--- title: "Transmission and y parameters, prove not reciprocal or symmetrical" subject: "Electrical Circuit Theory" chapter: 7 key: "semester_3/electrical_circuit_theory:transmission-and-y-parameters-neither-reciprocal" source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/transmission-and-y-parameters-neither-reciprocal site: BCT Engineering --- # Transmission and y parameters, prove not reciprocal or symmetrical Electrical Circuit Theory · Chapter 7: Two-port Parameter of Networks - Asked in: 2067 Asar (old course), 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/transmission-and-y-parameters-neither-reciprocal - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-7 ## Question Find the transmission and y-parameter of the two port network given in the following figure and also prove that the network is neither reciprocal nor symmetrical. [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)] ## Short answer Applying nodal analysis and solving for Y via Z, the admittance parameters are y11 = 0.28 S, y12 = −0.12 S, y21 = 0.12 S, y22 = 0.52 S. Transmission parameters: A = −13/3, B = −25/3 Ω, C = −4/3 S, D = −7/3. The network is neither reciprocal (y12 ≠ y21) nor symmetrical (y11 ≠ y22). ## Answer Transmission parameters: V1 = A V2 − B I2, I1 = C V2 − D I2. Admittance parameters: I1 = y11V1 + y12V2, I2 = y21V1 + y22V2. **Reading the figure (assumption).** 2 Ω series from port 1 to node a; 1 Ω shunt at node a; a dependent voltage source 2I1 in the series arm with + on the port 1 side (Va − Vb = 2I1), reaching node b; 3 Ω shunt at node b; 1 Ω series resistor to port 2. #### Equations Let Is be the current through the source (from a to b). ```text Va = V1 − 2 I1 (2 Ω drop) Vb = Va − 2 I1 = V1 − 4 I1 (controlled source) Node a: I1 = Va/1 + Is → Is = I1 − (V1 − 2 I1) = 3 I1 − V1 Node b: Is = Vb/3 + (Vb − V2)/1 Port 2: I2 = V2 − Vb (current through the 1 Ω into port 2) ``` #### Solving (first for Z) From port 2: V2 = Vb + I2. From node b: Is = Vb/3 − I2, so Vb = 3 Is + 3 I2 = 9 I1 − 3 V1 + 3 I2. Equate with Vb = V1 − 4 I1: ```text V1 − 4 I1 = 9 I1 − 3 V1 + 3 I2 → V1 = (13/4) I1 + (3/4) I2 V2 = Vb + I2 = V1 − 4 I1 + I2 → V2 = −(3/4) I1 + (7/4) I2 ``` So Z = [13/4 3/4; −3/4 7/4] Ω and Δz = (13/4)(7/4) + (3/4)(3/4) = 25/4. #### y-parameters (Y = Z⁻¹) ```text y11 = z22/Δz = 7/25 = 0.28 S y12 = −z12/Δz = −3/25 = −0.12 S y21 = −z21/Δz = 3/25 = 0.12 S y22 = z11/Δz = 13/25 = 0.52 S ``` #### Transmission parameters ```text A = −y22/y21 = −13/3 B = −1/y21 = −25/3 Ω C = −Δy/y21 = −4/3 S D = −y11/y21 = −7/3 (Δy = 4/25) ``` #### Reciprocity and symmetry - Reciprocal needs y12 = y21 (or AD − BC = 1). Here y12 = −0.12 ≠ y21 = 0.12, and AD − BC = (−13/3)(−7/3) − (−25/3)(−4/3) = 91/9 − 100/9 = −1 ≠ 1. **Not reciprocal.** - Symmetrical needs A = D (or y11 = y22): −13/3 ≠ −7/3. **Not symmetrical.** **Answer: T = [−13/3 −25/3 Ω; −4/3 S −7/3]; Y = [0.28 −0.12; 0.12 0.52] S; the network is neither reciprocal nor symmetrical.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With the source 2I1 (+ on port 1 side), what is Vb? A. Vb = V1 - 4I1 B. Vb = V1 + 4I1 C. Vb = V1 - 2I1 D. Vb = 2I1 Answer: A. Va = V1 - 2I1 and Vb = Va - 2I1 = V1 - 4I1. (Key point: Controlled source drop) 2. How is Y obtained in the answer? A. Solve for Z first, then Y = Z^-1 B. Short-circuit tests only C. Directly from T D. From the h-parameters Answer: A. Z is solved from the equations and then inverted. (Key point: Strategy) 3. Given y21 = 0.12 and y22 = 0.52, what is A? A. -13/3 B. 13/3 C. -7/3 D. -25/3 Answer: A. A = -y22/y21 = -0.52/0.12 = -13/3. (Key point: Y to T) 4. How does the answer show the network is not reciprocal? A. y12 = -0.12 differs from y21 = 0.12 and AD - BC = -1 B. AD - BC = 1 C. A = D D. y11 = y22 Answer: A. Reciprocity needs y12 = y21 or AD - BC = 1; both fail. (Key point: Reciprocity test) ## Close variants set in other exams - [Z and T parameters with dependent source 2I1](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Az-and-t-parameters-2i1-two) — Electrical Circuit Theory, chapter 7, asked 2× ## Related in Electrical Circuit Theory - [Z and g parameters, neither reciprocal nor symmetrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Az-and-g-parameters-neither-reciprocal) — Electrical Circuit Theory, chapter 7 - [Transmission and Y parameters with dependent current source](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Atransmission-and-y-parameters-0-5i3) — Electrical Circuit Theory, chapter 7 - [Y and g parameters with dependent source 4V1](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ay-and-g-parameters-1h-2f) — Electrical Circuit Theory, chapter 7 - [Transmission and Y parameters, reciprocity and symmetry](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Atransmission-admittance-parameters-reciprocity-symmetry-2v2) — Electrical Circuit Theory, chapter 7 - [Z and T' parameters with dependent sources](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Az-and-t-parameters-dependent-sources) — Electrical Circuit Theory, chapter 7 - [Reciprocal two-port and condition in T parameters](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Areciprocal-two-port-condition-t-parameters) — Electrical Circuit Theory, chapter 7, asked 2× ## Related in other subjects - [Two-port network model and parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atwo-port-network-model) — RF and Microwave Engineering, chapter 3