---
title: "G21(S) and Z21(S) of two-port with 0.5 F, 1 H"
subject: "Electrical Circuit Theory"
chapter: 5
key: "semester_3/electrical_circuit_theory:g21-z21-two-port-half-f"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/g21-z21-two-port-half-f
site: BCT Engineering
---

# G21(S) and Z21(S) of two-port with 0.5 F, 1 H

Electrical Circuit Theory · Chapter 5: Frequency Response of Network

- Asked in: 2071 Chaitra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/g21-z21-two-port-half-f
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-5

## Question

For the 2-port network shown in figure below, find voltage ratio transfer function, G21(S) and transfer admittance, Z21(S).
[Figure: series arm at port 1: a 0.5 F capacitor in parallel with a 1 Ω resistor, followed by a 2 Ω series resistor. Shunt branch: a 1 Ω resistor in series with a 1 H inductor to the bottom wire. Ports: V1, I1 and V2, I2. The scan prints "transfer admittance, Z21(S)".]

## Short answer

With port 2 open, series impedance Za=2(s+3)/(s+2), shunt Zb=s+1. Z21=Zb=s+1, Z11=Za+Zb=(s²+5s+8)/(s+2). Thus G21=(s+1)(s+2)/(s²+5s+8).

## Answer

G21 = V2/V1 and Z21 = V2/I1, both with port 2 open (I2 = 0). The scan calls Z21 a "transfer admittance", but Z21 is the **transfer impedance**; the transfer admittance Y21 = I2/V1 (V2 = 0) is also found below.

```text
         0.5F
      +---||---+     2Ω
 I1   |        +---/\/\---+-------o
 o----+        |          |
      +--/\/\--+         1Ω   (1 Ω + 1 H in series)
          1Ω              |
                         1H     V2
 V1                       |
 o------------------------+-------o
```

#### Series arm

```text
Za = (1 ∥ 0.5F) + 2 = [1·(2/s)] / [1 + 2/s] + 2 = 2/(s + 2) + 2 = 2(s + 3)/(s + 2)
```

#### Shunt arm

```text
Zb = 1 + s
```

#### Open circuit functions

```text
Z11 = Za + Zb = [2(s + 3) + (s + 1)(s + 2)] / (s + 2) = (s² + 5s + 8)/(s + 2)
V2  = I1·Zb   →   Z21 = Zb = s + 1
G21 = Z21/Z11 = (s + 1)(s + 2) / (s² + 5s + 8)
```

#### Transfer admittance (short-circuited port 2)

With V2 = 0 the shunt arm is shorted, so the current through the series arm is I2 = −V1/Za:

```text
Y21 = −1/Za = −(s + 2) / [2(s + 3)]
```

**Answer: G21(s) = (s + 1)(s + 2)/(s² + 5s + 8); Z21(s) = s + 1 (and Y21(s) = −(s + 2)/[2(s + 3)]).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The question says 'transfer admittance Z21'. What is Z21 actually?
   A. Transfer impedance V2/I1 with port 2 open
   B. Transfer admittance I2/V1 with port 2 shorted
   C. Voltage ratio V2/V1
   D. Driving point impedance V1/I1
   Answer: A. Z is an impedance; Z21 = V2/I1 with I2 = 0. Admittance would be Y21. (Key point: Z21 is transfer impedance)

2. What is Za = (1 Ω in parallel with 0.5 F) + 2 Ω?
   A. (s + 3)/(s + 2)
   B. 2(s + 2)/(s + 3)
   C. 2/(s + 2)
   D. 2(s + 3)/(s + 2)
   Answer: D. 1 ∥ (2/s) = 2/(s + 2); adding 2 Ω gives 2(s + 3)/(s + 2). (Key point: Series arm impedance)

3. Why is Z21 equal to the shunt arm Zb = 1 + s?
   A. The shunt arm is in series with the 2 Ω resistor and carries no current
   B. Zb is the sum Za + Zb
   C. With port 2 open, all of I1 passes through the shunt arm, so V2 = I1 Zb
   D. Port 2 is shorted so V2 = Zb
   Answer: C. No current flows in the 2 Ω resistor when I2 = 0, so V2 = I1(1 + s). (Key point: Why Z21 = Zb)

4. What is G21 = Z21/Z11 here?
   A. (s^2 + 5s + 8)/((s + 1)(s + 2))
   B. (s + 1)(s + 2)/(s^2 + 5s + 8)
   C. (s + 1)/(s + 2)
   D. (s + 2)/(s^2 + 5s + 8)
   Answer: B. Z11 = (s^2 + 5s + 8)/(s + 2) and Z21 = s + 1, so their ratio is as stated. (Key point: Resulting G21)

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