Equivalent Y-parameter of parallel two-ports

When two-port networks N_a and N_b are connected in parallel, their input ports are connected together and their output ports are connected together, so they share the same port voltages. The Y-parameters add.

        +------------------+
 I1a -->|       Na         |--> ... I2a
        +------------------+
 I1 o---+                  +---o I2
        +------------------+
 I1b -->|       Nb         |--> ... I2b
        +------------------+
 V1 (common)                  V2 (common)

Derivation

For each network (Y-parameter equations):

I1a = y11a V1 + y12a V2          I1b = y11b V1 + y12b V2
I2a = y21a V1 + y22a V2          I2b = y21b V1 + y22b V2

The connection gives V1a = V1b = V1 and V2a = V2b = V2, while the currents add:

I1 = I1a + I1b = (y11a + y11b) V1 + (y12a + y12b) V2
I2 = I2a + I2b = (y21a + y21b) V1 + (y22a + y22b) V2

Comparing with I1 = y11V1 + y12V2, I2 = y21V1 + y22V2:

[Y] = [Ya] + [Yb]

So y11 = y11a + y11b, y12 = y12a + y12b, y21 = y21a + y21b, y22 = y22a + y22b. For three networks in parallel: [Y] = [Ya] + [Yb] + [Yc] (and likewise for any number).

Condition. The result is valid only if the connection does not alter the individual Y-parameters (the port condition: the current entering one terminal of a port must equal the current leaving the other). When this fails (Brune test), isolating ideal 1:1 transformers are used.