Interconnection of two-port networks

Why it matters

Real systems are built from blocks: a sensor, a cable, a filter section, an amplifier and a feedback network. If you know each block's two-port parameters, the interconnection rules give the parameters of the whole chain directly, without re-solving the full circuit. Ladder filters, multi-section attenuators, bridged-T notch filters and feedback amplifiers are all analysed this way.

Key ideas

Five standard interconnections. In each case, pick the parameter set in which the quantities that are shared by both networks are the independent variables. The dependent quantities then add.

1. Cascade (tandem). The output port of network a feeds the input port of network b, so V2a = V1b and −I2a = I1b. The ABCD matrices multiply in order: [T] = [Ta]·[Tb]. Order matters, since matrix multiplication is not commutative.
2. Series–series. Both input ports are in series and both output ports are in series. The port currents are common and the voltages add, so [Z] = [Za] + [Zb]. This models current-sensing (series) feedback at both ports.
3. Parallel–parallel. Both input ports are in parallel and both output ports are in parallel. The port voltages are common and the currents add, so [Y] = [Ya] + [Yb]. Examples are a bridged-T, a twin-T, and shunt–shunt feedback.
4. Series–parallel. Inputs in series, outputs in parallel: [h] = [ha] + [hb]. This models voltage-series feedback amplifiers.
5. Parallel–series. Inputs in parallel, outputs in series: [g] = [ga] + [gb].

Validity: the port condition (Brune's test). The addition rules hold only if, after connection, each network still has equal and opposite currents at the two terminals of each port. Interconnecting can create circulating currents that violate this.

• Three-terminal networks (common ground between ports) connected in parallel–parallel or cascade always satisfy it.
• Series connections of three-terminal networks need care. One network's ground can short part of the other.
• When in doubt, insert an ideal 1:1 isolating transformer at a port; that always restores the port condition.

Cascade building blocks.

• Series impedance Z: [[1, Z], [0, 1]].
• Shunt admittance Y: [[1, 0], [Y, 1]].
• Ideal transformer N1:N2 = n: [[n, 0], [0, 1/n]].

Any ladder network is the product of these matrices, taken from input to output.

Terminated cascade. With load ZL at the output:

• Zin = (A·ZL + B)/(C·ZL + D)
• V2/V1 = ZL/(A·ZL + B)
• With the output open, V1/V2 = A.
• With the output shorted, Zin = B/D.

Identical networks. Two identical reciprocal networks in parallel–parallel give Y = 2·Ya, so the overall Z matrix is half of Za. In series–series, Z = 2·Za.

Formulas

• Cascade: [A B; C D] = [Aa Ba; Ca Da]·[Ab Bb; Cb Db]
• Series–series: [Z] = [Za] + [Zb] (Ω)
• Parallel–parallel: [Y] = [Ya] + [Yb] (S)
• Series–parallel: [h] = [ha] + [hb]. Parallel–series: [g] = [ga] + [gb].
• Series element: T = [[1, Z], [0, 1]]. Shunt element: T = [[1, 0], [Y, 1]].
• Terminated: Zin = (A·ZL + B)/(C·ZL + D), V2/V1 = ZL/(A·ZL + B).
• Reciprocity is preserved: if each block has AD − BC = 1, so does the cascade.

Worked examples

Example 1 (standard): cascade of two L-sections. Given: each section is a 2 Ω series resistor followed by a 0.5 S shunt conductance. Two identical sections are cascaded and loaded with RL = 4 Ω. Find the overall ABCD matrix, Zin and V2/V1.

1. One section: [[1, 2], [0, 1]]·[[1, 0], [0.5, 1]] = [[2, 2], [0.5, 1]].
2. Two sections: [[2, 2], [0.5, 1]]·[[2, 2], [0.5, 1]] = [[2·2 + 2·0.5, 2·2 + 2·1], [0.5·2 + 1·0.5, 0.5·2 + 1·1]] = [[5, 6], [1.5, 2]].
3. Check: AD − BC = 10 − 9 = 1 ✓ (reciprocal).
4. Input impedance: Zin = (5·4 + 6)/(1.5·4 + 2) = 26/8 = 3.25 Ω.
5. Voltage gain: V2/V1 = 4/(5·4 + 6) = 4/26 = 0.154.

Answer: T = [[5, 6 Ω], [1.5 S, 2]], Zin = 3.25 Ω, V2/V1 = 0.154.

Example 2 (GATE level): a bridged-T as a parallel–parallel connection. Given: a T network with series arms 2 Ω and 2 Ω and a shunt arm of 1 Ω, bridged by a 2 Ω resistor connected directly from input to output (common ground). Find the Y parameters, z11 and the open-circuit voltage ratio V2/V1.

1. T network: Za = [[3, 1], [1, 3]] with ΔZ = 8, so Ya = [[0.375, −0.125], [−0.125, 0.375]] S.
2. Bridging resistor (series element between the top terminals): Yb = [[0.5, −0.5], [−0.5, 0.5]] S.
3. Both are three-terminal networks with a common ground, so the port condition holds and Y = Ya + Yb = [[0.875, −0.625], [−0.625, 0.875]] S.
4. ΔY = 0.875² − 0.625² = 0.375, so z11 = y22/ΔY = 0.875/0.375 = 2.333 Ω.
5. Open-circuit voltage ratio: V2/V1 = −y21/y22 = 0.625/0.875 = 0.714.
6. Direct check: with the output open, the input sees (2 ∥ (2 + 2)) + 1 = 1.333 + 1 = 2.333 Ω ✓.

Answer: Y = [[0.875, −0.625], [−0.625, 0.875]] S, z11 = 2.33 Ω, V2/V1 = 0.714.

Common mistakes

• Adding ABCD matrices for a cascade, or multiplying them in the wrong order.
• Adding Z matrices for a parallel connection (add Y), or Y matrices for a series connection (add Z).
• Ignoring the port condition. Series connections of grounded networks often violate it.
• Treating "series connection" of two-ports as a cascade. They are different.
• Forgetting the sign convention for I2 when chaining ABCD (−I2a = I1b).

For GATE IN

Typical questions: the overall ABCD of a cascade of simple sections; Z of a series–series or Y of a parallel–parallel combination; Zin or the gain of a terminated cascade; recognising which parameter set adds for a given feedback topology; checking reciprocity of the result. Practise the elementary series, shunt and transformer matrices, since most cascade questions reduce to multiplying two or three of them.

Quick check

1. Which parameters add when two two-ports are connected in parallel at both ports?
2. What is the ABCD matrix of a single shunt admittance Y?
3. Two identical networks with Z = [[4, 2], [2, 4]] Ω are connected series–series. What is the overall Z?
4. A cascade has A = 5. What is V1/V2 with the output open?

Answers: 1. Y parameters. 2. [[1, 0], [Y, 1]]. 3. [[8, 4], [4, 8]] Ω. 4. 5.