left( \theta\_s - \frac{2 \pi}{3} \right) \right\} \right| \end{aligned} \tag{55}$$

$$\begin{aligned} A\_{(\delta n - 1)w} \operatorname{Sim} \left\{ \left. \left\{ \mathfrak{Im} \theta\_1 - \left( \delta n - 1 \right) \left( \theta\_s + \frac{2 \pi}{3} \right) \right\} \right| \\\ I\_{c2} = \sum\_{m} \sum\_{n} & + B\_{(\delta n - 3)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \delta n - 5 \right) \theta\_s + \frac{2 \pi}{3} \right\} \right\} \\\ I\_{c2} = \sum\_{m} \sum\_{n} & + B\_{(\delta n - 1)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \delta n - 1 \right) \left( \theta\_s + \frac{2 \pi}{3} \right) \right\} \right| \\\ & + B\_{(\delta n - 5)w} \operatorname{Sim} \left\{ \left. \mathfrak{Im} \theta\_1 - \left( \delta n - 5 \right) \theta\_s + \frac{2 \pi}{3} \right\} \right\} \end{aligned} \tag{56}$$

where m=1,2,....., n=1,2,......, θl =ωl t: ωl is the stator angular speed, θs=ωst: ωs is the slip angular speed.

Where it is difficult to put a mathematical expression to f(x), or where an analysis of an experimental or practical waveform obtained from a piece of equipment is required, graphical

**1.** The frequency of the harmonic current for the three-pulse and six-pulse cycloconverter is 6mfl ± (2n + 1)fs, where m is any integer from 1 to infinity, n is any integer from 0 to infinity, fl is the frequency of the power source, and fs is the output frequency of the cycloconverter

( ) { ( ) } ( ) { ( ) }

*n m s s n m*

6 61 6 65

+ -- + - - åå (54)

3

p

3

p

p

3

3

3

p

3

p

p

3

3

p

p

qq

qq

6 61 6 65

2 6 61

ì ü ï ï æ ö í ý -- - ç ÷ ï ï î þ è ø

 q

> q

åå (55)

 q

> q

 q

> q

åå (56)

 q

> q

2 6 65

ì ü ï ï æ ö

2 6 61

2 6 65

2 6 61

ì ü ï ï æ ö í ý -- + ç ÷ ï ï î þ è ø

2 6 65

ì ü ï ï æ ö

2 6 61

2 6 65

ì ü ï ï æ ö

ì ü ï ï æ ö

( ) { ( ) } ( ) { ( ) }

**3.** Zero-phase-sequence current (harmonic currents of multiple of 3) is neglected.

**2.** The analysis is carried out only on the harmonic components.

**4.** The stator is connected to the fundamental frequency voltage source.

Using these conditions, the three-phase rotor current can be defined as follows:

qq

qq

6 1 1

*A Sin m n*

6 5 1

*A Sin m n*

*B Sin m n*

*B Sin m n*

6 1 1

6 5 1

6 1 1

*A Sin m n*

6 5 1

*A Sin m n*

*B Sin m n*

*B Sin m n*

6 1 1

6 5 1

6 1 1 1 6 5

6 1 1 1 6 5

( ) ( )

*n m s*

q

*n m s*

q

*n m s*

+ í ý -- - ç ÷ ï ï î þ è ø

+ í ý -- - ç ÷ ï ï î þ è ø

q

*n m s*

q

( ) ( )

+ í ý -- - ç ÷ ï ï î þ è ø = ì ü ï ï æ ö

( ) ( )

( ) ( )

( ) ( )

*n m s*

q

*n m s*

q

*n m s*

+ í ý -- + ç ÷ ï ï î þ è ø

+ í ý -- + ç ÷ ï ï î þ è ø

q

*n m s*

q

( ) ( )

+ í ý -- + ç ÷ ï ï î þ è ø = ì ü ï ï æ ö

( ) ( )

( ) ( )

*m n n m s s n m*

*B Sin m n B Sin m n*

*A Sin m n A Sin m n*

analysis can be performed [30].

[32].

2

=

2

2

*m n*

*c*

*I*

*m n*

*b*

*I*

*a*

*I*

The conditions of analysis are as follows.

72 Induction Motors - Applications, Control and Fault Diagnostics

When Equations (54)–(56) are transformed to the d-q axis based on the stator voltage, (57) and (58) are derived:

$$A\_{\left(\delta n - 1\right)m} \operatorname{Sim}\left\{\left\{m\theta\_1 - \delta n\theta\_s\right\}\right\}$$

$$I\_{d2} = \sum\_m \sum\_n + A\_{\left(\delta n - 5\right)m} \operatorname{Sim}\left\{\left\{mn\theta\_1 - \left(\delta n - 1\right)\theta\_s\right\}\right\}$$

$$+ B\_{\left(\delta n - 1\right)m} \operatorname{Sim}\left\{\left\{mn\theta\_1 + \delta n\theta\_s\right\}\right\}$$

$$+ B\_{\left(\delta n - 3\right)m} \operatorname{Sim}\left\{\left\{mn\theta\_1 + \left(\delta n - 1\right)\theta\_s\right\}\right\}$$

$$\begin{aligned} A\_{\left(\mathfrak{s}n-1\right)w} \mathbb{C} \cos\left\{6m\theta\_1 - 6n\theta\_s\right\} \\\ I\_{q2} = \sum\_{m} & + A\_{\left(\mathfrak{s}n-5\right)w} \mathbb{C} \cos\left\{6m\theta\_1 - \left(6n-1\right)\theta\_s\right\} \\\ I\_{q\left\{4n-1\right\}w} & + B\_{\left(\mathfrak{s}n-1\right)w} \mathbb{C} \cos\left\{6m\theta\_1 + 6n\theta\_s\right\} \\\ & + B\_{\left(\mathfrak{s}n-5\right)w} \mathbb{C} \cos\left\{6m\theta\_1 + \left(6n-1\right)\theta\_s\right\} \end{aligned} \tag{58}$$

The characteristics of transmission from the rotor winding to the stator winding are analyzed by substituting into the fundamental equation of a wound rotor induction machine. The analysis uses the symmetrical coordinate method for simplification. In this case, positive phase sequence component value (F component value) and negative phase sequence component value (B component value) have a conjugate relationship. The F component value is used for the analysis in this paper, which gives the following fundamental equation of the wound rotor induction machine:

$$
\begin{bmatrix} V\_{F1} \\ V\_{F2} \end{bmatrix} = \begin{bmatrix} R\_1 + \left( p + jo\_1 \right) \left( L\_1 + L\_m \right) & \left( p + jo\_1 \right) L\_m \\ \left( p + jo\_0 \right) L\_m & R\_2 + \left( p + jo\_0 \right) \left( L\_2 + L\_m \right) \end{bmatrix} \begin{bmatrix} I\_{r1} \\ I\_{r2} \end{bmatrix} \tag{59}
$$

When Equation (59) is transformed relative to IF1, the following equation is obtained:

$$I\_{F1} = \frac{\left(p + jo\_1\right)L\_m}{R\_1 + \left(p + jo\_1\right)\left(L\_1 + L\_m\right)} I\_{F2} + V\_{F1} \tag{60}$$

In this analysis, stator voltage is the fundamental voltage source, then, VF1 = 0 is defined. When IF1 is expressed by Id2 and Iq2, using Equation (60) and IF2 = (1/√2)(Id2 + jIq2), the following expression is derived:

$$I\_{F1} = \frac{\left(p + jo\_1\right)L\_m}{R\_1 + \left(p + jo\_1\right)\left(L\_1 + L\_m\right)}\frac{1}{\sqrt{2}}\left(I\_{d2} + jI\_{q2}\right) \tag{61}$$

Equation (59) is substituted into Equations (57) and (58). Applying Laplace transformation, the stator current is calculated. A-phase stator current is derived as follows:

$$\mathbb{C}\_{\left(\delta n-1\right)m} \mathbb{C} \cos\left\{ \left( \left( \delta m + 1 \right) \theta\_1 - \delta n \theta\_s \right\} \right. $$

$$I\_{\omega 1} = \sum\_{m} \sum\_{n} \frac{+\mathbb{C}\_{\left(\delta n-3\right)m} \mathbb{C} \cos\left\{ \left( \left( \delta m - 1 \right) \theta\_1 - \left( \delta n - 1 \right) \theta\_s \right\} \right.}{+D\_{\left(\delta n-1\right)m} \mathbb{C} \cos\left\{ \left( \left( \delta m - 1 \right) \theta\_1 + \delta n \theta\_s \right) \right\}} \tag{62}$$

$$+D\_{\left(\delta n - 3\right)m} \mathbb{C} \cos\left\{ \left( \delta m + 1 \right) \theta\_1 + \left( \delta n - 1 \right) \theta\_s \right\}$$

where

$$\begin{split} \mathbf{C}\_{\left(\epsilon u - 1\right)m} &\approx A\_{\left(\epsilon u - 1\right)m} \left( \frac{L\_{m}}{\left(L\_{m} + L\_{1}\right)} \right) , D\_{\left(\epsilon u - 1\right)m} \approx B\_{\left(\epsilon u - 1\right)m} \left( \frac{L\_{m}}{\left(L\_{m} + L\_{1}\right)} \right) \\ \mathbf{C}\_{\left(\epsilon u - 5\right)m} &\approx A\_{\left(\epsilon u - 5\right)m} \left( \frac{L\_{m}}{\left(L\_{m} + L\_{1}\right)} \right) , D\_{\left(\epsilon u - 5\right)m} \approx B\_{\left(\epsilon u - 5\right)m} \left( \frac{L\_{m}}{\left(L\_{m} + L\_{1}\right)} \right) . \end{split}$$

The results of this analysis show that the harmonic currents fed to the rotors winding are transmitted to the stator windings by changing its frequency. The rotor harmonic currents at 6mω1±(6n - 5)ωs change to stator harmonic currents at (6m±1)ω1±6(n - 1)ωs. And the rotor harmonic currents at 6mω1±(6n - 1)ωs change to stator harmonic currents at (6m±1)ω1±6nωs. This is the effect of the rotating speed of the wound rotor induction generator. Moreover, the ratio between the amplitude of harmonic currents in the rotor and the stator is nearly 1:1.

Analysis results are verified by the experimental system shown in Fig. 6. Table 2 gives an example of experimental results. It is seen that experimental values well match with the theoretical ones. Thus, experiments confirmed that the analysis gives reliable results.

**Table 2.** Experimental results (ω1=50 Hz, ωs=50 Hz, m=1, N= 1400 r/min, I1=I2=25 A) [25]

It has been proved by experiment that this control system can control the active and reactive power independently and stably. In addition, it has been confirmed by analysis and experi‐ ment that the harmonic currents fed to the rotor windings of the generator are transmitted to the stator windings changing its frequency [25].

### **3.3. Proposed experimental setup**

( )

+

w

the stator current is calculated. A-phase stator current is derived as follows:

1

74 Induction Motors - Applications, Control and Fault Diagnostics

=

*a*

*I*

where

1 11

( )( ) ( ) 1 1 2 2

*m*

Equation (59) is substituted into Equations (57) and (58). Applying Laplace transformation,

6 1 1

*C Cos m n*

+ -+

+ - --

*C Cos m n*

6 1 1

*D Cos m n*

+ + +-

() () ( ) () () ( )

,

61 61 61 61

*nm nm nm nm*

*L L CA DB*

*L L CA DB*

65 65 65 65

*nm nm nm nm*

() () ( ) () () ( )

The results of this analysis show that the harmonic currents fed to the rotors winding are transmitted to the stator windings by changing its frequency. The rotor harmonic currents at 6mω1±(6n - 5)ωs change to stator harmonic currents at (6m±1)ω1±6(n - 1)ωs. And the rotor harmonic currents at 6mω1±(6n - 1)ωs change to stator harmonic currents at (6m±1)ω1±6nωs. This is the effect of the rotating speed of the wound rotor induction generator. Moreover, the ratio between the amplitude of harmonic currents in the rotor and the stator is nearly 1:1.

Analysis results are verified by the experimental system shown in Fig. 6. Table 2 gives an example of experimental results. It is seen that experimental values well match with the

theoretical ones. Thus, experiments confirmed that the analysis gives reliable results.

 Experimental results (ω1=50 Hz, ωs=50 Hz, m=1, N= 1400 r/min, I1=I2=25 A) [25]

**Rotor Harmonic Current Stator Harmonic Current**

,

*D Cos m n*

6 5 1

6 5 1

*m n n m s*

( ) {( ) }

*n m s*

61 6

q

+ -

 q

> q

åå (62)

1 1

*m m*

*L L L L*

+ + èø èø

*m m*

*L L L L*

+ + èø èø

*m m*

*m m*

1 1

 q

> q

6 1 61

q

61 6

q

6 1 61

q

( ) {( )( ) }

*n m s*

( ) {( ) }

( ) {( )( ) }

*n m s*

æö æö » » ç÷ ç÷

æö æö » » ç÷ ç÷

*m F d q*

*pj L I I jI R pj L L* w

= + ++ +

1 2

(61)

By using the computer and driver/buffer, the experimental setups to control active and reactive power of the wound rotor induction generator independently and stably are shown in Figs. 20 and 21 [28]. The terminals in experimental setups shown in Figs. 20 and 21 are numbered considering that the three-pulse cycloconverter with a total of 18 thyristors shown in Fig. 15a will be used.

**Figure 20.** Schematic block diagram of the experimental setup for computer-aided power control of wound rotor in‐ duction generator

**Figure 21.** Another schematic block diagram of the experimental setup for computer-aided power control of wound rotor induction generator

### **4. Conclusion**

New configurations for power control system of the doubly-fed wound rotor induction generator have been proposed. These configurations are based on a control method using a rotating reference frame fixed on the air-gap flux of the generator. By using them, the active and reactive power of generator can be controlled independently and stably. To achieve this purpose, power and current control that are fundamental subjects have been analyzed and as a result, a computer-aided circuit is given to achieve the power and current control. Using computers enables application of new technologies for easier control. For example, any new metaheuristic techniques or classification/identification techniques could be applied by just changing the code in the computer.

### **Author details**

Fevzi Kentli\*

Address all correspondence to<EMAIL_ADDRESS>
Marmara University- Faculty of Technology – Department of Mechatronics Engineering Kadıköy-Istanbul, Turkey

### **References**

[6] Nakra H L, Dube B. Slip power recovery induction generators for large vertical axis wind turbines. *IEEE Transact Energy Convers*. 1988;3(4):733-737. DOI: 10.1109/60.9346

**4. Conclusion**

**Author details**

Kadıköy-Istanbul, Turkey

Fevzi Kentli\*

**References**

552.

2015]

cessed: March 2015]

10.1109/TIE.2010.2103910

2001;16(3):390-399. DOI: 10.1109/63.923772

changing the code in the computer.

76 Induction Motors - Applications, Control and Fault Diagnostics

Address all correspondence to<EMAIL_ADDRESS>
New configurations for power control system of the doubly-fed wound rotor induction generator have been proposed. These configurations are based on a control method using a rotating reference frame fixed on the air-gap flux of the generator. By using them, the active and reactive power of generator can be controlled independently and stably. To achieve this purpose, power and current control that are fundamental subjects have been analyzed and as a result, a computer-aided circuit is given to achieve the power and current control. Using computers enables application of new technologies for easier control. For example, any new metaheuristic techniques or classification/identification techniques could be applied by just

Marmara University- Faculty of Technology – Department of Mechatronics Engineering

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[3] Nidec Motor Corporation. Induction Generator [Internet]. Available from: http:// www.usmotors.com/TechDocs/ProFacts/Induction-Generator.aspx [Accessed: March

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[22] Morel L, Godfroid H, Mirzaian A, Kauffmann J M. Double-fed induction machine: Converter optimisation and field oriented control without position sensor. *IEE Proc*

[23] Bogalecka E. Power control of a double fed induction generator without speed or po‐ sition sensor. In: Fifth European Conference on Power Electronics and Applications; 13-16 September 1993; Brighton, England. London, England: Institution of Electrical

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78 Induction Motors - Applications, Control and Fault Diagnostics

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## **Analysis and Methodology for Determining the Parasitic Capacitances in VSI-fed IM Drives Based on PWM Technique**

## Rudolf Ribeiro Riehl, Fernando de Souza Campos, Alceu Ferreira Alves and Ernesto Ruppert Filho

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61544

### **Abstract**

Three-phase induction motors present stray capacitances. The aim of this chapter is to present a methodology to experimentally determine these capacitances and also evaluate the effects of electromagnetic interference on motors in common mode. The proposed procedures for this methodology consist of: a) identifying the motor equivalent electrical circuit parameters through characteristic tests performed in the laboratory; b) setting up configurations between the PWM inverter and the motor for voltage and current meas‐ urements: common mode and shaft voltages, leakage and shaft (bearing) currents by us‐ ing a dedicated measuring circuit; c) calculating the parasitic capacitance values between stator and frame, stator and rotor, rotor and frame and bearings of the motor using the capacitance characteristic equation; d) using the dedicated software Pspice to simulate the system composed by the three-phase induction motor fed by PWM inverter with the equivalent electrical circuit parameters; e) determining the characteristic waveforms in‐ volved in the common mode phenomenon.

**Keywords:** Induction motors, parasitic capacitances, PWM inverter

### **1. Introduction**

The use of inverter controlled by pulse width modulation (PWM Inverter), on drive and control of the three-phase induction motors is increasingly common, especially for the power range of up to 10 Hp.

The recent developments in power semiconductor devices (IGBT, MOSFET, and others) have allowed these drives to achieve switching frequencies up to 20 kHz. In these frequencies, the

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

rise time of PWM voltage becomes very small and responsible for the appearance of phenom‐ ena, defined as electromagnetic interference (EMI) in the induction motor [12, 18, 25]. Due to the presence of parasitic capacitances in the motor caused by the free or isolated spaces between metal parts, capacitive couplings occur, which become flow paths of the high-frequency electric current between the motor phases (differential mode coupling) and between phases and the ground (common mode coupling). The higher the switching frequency, the greater are the extensions and consequences of these phenomena.

The differential mode phenomena are responsible for excessive heating of the motor, harming the electrical insulation characteristics, performance, and, consequently, their useful life, and can burn out the motor. This occurs because the PWM inverter does not feed the induction motor with a sinusoidal voltage, but by applying modulated or switched pulses, producing high-frequency harmonics, and high-voltage gradients values (high dV/dt) to the stator windings [3].

The phenomena due to the common mode are responsible for the appearance of circulating currents between motor and ground moving through the frame, bearings, and motor pedestals. As the common mode voltage is different from zero, it raises a shaft voltage between the bearing parts and the ground, which is dependent on both this common mode voltage and the parasitic capacitances that can circulate electric currents capacitive for multiple paths by the motor [10]. One of these paths passes through the bearings, and in this case the motor currents are known as bearing currents, which, due to discharges occurring at the dielectric break, can cause damages at their bearings and if the shaft locks, the motor would be forced to stop and it will cause burning of the windings. Another phenomenon that occurs is electric shock or electric discharge machining (EDM), due to the flow of leakage current motor to the frame when it is not grounded or this ground is not suitable [1, 4, 7].

The conventional method to determine these parasitic capacitances is by measuring the impedance of the induction motor using LCR Bridge [1 to 10]. In this method, the induction motor is disassembled for specific parameters to be measured. For the determination of the parameters between the stator and the motor frame, the rotor is removed and measurements are conducted with each phase bridge between the LCR and the frame and among the three phases (short-circuited in this case) and the frame. After that, the rotor is put back and its axis is short-circuited to the frame. Parameter values between stator and rotor are measured. The LCR Bridge is connected at the common point of the three phases (neutral) and the stator frame.

The parameters of the motor bearings are dynamic and depend on operating conditions (speed) of the three-phase induction motor and also the dielectric characteristics of resistivity of the lubricant temperature and the geometric construction of the motor [1 to 10]. The rotor parameters can only be determined after all previous parameters have been obtained. The LCR Bridge is connected between the rotor shaft and the frame of the induction motor.

In the absence of an LCR Bridge and an appropriate laboratory for this type of test, another methodology [40, 41] is used to determine the parameters of the equivalent circuit of the induction motor in steady state (nominal frequency) [26, 27] and for high frequency [1, 10, 14, 41], through laboratory measurements using features configurations of links between the PWM inverter and the motor for measuring the following parameters of interest: common mode voltage (VCM) and shaft voltage (VSHAFT); leakage (ILEAKAGE) and bearing (IB) currents, by measurement circuit designed for this purpose; and calculation of the values of the parasitic capacitances between the stator and motor frame (CSF), stator and rotor (CSR), rotor and motor frame (CRF) and bearings (CB) using characteristic equations as will be shown in this chapter. The computer application PSPICE [36, 40] is used to simulate the three-phase induction motor powered by a PWM inverter system using the high-frequency equivalent circuit of the motor. The characteristic waveforms that represent the common mode phenomena will be obtained to allow comparisons for validation of procedures to determine the capacitance.

rise time of PWM voltage becomes very small and responsible for the appearance of phenom‐ ena, defined as electromagnetic interference (EMI) in the induction motor [12, 18, 25]. Due to the presence of parasitic capacitances in the motor caused by the free or isolated spaces between metal parts, capacitive couplings occur, which become flow paths of the high-frequency electric current between the motor phases (differential mode coupling) and between phases and the ground (common mode coupling). The higher the switching frequency, the greater are

The differential mode phenomena are responsible for excessive heating of the motor, harming the electrical insulation characteristics, performance, and, consequently, their useful life, and can burn out the motor. This occurs because the PWM inverter does not feed the induction motor with a sinusoidal voltage, but by applying modulated or switched pulses, producing high-frequency harmonics, and high-voltage gradients values (high dV/dt) to the stator

The phenomena due to the common mode are responsible for the appearance of circulating currents between motor and ground moving through the frame, bearings, and motor pedestals. As the common mode voltage is different from zero, it raises a shaft voltage between the bearing parts and the ground, which is dependent on both this common mode voltage and the parasitic capacitances that can circulate electric currents capacitive for multiple paths by the motor [10]. One of these paths passes through the bearings, and in this case the motor currents are known as bearing currents, which, due to discharges occurring at the dielectric break, can cause damages at their bearings and if the shaft locks, the motor would be forced to stop and it will cause burning of the windings. Another phenomenon that occurs is electric shock or electric discharge machining (EDM), due to the flow of leakage current motor to the frame

The conventional method to determine these parasitic capacitances is by measuring the impedance of the induction motor using LCR Bridge [1 to 10]. In this method, the induction motor is disassembled for specific parameters to be measured. For the determination of the parameters between the stator and the motor frame, the rotor is removed and measurements are conducted with each phase bridge between the LCR and the frame and among the three phases (short-circuited in this case) and the frame. After that, the rotor is put back and its axis is short-circuited to the frame. Parameter values between stator and rotor are measured. The LCR Bridge is connected at the common point of the three phases (neutral) and the stator frame.

The parameters of the motor bearings are dynamic and depend on operating conditions (speed) of the three-phase induction motor and also the dielectric characteristics of resistivity of the lubricant temperature and the geometric construction of the motor [1 to 10]. The rotor parameters can only be determined after all previous parameters have been obtained. The LCR

In the absence of an LCR Bridge and an appropriate laboratory for this type of test, another methodology [40, 41] is used to determine the parameters of the equivalent circuit of the induction motor in steady state (nominal frequency) [26, 27] and for high frequency [1, 10, 14, 41], through laboratory measurements using features configurations of links between the

Bridge is connected between the rotor shaft and the frame of the induction motor.

the extensions and consequences of these phenomena.

82 Induction Motors - Applications, Control and Fault Diagnostics

when it is not grounded or this ground is not suitable [1, 4, 7].

windings [3].

In this methodology [40, 41], the three-phase induction motor does not need to be disassembled and reassembled as in the case of measurement with LCR Bridge. Only the bearings are insulated, and the determination of parasitic capacitances is performed while the motor is running. The results of testing for measurement and determination of two 1 Hp induction motor parasitic capacitances and simulations, using the Pspice program, are presented.

### **2. Equivalent circuit of three-phase induction motor at high frequencies**

At high frequencies, the capacitive reactance among the various parts of the three-phase induction motor are shown in Figure 1, which illustrates the equivalent circuit of the threephase induction motor fed by PWM inverter [1, 6, 15]. The distributed parameters R, L, and C represent the high-frequency coupling between the windings of the stator and the rotor.

ZRF is the impedance between the rotor and the motor frame, also called air gap impedance Zg; ZER is the impedance between the windings of the stator and the rotor; ZSF is the impedance between the stator windings and the frame; Zg is the air gap impedance; and ZB represents the impedance between the rotor and the bearing. RW and LW represent the equivalent impedance of the driver through which circulates the bearing current and Rg is the lead resistance connected between the frame and ground. Using the defined impedances shown by equations (1) through (4), one can obtain the high-frequency simplified circuit of the induction motor presented in Figure 2 [15]:

$$\mathbf{Z}\_{\rm rf} = \mathbf{Z}\_{\rm g} = \mathbf{jXC}\_{\rm g} \tag{1}$$

$$\mathbf{Z}\_{\rm sr} = \mathbf{R}\_{\rm sr} + \mathbf{jX}\mathbf{L}\_{\rm sr} + \mathbf{jX}\mathbf{C}\_{\rm sr} \tag{2}$$

$$\mathbf{Z}\_{sf} = \mathbf{R}\_{sf} + \mathbf{jX}\mathbf{L}\_{sf} + \mathbf{jX}\mathbf{C}\_{sf} \tag{3}$$

$$\mathbf{Z}\_b = \{\mathbf{R}\_l \mid \mathbf{/jXC\_b}\} + \mathbf{R}\_b + \mathbf{R}\_w + \mathbf{jXL\_w} \tag{4}$$

coupling between the windings of the stator and the rotor.

2. Equivalent circuit of three-phase induction motor at high frequencies

At high frequencies, the capacitive reactance among the various parts of the three-phase induction motor are shown in Figure 1, which illustrates the equivalent circuit of the three-phase induction motor

**Figure 1.** High-frequency equivalent circuit of induction motor

sfsf sf sf ++= jXCjXLRZ (2.3) **Figure 2.** High-frequency simplified circuit for three-phase induction motor according to its impedances

purely capacitive characteristic [10], according to the example shown in Figure 3.

capacitances at the high frequencies, as shown in Figure 4.

VCM

Figure 1: High-frequency equivalent circuit of induction motor

Simplifications of the circuit shown in Figure 2 can be done using the following considerations [1, 3, 6, 9, 10]: a) ZRF is purely capacitive; b) at frequencies lower than 200 kHz, ZSR assumes a capacitive characteristic; and c) ZSR represents a circuit with RC behavior. At frequencies of 4– 20 kHz range, which are typical switching frequencies of a PWM inverter, parasitic impedances lb wbb w = )//( +++ jXLRRjXCRZ (2.4) Simplifications of the circuit shown in Figure 2 can be done using the following considerations [1, 3, 6, 9, 10]: a) ZRF is purely capacitive; b) at frequencies lower than 200 kHz, ZSR assumes a capacitive characteristic; and c) ZSR represents a circuit with RC behavior. At frequencies of 4–20 kHz range, which

are typical switching frequencies of a PWM inverter, parasitic impedances of the induction motor assume

Figure 3: Capacitive impedance characteristic of the three-phase induction motor [10]

Thus, it is possible to simplify the equivalent circuit presented in Figure 2 keeping only the parasitic

VSHAFT

CB

Figure 4: Simplified equivalent circuit of high-frequency induction motor

CRF

R1

CSF

CSR

Figure 2: High-frequency simplified circuit for three-phase induction motor according to its impedances

Figure 2: High-frequency simplified circuit for three-phase induction motor according to its impedances

R1

VCM

ZSR

6, 9, 10]: a) ZRF is purely capacitive; b) at frequencies lower than 200 kHz, ZSR assumes a capacitive characteristic; and c) ZSR represents a circuit with RC behavior. At frequencies of 4–20 kHz range, which are typical switching frequencies of a PWM inverter, parasitic impedances of the induction motor assume

Simplifications of the circuit shown in Figure 2 can be done using the following considerations [1, 3,

ZRF

VSHAFT

ZSF ZB

of the induction motor assume purely capacitive characteristic [10], according to the example shown in Figure 3.

**Figure 3.** Capacitive impedance characteristic of the three-phase induction motor [10] Figure 3: Capacitive impedance characteristic of the three-phase induction motor [10]

capacitances at the high frequencies, as shown in Figure 4.

Thus, it is possible to simplify the equivalent circuit presented in Figure 2 keeping only the parasitic capacitances at the high frequencies, as shown in Figure 4. Thus, it is possible to simplify the equivalent circuit presented in Figure 2 keeping only the parasitic

Figure 4: Simplified equivalent circuit of high-frequency induction motor **Figure 4.** Simplified equivalent circuit of high-frequency induction motor

Simplifications of the circuit shown in Figure 2 can be done using the following considerations [1, 3, 6, 9, 10]: a) ZRF is purely capacitive; b) at frequencies lower than 200 kHz, ZSR assumes a capacitive characteristic; and c) ZSR represents a circuit with RC behavior. At frequencies of 4– 20 kHz range, which are typical switching frequencies of a PWM inverter, parasitic impedances

6, 9, 10]: a) ZRF is purely capacitive; b) at frequencies lower than 200 kHz, ZSR assumes a capacitive characteristic; and c) ZSR represents a circuit with RC behavior. At frequencies of 4–20 kHz range, which are typical switching frequencies of a PWM inverter, parasitic impedances of the induction motor assume

purely capacitive characteristic [10], according to the example shown in Figure 3.

capacitances at the high frequencies, as shown in Figure 4.

VCM

Figure 2: High-frequency simplified circuit for three-phase induction motor according to its impedances

**Figure 2.** High-frequency simplified circuit for three-phase induction motor according to its impedances

rf g g == jXCZZ (2.1)

srsr sr sr ++= jXCjXLRZ (2.2)

sfsf sf sf ++= jXCjXLRZ (2.3)

lb wbb w = )//( +++ jXLRRjXCRZ (2.4)

ZRF

ZSF ZB

Simplifications of the circuit shown in Figure 2 can be done using the following considerations [1, 3,

Figure 3: Capacitive impedance characteristic of the three-phase induction motor [10]

Thus, it is possible to simplify the equivalent circuit presented in Figure 2 keeping only the parasitic

VSHAFT

CB

Figure 4: Simplified equivalent circuit of high-frequency induction motor

CRF

R1

CSF

CSR

Figure 1: High-frequency equivalent circuit of induction motor

Frame

Cg LSF LSF

the impedance between the windings of the stator and the rotor; ZSF is the impedance between the stator windings and the frame; Zg is the air gap impedance; and ZB represents the impedance between the rotor and the bearing. RW and LW represent the equivalent impedance of the driver through which circulates the bearing current and Rg is the lead resistance connected between the frame and ground. Using the defined impedances shown by equations (2.1) through (2.4), one can obtain the high-frequency simplified circuit of

ZRF is the impedance between the rotor and the motor frame, also called air gap impedance Zg; ZER is

R1

ZSR

Rg

2. Equivalent circuit of three-phase induction motor at high frequencies

RSR

RSR

RSR

coupling between the windings of the stator and the rotor.

84 Induction Motors - Applications, Control and Fault Diagnostics

RSF RSF RSF

CSF CSF CSF

**Figure 1.** High-frequency equivalent circuit of induction motor

VCM

LSF

the induction motor presented in Figure 2 [15]:

At high frequencies, the capacitive reactance among the various parts of the three-phase induction motor are shown in Figure 1, which illustrates the equivalent circuit of the three-phase induction motor fed by PWM inverter [1, 6, 15]. The distributed parameters R, L, and C represent the high-frequency

LSR

CSR

CSR

CSR

Z ZRF=Zg SF ZB

RB

Rl

L

VSHAFT

CB

R

LSR

LSR

ZSR

In Figure 4, VCM is the common mode voltage, CSF is the capacitance between the stator winding per phase and the induction motor frame, CSR is the capacitance among the windings of the stator and the rotor, CRF is the capacitance between the rotor and motor frame, and CB is the capacitance of the bearing. Using this high-frequency equivalent circuit of the induction motor, the equations of both shaft voltage (VSHAFT) and leakage current (ILEAKAGE) are obtained:

$$
\upsilon\_{CM} = \frac{\upsilon\_a + \upsilon\_b + \upsilon\_c}{3} \tag{5}
$$

$$V\_{SHAT} = \; V\_{CM} \cdot \left(\frac{\mathbf{C}\_{SR}}{\mathbf{C}\_{SR} + \mathbf{C}\_{RF} + \mathbf{C}\_{B}}\right) \tag{6}$$

$$I\_{LEAKAGE} = \frac{V\_{CM}}{X\mathcal{C}\_{SF}} + \frac{V\_{SHAT}}{X\mathcal{C}\_{RF}} + \frac{V\_{SHAT}}{X\mathcal{C}\_{B}} \tag{7}$$

The capacitances CSF, CSR, CRF, and CB are defined according to the geometric characteristics of both the stator as the rotor of a three-phase induction motor and its bearings [1, 2, 11, 12, 13, 23]. In [1], these capacitances are set according to the following equations, depending on the geometrical dimensions of the induction motor shown in Figure 5.

$$\mathbf{C\_{SF}} = \frac{\mathbf{K\_{SF}} \cdot \mathbf{N\_S} \, \mathbf{c\_r} \, \mathbf{c\_0} \left(\mathcal{W}\_d + \mathcal{W}\_s\right) \mathbf{L\_S}}{d} \tag{8}$$

$$\mathbf{C}\_{SR} = \frac{K\_{SR} \mathcal{N}\_R \mathcal{E}\_0 \mathcal{W}\_r L\_r}{\mathcal{S}} \tag{9}$$

$$\mathbf{C}\_{\rm RF} = \frac{\mathbf{K}\_{\rm RF} \cdot \pi \,\mathcal{E}\_0 \cdot \mathbf{L}\_r}{\ln \left( \mathcal{R}\_s / \mathcal{R}\_r \right)} \tag{10}$$

$$\mathbf{C}\_{B} = \frac{\mathbf{N}\_{b}\mathbf{4}.\pi\varepsilon\_{0}.\varepsilon\_{r}}{\left(\mathbf{R}\_{b} + \mathbf{R}\_{c}\,/\,\mathbf{R}\_{b} - \mathbf{1}\right)}\tag{11}$$

In the above equations, KSF, KSR, and KRF factors are stacked packages of magnetic stator and rotor, NS and NR are the number of slots of the stator winding and the rotor, Ws and Wd are the width and depth of the groove stator, Wr is the width of the rotor slot, Ls and Lr are the lengths of the stator and rotor slots, Rr and Rs are the radii of the stator and rotor, d is the thickness of the insulating dielectric material of the stator channel, g is the gap length, Nb is the number of bearing balls, Rb and Rc are the ball lightning and the raceway, ε0 and εr are the permittivity of the medium (air and insulation). The parasitic capacitances become important when, besides the common mode voltage is different from zero, the frequency of the phase voltages becomes high, resulting in small capacitive reactance and the circuit in parallel with the remaining equivalent.

When the three-phase induction motor is fed by pure sinusoidal voltages, at the power grid frequency, the effect of these capacitances is minimal or nonexistent. If there is an unbalance in phase voltages, the common mode voltage becomes nonzero, establishing current flow through these capacitances (Figure 4) which will be significant, if the amplitudes of phase voltages are high. Thus, assuming that the motor is supplied by balanced phase voltages, the common-mode voltage (5) is zero. When the three-phase induction motor is fed through a PWM inverter, it establishes a "capacitive coupling" created by the modulated phase voltages of the inverter output. These voltages have trapezoidal characteristic value with high dV/dt of (VSHAFT) and leakage current (ILEAKAGE) are obtained:

In Figure 4, VCM is the common mode voltage, CSF is the capacitance between the stator winding per

3

SR RF B

RF

SHAFT

XC

SR

v v v

v a b c

SHAFT CM C C C

+ + =

+ + =

V

CM

C V V .

The capacitances CSF, CSR, CRF, and CB are defined according to the geometric characteristics of both

SF

SR

Figure 5: Dimensions – induction motor and bearing [1]

the stator as the rotor of a three-phase induction motor and its bearings [1, 2, 11, 12, 13, 23]. In [1], these

SF

LEAKAGE XC

CM

XC V I = + +

capacitances are set according to the following equations, depending on the geometrical dimensions of the

(2.5)

(2.6)

(2.7)

(2.8)

(2.11)

( )

d

. . . . 0 ε = (2.9)

... 0 π ε = (2.10)

K N W W L C SF S r d s S

g

( ) s r

( ) / 1

+ − = b c b

B R R R N C π ε ε

...4. 0

b r

RF r

K N W L C SR R r r

B

SHAFT

V

RF R R K L C

ln /

. ... . 0 + = ε ε

phase and the induction motor frame, CSR is the capacitance among the windings of the stator and the

rotor, CRF is the capacitance between the rotor and motor frame, and CB is the capacitance of the bearing.

Using this high-frequency equivalent circuit of the induction motor, the equations of both shaft voltage

assuming that the motor is supplied by balanced phase voltages, the common-mode voltage (2.5) is zero. When the three-phase induction motor is fed through a PWM inverter, it establishes a "capacitive **Figure 5.** Dimensions – induction motor and bearing [1]

as shown in Figure 6 (a, b) [12, 18, 22].

kHz.

*CM SHAFT SHAFT*

=+ + (7)

+ = (8)

= (9)

= (10)

= + - (11)

*VV V I*

*SF RF B*

*XC XC XC*

The capacitances CSF, CSR, CRF, and CB are defined according to the geometric characteristics of both the stator as the rotor of a three-phase induction motor and its bearings [1, 2, 11, 12, 13, 23]. In [1], these capacitances are set according to the following equations, depending on the

*SF S r d s S* . ... . 0 ( )

*KN W WL*

e e

*K N WL C g* e

*K L C*

*d*

0 . .. . *SR R r r*

( ) 0 .. . ln / *RF r*

*R R* p e

( ) 0 .4. . . / 1 *b r*

In the above equations, KSF, KSR, and KRF factors are stacked packages of magnetic stator and rotor, NS and NR are the number of slots of the stator winding and the rotor, Ws and Wd are the width and depth of the groove stator, Wr is the width of the rotor slot, Ls and Lr are the lengths of the stator and rotor slots, Rr and Rs are the radii of the stator and rotor, d is the thickness of the insulating dielectric material of the stator channel, g is the gap length, Nb is the number of bearing balls, Rb and Rc are the ball lightning and the raceway, ε0 and εr are the permittivity of the medium (air and insulation). The parasitic capacitances become important when, besides the common mode voltage is different from zero, the frequency of the phase voltages becomes high, resulting in small capacitive reactance and the circuit in parallel with

When the three-phase induction motor is fed by pure sinusoidal voltages, at the power grid frequency, the effect of these capacitances is minimal or nonexistent. If there is an unbalance in phase voltages, the common mode voltage becomes nonzero, establishing current flow through these capacitances (Figure 4) which will be significant, if the amplitudes of phase voltages are high. Thus, assuming that the motor is supplied by balanced phase voltages, the common-mode voltage (5) is zero. When the three-phase induction motor is fed through a PWM inverter, it establishes a "capacitive coupling" created by the modulated phase voltages of the inverter output. These voltages have trapezoidal characteristic value with high dV/dt of

*b cb*

*R RR* pe e

*s r*

*LEAKAGE*

86 Induction Motors - Applications, Control and Fault Diagnostics

geometrical dimensions of the induction motor shown in Figure 5.

*SR*

*RF*

*N C*

*B*

the remaining equivalent.

*SF*

*C*

the inverter while the semiconductor switches are turned on or off, as shown in Figure 6 (a, b) [12, 18, 22]. coupling" created by the modulated phase voltages of the inverter output. These voltages have trapezoidal characteristic value with high dV/dt of the inverter while the semiconductor switches are turned on or off, In the above equations, KSF, KSR, and KRF factors are stacked packages of magnetic stator and rotor,

Figure 6: a) Terminal voltage of the PWM inverter , b) dV/dt voltage

As the supply voltages are not sinusoidal, the common mode voltage (VCM) takes nonzero values.

a)

 Time 71.0ms 71.1ms 71.2ms 71.3ms 71.4ms 71.5ms 71.6ms 71.7ms 71.8ms 71.9ms 72.0ms

0 2.0

a) t(ms)

Thus, this voltage and also the switching frequency (fS) of the PWM inverter now have an important role on common mode capacitive currents. For example, Figures 7a and 7b present the waveforms of the output phase voltages of the inverter and the common-mode voltage for a switching frequency (fS) of 16

**Figure 6.** a) Terminal voltage of the PWM inverter, b) dV/dt voltage

V(T,0) -200V -100V 0V 100V 200V

V(S,0) -200V -100V 0V 100V 200V V(R,0) -200V -100V 0V 100V V 200V RN(V) 200

SEL>>

0 -200

0

0 -200 VTN(V) 200 kHz.

relationships:

As the supply voltages are not sinusoidal, the common mode voltage (VCM) takes nonzero values. Thus, this voltage and also the switching frequency (fS) of the PWM inverter now have an important role on common mode capacitive currents. For example, Figures 7a and 7b present the waveforms of the output phase voltages of the inverter and the common-mode voltage for a switching frequency (fS) of 16 kHz. As the supply voltages are not sinusoidal, the common mode voltage (VCM) takes nonzero values. Thus, this voltage and also the switching frequency (fS) of the PWM inverter now have an important role on common mode capacitive currents. For example, Figures 7a and 7b present the waveforms of the output phase voltages of the inverter and the common-mode voltage for a switching frequency (fS) of 16

**Figure 7.** a) Phase voltage, b) a common mode voltage (VCM) - fS = 16 kHz

of the motor for high frequency presented in Figure 8b [1 to 10].

The higher the switching frequency, the better is the characteristic of the waveform of the current applied to the induction motor. Besides that, the frequency raise often implies an increase in the switching times of the IGBTs increasingly smaller, providing both increased feature dV/dt, which is directly related to the values of the capacitive currents of common mode current, as the reduction of the capacitive reactance significantly increased the amplitudes thereof. The following equations show these The higher the switching frequency, the better is the characteristic of the waveform of the current applied to the induction motor. Besides that, the frequency raise often implies an increase in the switching times of the IGBTs increasingly smaller, providing both increased feature dV/dt, which is directly related to the values of the capacitive currents of common mode current, as the reduction of the capacitive reactance significantly increased the ampli‐ tudes thereof. The following equations show these relationships:

> *I* 2.

*dt*

Observing equations (2.12) and (2.13), it is possible to conclude that V and I are effective values of

voltage and common mode current. Thus, from the down-movement of the common-mode electrical currents (also called capacitive currents) as shown in Figure 8a [18], one can obtain the equivalent circuit

**Motor**

Figure 7: a) Phase voltage, b) a common mode voltage (VCM) - fS = 16 kHz

$$
\dot{u} = C. \frac{dv}{dt} \tag{12}
$$

**Load**

.*fS* .*C*.*V* (2.13)

relationships: dv i = C. (2.12) Analysis and Methodology for Determining the Parasitic Capacitances in VSI-fed IM Drives Based on PWM Technique http://dx.doi.org/10.5772/61544 89

dt

Figure 7: a) Phase voltage, b) a common mode voltage (VCM) - fS = 16 kHz

 Time 71.0ms 71.1ms 71.2ms 71.3ms 71.4ms 71.5ms 71.6ms 71.7ms 71.8ms 71.9ms 72.0ms

0 2.0

The higher the switching frequency, the better is the characteristic of the waveform of the current

applied to the induction motor. Besides that, the frequency raise often implies an increase in the switching times of the IGBTs increasingly smaller, providing both increased feature dV/dt, which is directly related to the values of the capacitive currents of common mode current, as the reduction of the capacitive reactance significantly increased the amplitudes thereof. The following equations show these

$$I = 2.\pi.f\_{\mathbb{S}}.\text{C.V.}\tag{13}$$

b) t(ms)

Observing equations (12) and (13), it is possible to conclude that V and I are effective values of voltage and common mode current. Thus, from the down-movement of the common-mode electrical currents (also called capacitive currents) as shown in Figure 8a [18], one can obtain the equivalent circuit of the motor for high frequency presented in Figure 8b [1 to 10]. Observing equations (2.12) and (2.13), it is possible to conclude that V and I are effective values of voltage and common mode current. Thus, from the down-movement of the common-mode electrical currents (also called capacitive currents) as shown in Figure 8a [18], one can obtain the equivalent circuit

of the motor for high frequency presented in Figure 8b [1 to 10].

(V(R,0)+V(S,0)+V(T,0))/3

200

0V

0

100V

V 200V CM(V) 200

As the supply voltages are not sinusoidal, the common mode voltage (VCM) takes nonzero values. Thus, this voltage and also the switching frequency (fS) of the PWM inverter now have an important role on common mode capacitive currents. For example, Figures 7a and 7b present the waveforms of the output phase voltages of the inverter and the common-mode

Thus, this voltage and also the switching frequency (fS) of the PWM inverter now have an important role on common mode capacitive currents. For example, Figures 7a and 7b present the waveforms of the output phase voltages of the inverter and the common-mode voltage for a switching frequency (fS) of 16

As the supply voltages are not sinusoidal, the common mode voltage (VCM) takes nonzero values.

a)

 Time 71.0ms 71.1ms 71.2ms 71.3ms 71.4ms 71.5ms 71.6ms 71.7ms 71.8ms 71.9ms 72.0ms

0 2.0

a) t(ms)

b) t(ms)

Figure 7: a) Phase voltage, b) a common mode voltage (VCM) - fS = 16 kHz

 Time 71.0ms 71.1ms 71.2ms 71.3ms 71.4ms 71.5ms 71.6ms 71.7ms 71.8ms 71.9ms 72.0ms

0 2.0

The higher the switching frequency, the better is the characteristic of the waveform of the current

*dt*

Observing equations (2.12) and (2.13), it is possible to conclude that V and I are effective values of

voltage and common mode current. Thus, from the down-movement of the common-mode electrical currents (also called capacitive currents) as shown in Figure 8a [18], one can obtain the equivalent circuit

**Motor**

*dv i C*. (2.12)

*dt* = (12)

**Load**

.*fS* .*C*.*V* (2.13)

applied to the induction motor. Besides that, the frequency raise often implies an increase in the switching times of the IGBTs increasingly smaller, providing both increased feature dV/dt, which is directly related to the values of the capacitive currents of common mode current, as the reduction of the capacitive reactance significantly increased the amplitudes thereof. The following equations show these

The higher the switching frequency, the better is the characteristic of the waveform of the current applied to the induction motor. Besides that, the frequency raise often implies an increase in the switching times of the IGBTs increasingly smaller, providing both increased feature dV/dt, which is directly related to the values of the capacitive currents of common mode current, as the reduction of the capacitive reactance significantly increased the ampli‐

> *I* 2.

. *dv i C*

of the motor for high frequency presented in Figure 8b [1 to 10].

voltage for a switching frequency (fS) of 16 kHz.

88 Induction Motors - Applications, Control and Fault Diagnostics

V(T,0) -200V -100V 0V 100V 200V

(V(R,0)+V(S,0)+V(T,0))/3

**Figure 7.** a) Phase voltage, b) a common mode voltage (VCM) - fS = 16 kHz

tudes thereof. The following equations show these relationships:

V(S,0) -200V -100V 0V 100V 200V V(R,0) -200V -100V 0V 100V

SEL>>

200

0V

0

100V

V 200V CM(V) 200

0 -200

0

0

V 200V RN(V) 200

kHz.

relationships:

Figure 8: a) Circulating currents in the motor, b) high-frequency equivalent circuit and circulating currents [10] **Figure 8.** a) Circulating currents in the motor, b) high-frequency equivalent circuit and circulating currents [10]

The electric currents are established as follows: [10] a) II is the current flowing through the common mode voltage point (VCM), which passes through the pump capacitance for the motor frame, ground or neutral and returns to the system to VCM point. It is the largest component of the leakage current (ILEAKAGE), compared to other capacitive currents because CSF capacitance is much larger than others. This current is primarily responsible for the electrical discharge of the motor frame to the "ground." If the motor is not grounded to the frame satisfactorily, an important electrical discharge at various parts of it and even at the load may occur and cause an "electric shock" if someone leans on the motor frame. b) III is the current flowing through the common mode voltage point (VCM), which passes through the CSR and capacitances CRF, motor frame, ground or neutral system and returns to the VCM point. c) III is the current flowing through the common mode voltage point (VCM), which passes through the CSR capacitance motor shaft bearing capacitance CB motor frame, ground or neutral system and returns to the VCM point. d) IV is the current flowing shaft voltage point (VSHAFT); CRF capacitance thus stores energy through the motor shaft, the switch SH, the motor frame, and returns to the CRF capacitance.

The switch SH in Figure 8b is, when closed, the breaking of the bearing insulation dielectric (grease film). When this occurs, and CSR is much smaller than CRF, a new mesh current flow given by IV is then established. The IV mesh is responsible for the electrical discharge in the motor bearings due to the charge stored in the capacitor CRF.

### **3. Experimental method for determination of parasitic capacitances rotor cage three-phase induction motor [40]**

This item presents a methodology [40] for the determination of the parasitic capacitances of the equivalent high-frequency rotor three-phase induction motor circuit cage and the effects of electromagnetic interference caused in the same common mode, when it is driven by a PWM inverter. This is a methodology that uses an electronic circuit to measure variables needed to calculate these parameters.

The procedures proposed for the development of this methodology are: a) determination of the equivalent circuit parameters of the three-phase induction motor in steady state and high frequency and [1, 10, 15, 31] through typical laboratory test; b) establish settings of links between the PWM inverter and the motor for measurements of quantities of interest: common mode (VCM) and shaft (VSHAFT) voltages, leakage (ILEAKAGE) and shaft (ISHAFT) currents, developed by measuring circuit for this purpose [13, 14]; c) calculate the values of the parasitic capaci‐ tances between stator and frame (CSF), stator and rotor (CSR), rotor and frame (CRF) and bearings (CB), using their characteristic equations [10, 15]; d) using PSPICE [36] to simulate the system (three-phase induction motor fed by PWM inverter) with the high-frequency equivalent circuit in the same [6, 16]; e) to obtain the waveforms characteristics of the EMI phenomena.

### **3.1. Methodology determination procedure**

The methodology determines the high-frequency equivalent circuit parameters of rotor cage three-phase induction motor, through direct measurement of quantities of interest and, using equation (14), calculate the values of the parasitic capacitances. The quantities of interest are common mode voltage (VCM), shaft voltage (VSHAFT), leakage current (ILEAKAGE), and shaft current (ISHAFT).

$$\text{C} = \frac{I\_{\text{C}}}{2.\pi f\_{\text{s}} f\_{\text{s}} V\_{\text{C}}} \tag{14}$$

In equation (14), IC and VC represent the current and the effective voltage across the capacitor, respectively, and fS the switching frequency of the PWM inverter. The schematic diagram of the methodology is shown in Figure 9.

In equation (3.1), IC and VC represent the current and the effective voltage across the capacitor, respectively, and fS the switching frequency of the PWM inverter. The schematic diagram of the Analysis and Methodology for Determining the Parasitic Capacitances in VSI-fed IM Drives Based on PWM Technique http://dx.doi.org/10.5772/61544 91

(VCM), shaft voltage (VSHAFT), leakage current (ILEAKAGE), and shaft current (ISHAFT).

C

The methodology determines the high-frequency equivalent circuit parameters of rotor cage threephase induction motor, through direct measurement of quantities of interest and, using equation (3.1), calculate the values of the parasitic capacitances. The quantities of interest are common mode voltage

s C

2π = (3.1)

C V.f.. I

3.1. Methodology determination procedure

**Figure 9.** Schematic diagram of the methodology

energy through the motor shaft, the switch SH, the motor frame, and returns to the CRF

The switch SH in Figure 8b is, when closed, the breaking of the bearing insulation dielectric (grease film). When this occurs, and CSR is much smaller than CRF, a new mesh current flow given by IV is then established. The IV mesh is responsible for the electrical discharge in the

**3. Experimental method for determination of parasitic capacitances rotor**

This item presents a methodology [40] for the determination of the parasitic capacitances of the equivalent high-frequency rotor three-phase induction motor circuit cage and the effects of electromagnetic interference caused in the same common mode, when it is driven by a PWM inverter. This is a methodology that uses an electronic circuit to measure variables needed to

The procedures proposed for the development of this methodology are: a) determination of the equivalent circuit parameters of the three-phase induction motor in steady state and high frequency and [1, 10, 15, 31] through typical laboratory test; b) establish settings of links between the PWM inverter and the motor for measurements of quantities of interest: common mode (VCM) and shaft (VSHAFT) voltages, leakage (ILEAKAGE) and shaft (ISHAFT) currents, developed by measuring circuit for this purpose [13, 14]; c) calculate the values of the parasitic capaci‐ tances between stator and frame (CSF), stator and rotor (CSR), rotor and frame (CRF) and bearings (CB), using their characteristic equations [10, 15]; d) using PSPICE [36] to simulate the system (three-phase induction motor fed by PWM inverter) with the high-frequency equivalent circuit

in the same [6, 16]; e) to obtain the waveforms characteristics of the EMI phenomena.

The methodology determines the high-frequency equivalent circuit parameters of rotor cage three-phase induction motor, through direct measurement of quantities of interest and, using equation (14), calculate the values of the parasitic capacitances. The quantities of interest are common mode voltage (VCM), shaft voltage (VSHAFT), leakage current (ILEAKAGE), and shaft

> 2. . . *C s C*

In equation (14), IC and VC represent the current and the effective voltage across the capacitor, respectively, and fS the switching frequency of the PWM inverter. The schematic diagram of

*f V* = (14)

p

*C*

*I*

motor bearings due to the charge stored in the capacitor CRF.

**cage three-phase induction motor [40]**

90 Induction Motors - Applications, Control and Fault Diagnostics

**3.1. Methodology determination procedure**

the methodology is shown in Figure 9.

calculate these parameters.

current (ISHAFT).

capacitance.

The structure consists of the following equipment: three-phase power supply 220V/60Hz, 1 Hp three-phase PWM inverter, two three-phase 1Hp induction motors, measuring circuit of the quantities of interest, data acquisition board LabView, notebook PWM inverter management and signal processing through dedicated software LabView 8.5 and MOVITOOLS, base for sustaining fully insulated equipment to allow measurements, especially the currents (as close as possible to an actual situation), connecting cable with the neutral system interconnected to the PWM inverter ground terminals and the induction motor providing a circulating path to the induction motor current. The structure consists of the following equipment: three-phase power supply 220V/60Hz, 1 Hp three-phase PWM inverter, two three-phase 1Hp induction motors, measuring circuit of the quantities of interest, data acquisition board LabView, notebook PWM inverter manage‐ ment and signal processing through dedicated software LabView 8.5 and MOVITOOLS, base for sustaining fully insulated equipment to allow measurements, especially the currents (as close as possible to an actual situation), connecting cable with the neutral system intercon‐ nected to the PWM inverter ground terminals and the induction motor providing a circulating path to the induction motor current.

The magnitudes of the measuring circuit (VCM, VSHAFT, ILEAKAGE, and ISHAFT) are shown in Figure 10. The common mode voltage (VCM) is the common point of the measuring voltage (neutral) to the motor frame when it uses star connection (Y). When the same connection uses delta (∆), it should be equivalent The magnitudes of the measuring circuit (VCM, VSHAFT, ILEAKAGE, and ISHAFT) are shown in Figure 10. The common mode voltage (VCM) is the common point of the measuring voltage (neutral) to the motor frame when it uses star connection (Y). When the same connection uses delta (Δ), it should be equivalent to a star connection using high-value resistors (1MΩ) connected to each phase of the motor with a common point (neutral). In the adjustment of the common mode voltage measurement (VCM) to the acquisition board, a resistive divider (R1, R2, R3, and R4) is added so that the measured voltage does not exceed the maximum allowed by the acquisition board, that is, +/- 10V, as shown in Figure 10.

For measuring the shaft voltage (VSHAFT), which is the voltage measured between the shaft and the induction motor frame, a copper ring carbon brush system is added to the induction motor shaft.

For leakage current (ILEAKAGE) and shaft current (ISHAFT) measurements, a Rogowski coil current sensor, developed for this methodology, was used [40]. The leakage current is the current measured in the connection cable between the induction motor frame and the metal frame of

the PWM inverter connected to the neutral system. It is composed of the sum of the currents flowing through the stator-frame capacitances (CSF), rotor-frame (CRF), and bearings (CB). to the neutral system. It is composed of the sum of the currents flowing through the stator-frame capacitances (CSF), rotor-frame (CRF), and bearings (CB).

connection cable between the induction motor frame and the metal frame of the PWM inverter connected

For measuring the shaft voltage (VSHAFT), which is the voltage measured between the shaft and the

For leakage current (ILEAKAGE) and shaft current (ISHAFT) measurements, a Rogowski coil current

to a star connection using high-value resistors (1MΩ) connected to each phase of the motor with a common point (neutral). In the adjustment of the common mode voltage measurement (VCM) to the acquisition board, a resistive divider (R1, R2, R3, and R4) is added so that the measured voltage does not

exceed the maximum allowed by the acquisition board, that is, +/- 10V, as shown in Figure 10.

induction motor frame, a copper ring carbon brush system is added to the induction motor shaft.

The shaft current (ISHAFT) is the current measured in the conductor which is connected to the brush and corresponding contributions from currents flowing through the rotor-frame capacitances (CRF) and bearings (CB). The shaft current (ISHAFT) is the current measured in the conductor which is connected to the brush and corresponding contributions from currents flowing through the rotor-frame capacitances (CRF) and bearings (CB).

Figure 10: Measuring circuit of the quantities of interest **Figure 10.** Measuring circuit of the quantities of interest

### *3.1.1. Induction motor preparation*

For the application of the proposed method, the motor rotor three-phase induction cage should be prepared so that the capacitance can be determined [1, 10, 14]. Under normal conditions, the motor bearings are directly connected to both the shaft and the motor frame. It causes the capacitance existing between the rotor and the frame (CRF) to short-circuit.

So the following changes are made. (a) Motor bearings are insulated by a nylon cover (tecnil) high electrical resistance. Thus, there will be no current flow by bearings (IB) and the contri‐ So the following changes are made. (a) Motor bearings are insulated by a nylon cover (tecnil) high

For the application of the proposed method, the motor rotor three-phase induction cage should be prepared so that the capacitance can be determined [1, 10, 14]. Under normal conditions, the motor

electrical resistance. Thus, there will be no current flow by bearings (IB) and the contribution branch rotor

frame can be evaluated, as shown in Figure 11 [10]. (b) A conductor is connected between the external

So the following changes are made. (a) Motor bearings are insulated by a nylon cover (tecnil) high

For the application of the proposed method, the motor rotor three-phase induction cage should be

prepared so that the capacitance can be determined [1, 10, 14]. Under normal conditions, the motor

bearings are directly connected to both the shaft and the motor frame. It causes the capacitance existing

bution branch rotor frame can be evaluated, as shown in Figure 11 [10]. (b) A conductor is connected between the external surface of one of the bearings and closest to the motor frame by a switch point. When the switch is open, there is the condition set out in item (a). When the switch is closed, then one has the rolling movement for the current frame and the returning to the bearing [10]. (c) The motor shaft settles in a ring and brush assembly, for measuring the shaft voltage (VSHAFT). This system is also used to measure the shaft current (ISHAFT) [14]. Figure 12 shows the items (b) and (c). electrical resistance. Thus, there will be no current flow by bearings (IB) and the contribution branch rotor frame can be evaluated, as shown in Figure 11 [10]. (b) A conductor is connected between the external surface of one of the bearings and closest to the motor frame by a switch point. When the switch is open, there is the condition set out in item (a). When the switch is closed, then one has the rolling movement for the current frame and the returning to the bearing [10]. (c) The motor shaft settles in a ring and brush assembly, for measuring the shaft voltage (VSHAFT). This system is also used to measure the shaft current the current frame and the returning to the bearing [10]. (c) The motor shaft settles in a ring and brush assembly, for measuring the shaft voltage (VSHAFT). This system is also used to measure the shaft current (ISHAFT) [14]. Figure 12 shows the items (b) and (c). Insulated

Figure 11 a, b): Insulation of the motor shaft bearings

b

(ISHAFT) [14]. Figure 12 shows the items (b) and (c).

bearings with

Figure 11 a, b): Insulation of the motor shaft bearings **Figure 11.** a, b): Insulation of the motor shaft bearings a) b)

3.1.1. Induction motor preparation

between the rotor and the frame (CRF) to short-circuit.

3.1.1. Induction motor preparation

the PWM inverter connected to the neutral system. It is composed of the sum of the currents flowing through the stator-frame capacitances (CSF), rotor-frame (CRF), and bearings (CB).

sensor, developed for this methodology, was used [40]. The leakage current is the current measured in the connection cable between the induction motor frame and the metal frame of the PWM inverter connected to the neutral system. It is composed of the sum of the currents flowing through the stator-frame

For measuring the shaft voltage (VSHAFT), which is the voltage measured between the shaft and the

For leakage current (ILEAKAGE) and shaft current (ISHAFT) measurements, a Rogowski coil current

to a star connection using high-value resistors (1MΩ) connected to each phase of the motor with a common point (neutral). In the adjustment of the common mode voltage measurement (VCM) to the acquisition board, a resistive divider (R1, R2, R3, and R4) is added so that the measured voltage does not

exceed the maximum allowed by the acquisition board, that is, +/- 10V, as shown in Figure 10.

induction motor frame, a copper ring carbon brush system is added to the induction motor shaft.

The shaft current (ISHAFT) is the current measured in the conductor which is connected to the brush and corresponding contributions from currents flowing through the rotor-frame

and corresponding contributions from currents flowing through the rotor-frame capacitances (CRF) and

3Φ PWM INVERTER V/f GINV

R1

R2

The shaft current (ISHAFT) is the current measured in the conductor which is connected to the brush

IM 3Φ 220V/60 Hz

CRF CB

SH

ISHAFT

R = 1MΩ

R3

R4

Figure 10: Measuring circuit of the quantities of interest

For the application of the proposed method, the motor rotor three-phase induction cage should be prepared so that the capacitance can be determined [1, 10, 14]. Under normal conditions, the motor bearings are directly connected to both the shaft and the motor frame. It causes the

So the following changes are made. (a) Motor bearings are insulated by a nylon cover (tecnil) high electrical resistance. Thus, there will be no current flow by bearings (IB) and the contri‐

capacitance existing between the rotor and the frame (CRF) to short-circuit.

CSF

CSR

VCMVSHAFT

capacitances (CRF) and bearings (CB).

LABVIEW

ACQUISITION & MEASUREMENT BOARD

CHANNELS

**Figure 10.** Measuring circuit of the quantities of interest

*3.1.1. Induction motor preparation*

a0+ a0 a1+ a1 a2+ a2 a3+ a3- GND

bearings (CB).

capacitances (CSF), rotor-frame (CRF), and bearings (CB).

92 Induction Motors - Applications, Control and Fault Diagnostics

N R SS

T

ILEAKAGE

R S T

Figure 12: Switch (b) and ring-brush assembly (c) for measuring **Figure 12.** Switch (b) and ring-brush assembly (c) for measuring

### *3.1.2. Data acquisition board and measuring*

In order to acquire the quantities and accomplish the measurements, LabView [32, 33, 34] platform is used. Four channels for the different measurements are defined as follows: (a) a0 channel: leakage current (ILEAKAGE); (b) A1 channel: the common mode voltage (VCM), which is set to the voltage of -10V to + 10V limits; (c) a2 channel: shaft voltage (VSHAFT); and (d) a3 channel: shaft current (ISHAFT). 3.1.2. Data acquisition board and measuring In order to acquire the quantities and accomplish the measurements, LabView [32, 33, 34] platform

Using dedicated software LabView 8.5, [35] created a block diagram (plant) of the measuring system for the magnitudes of interest. Figure 13 shows the block diagram for leakage current (ILEAKAGE). The configuration for each quantity to be measured using channels with differential inputs is shown in Figure 14 to minimize the effects of common-mode voltages (noise). is used. Four channels for the different measurements are defined as follows: (a) a0 channel: leakage current (ILEAKAGE); (b) A1 channel: the common mode voltage (VCM), which is set to the voltage of -10V to + 10V limits; (c) a2 channel: shaft voltage (VSHAFT); and (d) a3 channel: shaft current (ISHAFT). Using dedicated software LabView 8.5, [35] created a block diagram (plant) of the measuring

Observations (Figure 13): system for the magnitudes of interest. Figure 13 shows the block diagram for leakage current

Block A - CORRENTE DE FUGA - Ifuga <=> LEAKAGE CURRENT - Ileakage (ILEAKAGE). The configuration for each quantity to be measured using channels with differential inputs

Block D - FILTRO RUÍDO <=> NOTCH FILTER is shown in Figure 14 to minimize the effects of common-mode voltages (noise). Observations (Figure 13):

Block E - FILTRO PASSA-BAIXA <=> LOW-PASS FILTER Block A - CORRENTE DE FUGA - Ifuga <=> LEAKAGE CURRENT - Ileakage

Block F - ESPECTRO - Ifuga <=> ESPECTRUM - Ileakage Block D - FILTRO RUÍDO <=> NOTCH FILTER

Block G - RMS Ifuga\_fs <=> RMS Ileakage\_fs Block E - FILTRO PASSA-BAIXA <=> LOW-PASS FILTER Block F - ESPECTRO - Ifuga <=> ESPECTRUM - Ileakage

SAÍDA DO FILTRO - Ifuga <=> OUTPUT FILTER – Ileakage Block G - RMS Ifuga\_fs <=> RMS Ileakage\_fs

SAÍDA DO FILTRO - Ifuga <=> OUTPUT FILTER - Ileakage

**Figure 13.** Block diagram for reading the leakage current (ILEAKAGE)

CHANNEL R VSIGNAL Using the DAQ block (A), the following items are defined for each channel: a) type of quantity (voltage or current), b) name of greatness and sample rate, c) type of sampling, d) connection diagram channel and other settings which depend on the desired type of measurement. Can be used up to 16 signals for reading.

LABVIEW

Figure 14: Differential input of the A/D channels

100kΩ

Figure 13: Block diagram for reading the leakage current (ILEAKAGE)

C

F

In order to acquire the quantities and accomplish the measurements, LabView [32, 33, 34] platform is used. Four channels for the different measurements are defined as follows: (a) a0 channel: leakage current (ILEAKAGE); (b) A1 channel: the common mode voltage (VCM), which is set to the voltage of -10V to +

Using dedicated software LabView 8.5, [35] created a block diagram (plant) of the measuring system for the magnitudes of interest. Figure 13 shows the block diagram for leakage current (ILEAKAGE). The configuration for each quantity to be measured using channels with differential inputs

10V limits; (c) a2 channel: shaft voltage (VSHAFT); and (d) a3 channel: shaft current (ISHAFT).

is shown in Figure 14 to minimize the effects of common-mode voltages (noise).

Block A - CORRENTE DE FUGA - Ifuga <=> LEAKAGE CURRENT - Ileakage

Figure 13: Block diagram for reading the leakage current (ILEAKAGE)

D

**Figure 14.** Differential input of the A/D channels

3.1.2. Data acquisition board and measuring

Observations (Figure 13):

A

Block D - FILTRO RUÍDO <=> NOTCH FILTER

Block G - RMS Ifuga\_fs <=> RMS Ileakage\_fs

Block E - FILTRO PASSA-BAIXA <=> LOW-PASS FILTER Block F - ESPECTRO - Ifuga <=> ESPECTRUM - Ileakage

SAÍDA DO FILTRO - Ifuga <=> OUTPUT FILTER - Ileakage

*3.1.2. Data acquisition board and measuring*

94 Induction Motors - Applications, Control and Fault Diagnostics

3.1.2. Data acquisition board and measuring

Block D - FILTRO RUÍDO <=> NOTCH FILTER

Block G - RMS Ifuga\_fs <=> RMS Ileakage\_fs

Block G - RMS Ifuga\_fs <=> RMS Ileakage\_fs

B

be used up to 16 signals for reading.

Block D - FILTRO RUÍDO <=> NOTCH FILTER

Block E - FILTRO PASSA-BAIXA <=> LOW-PASS FILTER

is shown in Figure 14 to minimize the effects of common-mode voltages (noise).

Block A - CORRENTE DE FUGA - Ifuga <=> LEAKAGE CURRENT - Ileakage

Block F - ESPECTRO - Ifuga <=> ESPECTRUM - Ileakage

Block E - FILTRO PASSA-BAIXA <=> LOW-PASS FILTER Block F - ESPECTRO - Ifuga <=> ESPECTRUM - Ileakage

SAÍDA DO FILTRO - Ifuga <=> OUTPUT FILTER – Ileakage

SAÍDA DO FILTRO - Ifuga <=> OUTPUT FILTER - Ileakage

shaft current (ISHAFT).

Observations (Figure 13):

Observations (Figure 13):

A

In order to acquire the quantities and accomplish the measurements, LabView [32, 33, 34] platform is used. Four channels for the different measurements are defined as follows: (a) a0 channel: leakage current (ILEAKAGE); (b) A1 channel: the common mode voltage (VCM), which is set to the voltage of -10V to + 10V limits; (c) a2 channel: shaft voltage (VSHAFT); and (d) a3 channel:

Using dedicated software LabView 8.5, [35] created a block diagram (plant) of the measuring system for the magnitudes of interest. Figure 13 shows the block diagram for leakage current (ILEAKAGE). The configuration for each quantity to be measured using channels with differential inputs is shown in Figure 14 to minimize the effects of common-mode voltages (noise).

In order to acquire the quantities and accomplish the measurements, LabView [32, 33, 34] platform is used. Four channels for the different measurements are defined as follows: (a) a0 channel: leakage current (ILEAKAGE); (b) A1 channel: the common mode voltage (VCM), which is set to the voltage of -10V to +

Using dedicated software LabView 8.5, [35] created a block diagram (plant) of the measuring

Figure 13: Block diagram for reading the leakage current (ILEAKAGE)

Using the DAQ block (A), the following items are defined for each channel: a) type of quantity (voltage or current), b) name of greatness and sample rate, c) type of sampling, d) connection diagram channel and other settings which depend on the desired type of measurement. Can

E

D

R 100kΩ

VSIGNAL

**Figure 13.** Block diagram for reading the leakage current (ILEAKAGE)

C

F

G

LABVIEW CHANNEL

Figure 14: Differential input of the A/D channels

Block A - CORRENTE DE FUGA - Ifuga <=> LEAKAGE CURRENT - Ileakage

system for the magnitudes of interest. Figure 13 shows the block diagram for leakage current (ILEAKAGE). The configuration for each quantity to be measured using channels with differential inputs

10V limits; (c) a2 channel: shaft voltage (VSHAFT); and (d) a3 channel: shaft current (ISHAFT).

Signals Using Split (B) block quantities of simultaneous readings can be made. In this case, there are 4 outputs. The current ILEAKAGE a function of time is shown on the oscilloscope ILEAKAGE (t).

The Amplitude and Level Measurements block (C) allows for the signs readings DC values (DC) and effective (RMS). The value is shown on the display "RMS ILEAKAGE."

The blocks Noise Filter (D) and Pass Filter Range MFB (E) treat the signal so that it can be measured on the switching frequency of the PWM inverter. The filtered signal is displayed on the oscilloscope FILTER OUT – ILEAKAGE.

Noise Filter (NOTCH) has the function of not allowing passage of signals of a frequency band or a specific frequency. The Band Pass Filter MFB allows only the frequency band specified signal to pass. This filtering block allows you to define: a) type of filter: low-pass, high-pass, band-pass, band-reject; b) frequency band; c) the type of response: finite impulse response (FIR) or infinite impulse response (IIR); d) topology: Butterworth, Chebyshev, Inverse Chebyshev, Elliptic, and Bessel.

Spectral Measurement block (F) shows the oscilloscope through the spectrum ILEAKAGE already filtered at the desired frequency. And the block (G) similar to block (C) provides effective current at switching frequency fS, available in display RMS\_ILEAKAGE\_fS. This entire procedure aims to ensure that measurements are made in order to respect the equation for calculating the capacitance (14).

### *3.1.3. PWM inverter*

The PWM inverter used in this study is a new generation of static power converters that feature improvements in its modular structure, providing better functions in the lower-frequency range and greater overload capacity. Introducing control functions integrated with the possibility of use of communication accessories, this enables applications to AC inverter drive systems requiring high efficiency in a power range 0.55–160 kW.

The Inverter MOVIDRIVE MDX60B/61B [29] has the following main characteristics: a) The MOVIDRIVE MDX61B model allows application for asynchronous motors (induction motor) with or without encoder feedback, synchronous and asynchronous servomotors. b) Control modes: VFC (Voltage Flux Control): to control induction motors. Using encoder feedback, it operates with vector control activated. If there is no feedback loop, it operates with V/f control (scalar) and CFC (Current Flux Control); for controlling synchronous and asynchronous servomotors it always operates with encoder feedback. The inverter model used in this work is the Movidrive MDX61B 0037-2A3-4-0 with the following characteristics: Power 5 Hp (3.7 kw); phase power supply: 220 V/50–60 Hz; rated output current: 15.2 A and PWM switching frequency: 4, 8, 12, and 16 kHz. Using MOVITOOLS [31] dedicated program, both the switch‐ ing frequency fS of the inverter induction motor as well as the rotational speed can be changed, allowing a more complete analysis of the behavior of the parasitic capacitances to be realized.

### **3.2. Tests laboratory procedure**

A procedure was adopted within the Electrical Machines Laboratory for the measurement tests so that determining the high-frequency capacitance of the induction motors could be carried out in conditions where there is the least possible interference in the results thereof. These procedures are described as follows: a) Ensure the total isolation of the set: threephase induction motor being tested and PWM inverter from the base where they are supported in order that there is no current flow to the system from the motor frame and the inverter. b) Verification of the connections between the motor frame and the inverter with the NEUTRAL of three-phase power supply, allowing you to create a single move‐ ment path of the motor leakage current and hence its correct measurement. c) The entire measurement system (LABVIEW acquisition board) and the computer (notebook) used in the execution of dedicated programs and measured data storage must be the same reference system NEUTRAL. This ensures reliable measurements. d) Correct positioning and enough brush pressure in the ring, the induction motor shaft, ensuring good contact and, there‐ fore, reliable measurements of shaft voltage and current (VSHAFT and ISHAFT). e) Check all connections of the acquisition system/measurement and inverter with the computer. f) carrying out the tests in air-conditioned environment, to ensure the same conditions for the other tests, and g) if possible, work with a fully isolated power system, not only using an isolating transformer, but also ensuring electrical insulation. This way, voltage and current of the supply system will always be the same in any situation and time, providing measurements and better results. Following these procedures, tests of induction motors to determine the parasitic capacitances can be started.

### **3.3. Parasitic capacitances determination of the rotor cage three-phase induction motor**

The high-frequency equivalent circuit is shown again in Figure 15.

When determining the capacitance the following procedure should be followed: a) Switch SH open: Motor bearings are isolated. So there is no current flowing through the bearings and the CB capacitance does not contribute to the value of BVR (Bearing Voltage Ratio). b) Switch SH closed: a circulating current starts to flow through the bearings, and the CB capacitance becomes to influence the value of BVR.

Measurements are performed in the following quantities of interest: common mode voltage (VCM), shaft voltage (VSHAFT), leakage current (ILEAKAGE) shaft current with SH off (ISHAFT-OFF),

motor Analysis and Methodology for Determining the Parasitic Capacitances in VSI-fed IM Drives Based on PWM Technique http://dx.doi.org/10.5772/61544 97

3.3. Parasitic capacitances determination of the rotor cage three-phase induction

tests of induction motors to determine the parasitic capacitances can be started.

The high-frequency equivalent circuit is shown again in Figure 15.

switching frequency fS of the inverter induction motor as well as the rotational speed can be changed,

A procedure was adopted within the Electrical Machines Laboratory for the measurement tests so that determining the high-frequency capacitance of the induction motors could be carried out in conditions where there is the least possible interference in the results thereof. These procedures are described as follows: a) Ensure the total isolation of the set: three-phase induction motor being tested and PWM inverter from the base where they are supported in order that there is no current flow to the system from the motor frame and the inverter. b) Verification of the connections between the motor frame and the inverter with the NEUTRAL of three-phase power supply, allowing you to create a single movement path of the motor leakage current and hence its correct measurement. c) The entire measurement system (LABVIEW acquisition board) and the computer (notebook) used in the execution of dedicated programs and measured data storage must be the same reference system NEUTRAL. This ensures reliable measurements. d) Correct positioning and enough brush pressure in the ring, the induction motor shaft, ensuring good contact and, therefore, reliable measurements of shaft voltage and current (VSHAFT and ISHAFT). e) Check all connections of the acquisition system/measurement and inverter with the computer. f) carrying out the tests in air-conditioned environment, to ensure the same conditions for the other tests, and g) if possible, work with a fully isolated power system, not only using an isolating transformer, but also ensuring electrical insulation. This way, voltage and current of the supply system will always be the same in any situation and time, providing measurements and better results. Following these procedures,

allowing a more complete analysis of the behavior of the parasitic capacitances to be realized.

3.2. Tests laboratory procedure

**Figure 15.** Simplified equivalent circuit of high-frequency induction motor