Skip to main content

Voltage ratio transfer function of LC ladder

Short answer

Each series and shunt arm equals X=1+s/(s²+1). Cascade voltage dividers yield V2/V1=(2/5)*(1/2)=1/5, so G21(s)=1/5, constant.

For the Two-Port network shown in figure below. Find the voltage ratio transfer function. [Figure: port 1 with a 1 Ω series resistor, then a series arm of 1 H in parallel with 1 F, then a node A. Shunt branch at A: a 1 Ω resistor in series with (1 H in parallel with 1 F) to the bottom wire. Then a series 1 Ω resistor, a series arm of (1 H in parallel with 1 F), then a node B with the same shunt branch (1 Ω in series with 1 H ∥ 1 F). Then a series 1 Ω resistor and a final series arm of (1 H in parallel with 1 F) to port 2 (V2, I2).]

The voltage ratio transfer function is G21 = V2/V1 with port 2 open (I2 = 0), the usual meaning when no load is given.

Notation. Let Zp = (1 H ∥ 1 F) = sL·(1/sC)/(sL + 1/sC) = s/(s² + 1). The network is three identical series arms (1 Ω + Zp) separated by two identical shunt arms (1 Ω + Zp).

X = 1 + Zp = (s² + s + 1)/(s² + 1)
series arm  = X          shunt arm = X

Because the series arm and the shunt arm are both X, the result is independent of s.

Last section (node B)

Port 2 is open, so the last series arm carries no current and V2 = VB. Node B is loaded only by its shunt arm X:

V2/VA = X / (X + X) = 1/2          (VA = voltage at node A)

Node A

The load at A is the shunt arm X in parallel with the series arm plus the node B load (X + X = 2X):

Z_A = X ∥ 2X = 2X/3
VA/V1 = (2X/3) / (X + 2X/3) = 2/5

Overall

G21 = (VA/V1)(V2/VA) = (2/5)(1/2)

Answer: G21(s) = V2/V1 = 1/5 (a constant, independent of frequency).

The magnitude is flat at 0.2 (−14 dB) and the phase is 0° at every frequency, because each series arm equals the shunt arm that follows it. (Confirmed with a cascade-matrix calculation in Python.)

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Electrical Circuit Theory chapter 5 questions →