- Asked 2/12
- 2070 Asar (old course) · 8 marks
Z and T parameters with dependent source 2I1
Short answer
Formulating mesh equations with the dependent source yields Z11 = 13/4 Ω, Z12 = 3/4 Ω, Z21 = −3/4 Ω, Z22 = 7/4 Ω. The corresponding T-parameters are A = −13/3, B = −25/3 Ω, C = −4/3 S, D = −7/3.
Find the Z-parameter and T-parameter for the two-port network given in the following figure. [Figure: port 1 (V1, I1) with a 2 Ω series resistor; a 1 Ω shunt resistor; a dependent voltage source 2I1 (+ on left, − on right) in the series arm; a 3 Ω shunt resistor; a 1 Ω series resistor to port 2 (V2, I2)]
Z-parameters: V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2. T-parameters: V1 = A V2 − B I2, I1 = C V2 − D I2.
Reading the figure (assumption). 2 Ω series at port 1, then a 1 Ω shunt at node a, then the dependent voltage source 2I1 in the series arm (+ on the port 1 side, Va − Vb = 2I1) reaching node b, a 3 Ω shunt at node b, and a 1 Ω series resistor to port 2.
Equations
Let Is be the current through the 2I1 source (from a to b).
V1 = Va + 2 I1
Vb = Va − 2 I1
Node a: I1 = Va + Is (1 Ω shunt)
Node b: Is = Vb/3 + (Vb − V2) (3 Ω shunt and the 1 Ω series)
Port 2: I2 = V2 − Vb
From port 2: V2 = Vb + I2. Node b: Is = Vb/3 − I2, so Vb = 3 Is + 3 I2. Node a: Is = I1 − Va, and Va = V1 − 2 I1, so Is = 3 I1 − V1. Hence Vb = 9 I1 − 3 V1 + 3 I2. Also Vb = Va − 2 I1 = V1 − 4 I1. Equate:
V1 − 4 I1 = 9 I1 − 3 V1 + 3 I2 → 4 V1 = 13 I1 + 3 I2
V1 = (13/4) I1 + (3/4) I2
V2 = I2 + Vb = I2 + V1 − 4 I1 = −(3/4) I1 + (7/4) I2
Z-parameters
z11 = 13/4 Ω z12 = 3/4 Ω
z21 = −3/4 Ω z22 = 7/4 Ω
Δz = (13/4)(7/4) + (3/4)(3/4) = 100/16 = 25/4
T-parameters (from Z)
A = z11/z21 = −13/3 B = Δz/z21 = −25/3 Ω
C = 1/z21 = −4/3 S D = z22/z21 = −7/3
Check: AD − BC = 91/9 − 100/9 = −1 (not 1, so not reciprocal), and A ≠ D, so not symmetrical.
Answer: Z = [13/4 3/4; −3/4 7/4] Ω; T: A = −13/3, B = −25/3 Ω, C = −4/3 S, D = −7/3.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Transmission and y parameters, prove not reciprocal or symmetrical · Electrical Circuit Theory, chapter 7 · asked 2×
- Transmission and Y parameters with dependent current source · Electrical Circuit Theory, chapter 7
- Y and g parameters with dependent source 4V1 · Electrical Circuit Theory, chapter 7
- Laplace: i1, i2 in unenergised two-mesh network · Electrical Circuit Theory, chapter 4 · asked 3×
- Z and T' parameters with dependent sources · Electrical Circuit Theory, chapter 7
- Transmission and Y parameters, reciprocity and symmetry · Electrical Circuit Theory, chapter 7
- Voltage ratio transfer function of two-section LC network · Electrical Circuit Theory, chapter 5
- Two-port network model and parameters · RF and Microwave Engineering, chapter 3
- Series-parallel circuit with I1, I2, I3 phasors · Basic Electrical Engineering, chapter 6 · asked 2×
- Importance of S-parameters and two-port derivation · RF and Microwave Engineering, chapter 3 · asked 3×
- S-parameters and three-port network S-matrix · RF and Microwave Engineering, chapter 3 · asked 4×