Figure Q2(a) shows the schematic of a passive LC (this means inductor and capacitor) low-pass filter (LPF). The LPF is a two-port network, consisting of the input and output ports, the port voltages are as indicated in Figure Q2(a). The behavior of the network depends on the frequency of the voltage source driving the input, and is typically expressed as a ratio VL/VS. Since the source is a sinusoidal voltage source, the currents and voltages within this network can be expressed as Phasors. Thus in general the ratio VL/VS is a complex number which varies with frequency. A plot of the magnitude and phase of VL/VS is generally known as the Frequency Response of the network. For a LPF, the magnitude of |VL/VS| approaches unity when the frequency is approaching DC, and rapidly drops to zero above a certain frequency limits (the cut-off frequency).

Figure Q2(a) – An LC low-pass filter with load (RL) and source (RS) resistors.

(a) Using phasor notation and assuming Rs = RL = R, C1=C2=C, derive the expression for VL/VIN in terms of the elements of the LPF (e.g. L1, L2 and C1) and w (w = 2pf). What is the value for VL/VS when f = 0? Briefly explain why.
Hint: Use Nodal Analysis or other valid circuit analysis methods.

(b) Given R = 100, suggest suitable values of L1, L2 and C1 such that |VL/VS| with properties as shown in Figure Q2(b) is obtained. Using Excel or other equivalent software, with suitable Y and X axis scale, plot the magnitude response |VL/VS| versus frequency from DC to 3 MHz.
Hint: The most direct way is to use the ‘brute-force’ method. Assume a suitable set of values for L1, C1 and C2 (take C1=C2). Then plot |VL/VS| from the expression you obtain in part (a). Repeat until you get something close to Figure Q2(b).

Figure Q2(b) – The required magnitude versus frequency response.

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