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Equivalent Z-parameter of three series two-ports

Short answer

In series connection, the same currents I1 and I2 flow through each network, so port voltages add. Equivalent Z-parameters are the sum: [Z] = [Za] + [Zb] + [Zc]. The port condition must be preserved; isolation transformers may be needed.

With necessary circuit diagram, obtain the equivalent Z-parameter if three two port networks are connected in series.

When two-port networks are connected in series, their input ports are in series and their output ports are in series: the port currents are common and the port voltages add. The Z-parameters add.

      I1       +------+        I2
  o-->---------|  Na  |-----<--------o
   +           +------+           +
  V1           +------+          V2
               |  Nb  |
               +------+
               +------+
               |  Nc  |
               +------+
  (I1 flows through all three at port 1, I2 through all three at port 2)
  o------------------------------------o

Derivation

For each network k = a, b, c: V1k = z11k I1 + z12k I2, V2k = z21k I1 + z22k I2, because the same currents I1 and I2 flow in every network.

The terminal voltages are the sums:

V1 = V1a + V1b + V1c = (z11a + z11b + z11c) I1 + (z12a + z12b + z12c) I2
V2 = V2a + V2b + V2c = (z21a + z21b + z21c) I1 + (z22a + z22b + z22c) I2

Comparing with V1 = z11 I1 + z12 I2, V2 = z21 I1 + z22 I2:

[Z] = [Za] + [Zb] + [Zc]

that is, z11 = z11a + z11b + z11c, z12 = z12a + z12b + z12c, z21 = z21a + z21b + z21c, z22 = z22a + z22b + z22c.

Condition. The individual Z-parameters stay unchanged only if the series connection keeps the port condition at every network (same current into one terminal of a port as out of the other); otherwise ideal isolating transformers are used.

✓ Answer checked by Claude Opus 5.5 (high effort)

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