Current and voltage ratio transfer functions, 2 H shunt

The current ratio transfer function is α21 = I2/I1 and the voltage ratio transfer function is G21 = V2/V1, with the 2 Ω load across port 2. I2 is taken as the current into the load (I2 = V2/2, opposite to the arrow drawn); with the arrow as drawn, α21 changes sign.

Figure. Series arm: 1 Ω, then 1 Ω; a ½ F capacitor bridging across both resistors; a 2 H inductor in shunt at the junction M; 2 Ω load across port 2.

      +---------||---------+   ½ F
      |                    |
 o----+---/\/\---+---/\/\--+----o
           1Ω    |    1Ω        |
                2H             2Ω load
 o---------------+--------------+--o

Node equations

Admittances: 1 Ω → 1, 2 H → 1/(2s), ½ F → s/2, load → 1/2.

M: (VM − V1) + (VM − V2) + VM/(2s) = 0
O: (V2 − VM) + (s/2)(V2 − V1) + V2/2 = 0
I1 = (V1 − VM) + (s/2)(V1 − V2)

From M: VM(2 + 1/(2s)) = V1 + V2, i.e. VM = 2s(V1 + V2)/(4s + 1). Substituting into the O equation and simplifying (checked in Python):

G21 = V2/V1 = s(4s + 5) / (4s² + 9s + 3)

Current ratio

I2 = V2/2
α21 = I2/I1 = s(4s + 5) / (4s² + 9s + 6)

(At s → ∞ the capacitor shorts the bridge, and G21 → 1.)

Answer: G21(s) = s(4s + 5)/(4s² + 9s + 3); α21(s) = s(4s + 5)/(4s² + 9s + 6).

Both are zero at s = 0 (the 2 H shunt inductor short-circuits the output at d.c.).