Units of Chapter 24 Alternating Voltages and Currents Capacitors in AC Circuits RC Circuits Inductors in AC Circuits RLC Circuits Resonance in Electrical Circuits
24-1 Alternating Voltages and Current
Units of Chapter 24 Alternating Voltages and Currents Capacitors in AC Circuits RC Circuits Inductors in AC Circuits RLC Circuits Resonance in Electrical Circuits
24-1 Alternating Voltages and Currents Wall sockets provide current and voltage that vary sinusoidally with time. Here is a simple ac circuit:
24-1 Alternating Voltages and Currents The voltage as a function of time is:
24-1 Alternating Voltages and Currents Since this circuit has only a resistor, the current is given by: Here, the current and voltage have peaks at the same time – they are in phase.
24-1 Alternating Voltages and Currents In order to visualize the phase relationships between the current and voltage in ac circuits, we define phasors – vectors whose length is the maximum voltage or current, and which rotate around an origin with the angular speed of the oscillating current. The instantaneous value of the voltage or current represented by the phasor is its projection on the y axis.
24-1 Alternating Voltages and Currents The voltage and current in an ac circuit both average to zero, making the average useless in describing their behavior. We use instead the root mean square (rms); we square the value, find the mean value, and then take the square root: 120 volts is the rms value of household ac.
24-1 Alternating Voltages and Currents By calculating the power and finding the average, we see that:
24-1 Alternating Voltages and Currents Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance. A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts. A circuit breaker is similar, except that it is a bimetallic strip that bends enough to break the connection when it becomes too hot. When it cools, it can be reset.
24-1 Alternating Voltages and Currents A ground fault circuit interrupter can cut off the current in a short circuit within a millisecond.
24-2 Capacitors in AC Circuits How is the rms current in the capacitor related to its capacitance and to the frequency? The answer, which requires calculus to derive:
24-2 Capacitors in AC Circuits In analogy with resistance, we write:
24-2 Capacitors in AC Circuits The voltage and current in a capacitor are not in phase. The voltage lags by 90°.
24-3 RC Circuits In an RC circuit, the current across the resistor and the current across the capacitor are not in phase. This means that the maximum current is not the sum of the maximum resistor current and the maximum capacitor current; they do not peak at the same time.
24-3 RC Circuits This phasor diagram illustrates the phase relationships. The voltages across the capacitor and across the resistor are at 90° in the diagram; if they are added as vectors, we find the maximum.
24-3 RC Circuits This has the exact same form as V = IR if we define the impedance, Z :
24-3 RC Circuits There is a phase angle between the voltage and the current, as seen in the diagram.
24-3 RC Circuits The power in the circuit is given by: Because of this, the factor cos φ is called the power factor.
24-4 Inductors in AC Circuits Just as with capacitance, we can define inductive reactance:
24-4 Inductors in AC Circuits The voltage across an inductor leads the current by 90°.
24-4 Inductors in AC 