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Z parameters, reciprocity and symmetricity with 3V1

Short answer

The 1 H shunt and dependent source 3V1 give V1 = −s I1, so z11 = −s, z12 = 0. KCL at port 2 with 1 F capacitor yields z21 = −3s³/(s²+1), z22 = s/(s²+1). Not reciprocal (z12 ≠ z21) and not symmetrical (z11 ≠ z22).

Find the Z-parameters in the network shown below and also check for its reciprocity and symmetricity. [Figure: port 1 (V1, I1); a 1 H inductor as a shunt at port 1; a 1 H inductor in the series arm; a dependent voltage source 3V1 (+ at top) as a middle shunt branch; a 1 F capacitor in the series arm; a 1 H inductor as a shunt at port 2 (V2)]

Z-parameters: V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2.

Circuit (from the figure). A 1 H shunt inductor at port 1; a 1 H inductor in the series arm; a dependent voltage source 3V1 (+ at top) as the middle shunt branch; a 1 F capacitor in the series arm; a 1 H shunt inductor at port 2. The middle node M is therefore held at VM = 3V1.

Port 1

The current through the series 1 H into M is (V1 − VM)/s, and the shunt inductor takes V1/s:

I1 = V1/s + (V1 − 3V1)/s = −V1/s      →   V1 = −s I1

So z11 = −s and z12 = 0 (V1 does not depend on I2).

Port 2

The 1 F capacitor (impedance 1/s) carries current (VM − V2)·s towards port 2. KCL at the port 2 node (I2 flows in) gives I2 + s(3V1 − V2) = V2/s:

V2 (s + 1/s) = I2 + 3s V1 = I2 − 3s² I1
V2 = [ s/(s² + 1) ] I2 − [ 3s³/(s² + 1) ] I1

z11 = −s                 z12 = 0
z21 = −3s³/(s² + 1)      z22 = s/(s² + 1)

Reciprocity and symmetry

  • Reciprocal if z12 = z21: here 0 ≠ −3s³/(s² + 1), so the network is not reciprocal (the dependent source).
  • Symmetrical if z11 = z22 (and reciprocal): here −s ≠ s/(s² + 1), so it is not symmetrical.

Answer: Z = [−s 0; −3s³/(s²+1) s/(s²+1)] Ω; neither reciprocal nor symmetrical.

✓ Answer checked by Claude Opus 5.5 (high effort)

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