Question E7C01:
How are the capacitors and inductors of a low-pass filter Pi-network arranged between the network’s input and output?  

Choices:  
A: Two inductors are in series between the input and output, and a capacitor is connected between the two inductors and ground  
B: Two capacitors are in series between the input and output, and an inductor is connected between the two capacitors and ground  
C: An inductor is connected between the input and ground, another inductor is connected between the output and ground, and a capacitor is connected between the input and output  
D: A capacitor is connected between the input and ground, another capacitor is connected between the output and ground, and an inductor is connected between the input and output    

Correct Answer is  D: A capacitor is connected between the input and ground, another capacitor is connected between the output and ground, and an inductor is connected between the input and output  

Correct answer: D

1) Why D is correct
- A low-pass Pi-network looks like the Greek letter π: shunt capacitor at the input, shunt capacitor at the output, and a series inductor between input and output.
- Frequency behavior:
  - Inductor series path: $X_L = 2\pi f L$ increases with $f$, so low frequencies see low series reactance (pass), high frequencies see high series reactance (blocked).
  - Capacitors to ground: $X_C = \dfrac{1}{2\pi f C}$ decreases with $f$, so high frequencies are shunted to ground, while low frequencies are not.
- Together, series L and shunt Cs form a classic low-pass response.

2) Why the other choices are incorrect
- A: “Two inductors in series with a shunt capacitor between them” is a T-network low-pass, not a Pi-network.
- B: “Two capacitors in series with a shunt inductor between them” is a T-network high-pass (series Cs pass high, shunt L shorts low), not a low-pass Pi.
- C: “Shunt inductors at input and output with a series capacitor” is a Pi-network high-pass (series C passes high, shunt Ls short low), not low-pass.

3) Useful formulas and rules of thumb
- Reactances:
  - $X_L = 2\pi f L$
  - $X_C = \dfrac{1}{2\pi f C}$
- Rules of thumb:
  - Low-pass filters use series inductors and shunt capacitors.
  - High-pass filters use series capacitors and shunt inductors.
  - Pi-network: shunt–series–shunt. For low-pass, that’s C–L–C.
- A rough indicator of the corner region is where magnitudes of series and shunt reactances become comparable (e.g., $|X_L| \approx |X_C|$), though exact cutoff depends on terminations and design.