Two-port network parameters: Z, Y, h and ABCD

Z, Y, h and ABCD two-port parameters: definitions, measurement terminations, conversions, reciprocity, symmetry and terminated two-ports.

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Why it matters

Amplifiers, filters, attenuators, transmission lines, transducers and even whole instruments are black boxes with an input port and an output port. Two-port parameters describe such a box with four numbers measured at its terminals. You can then predict gain, input and output impedance and loading for any source and load without knowing the inside. Transistor datasheets give h-parameters, RF parts give Y- or S-parameters, and line engineers use ABCD.

Key ideas

Two-port conventions. Port 1 has (V1, I1) and port 2 has (V2, I2). Both currents are taken as entering the + terminal of their port. The network must be linear and contain no independent sources. Each port must satisfy the port condition: the current entering one terminal leaves by the other.

Z (open-circuit impedance) parameters.

  • V1 = z11·I1 + z12·I2 and V2 = z21·I1 + z22·I2.
  • Measure with one port open: z11 = V1/I1 with I2 = 0, z21 = V2/I1 with I2 = 0, z12 = V1/I2 with I1 = 0, and z22 = V2/I2 with I1 = 0.
  • For a T network with series arms Za (port 1 side) and Zb (port 2 side) and shunt arm Zc: z11 = Za + Zc, z22 = Zb + Zc, z12 = z21 = Zc.

Y (short-circuit admittance) parameters.

  • I1 = y11·V1 + y12·V2 and I2 = y21·V1 + y22·V2.
  • Measure with one port shorted: for example y11 = I1/V1 with V2 = 0.
  • [Y] = [Z]⁻¹.
  • For a π network with series arm Yb and shunt arms Ya (port 1) and Yc (port 2): y11 = Ya + Yb, y22 = Yc + Yb, y12 = y21 = −Yb.

h (hybrid) parameters.

  • V1 = h11·I1 + h12·V2 and I2 = h21·I1 + h22·V2.
  • h11 is the input impedance with the output shorted (Ω). h12 is the reverse voltage ratio with the input open (dimensionless). h21 is the forward current gain with the output shorted (dimensionless). h22 is the output admittance with the input open (S).
  • Natural for BJTs: in the CE configuration, h21 = hfe ≈ β.
  • The g-parameters are the inverse hybrid set.

ABCD (transmission) parameters.

  • V1 = A·V2 − B·I2 and I1 = C·V2 − D·I2. The minus signs appear because I2 is defined as entering port 2; with I2 taken as leaving, the signs become +.
  • A = V1/V2 with port 2 open (dimensionless). B = −V1/I2 with port 2 shorted (Ω). C = I1/V2 with port 2 open (S). D = −I1/I2 with port 2 shorted (dimensionless).
  • Used for cascades, because the overall matrix is the product of the individual matrices.

Reciprocity and symmetry.

  • Reciprocal (passive RLC networks, transformers): z12 = z21, y12 = y21, h12 = −h21, AD − BC = 1.
  • Symmetric (the ports can be swapped without change): z11 = z22, y11 = y22, A = D, h11·h22 − h12·h21 = 1.
  • Symmetry and reciprocity are separate conditions. Every passive RLC network is reciprocal, but only some are symmetric; an active network can be symmetric (z11 = z22) and still non-reciprocal.

Existence. A parameter set may not exist for some networks. An ideal transformer has no Z or Y parameters. A single series impedance has no Z parameters. A single shunt admittance has no Y parameters.

Terminated two-port. With source Vs, Rs at port 1 and load ZL at port 2 (V2 = −ZL·I2):

  • Zin = z11 − z12·z21/(z22 + ZL)
  • Zout = z22 − z12·z21/(z11 + Rs)
  • V2/V1 = z21·ZL/(z11·(z22 + ZL) − z12·z21)

Formulas

  • ΔZ = z11·z22 − z12·z21.
  • Y from Z: y11 = z22/ΔZ, y12 = −z12/ΔZ, y21 = −z21/ΔZ, y22 = z11/ΔZ.
  • h from Z: h11 = ΔZ/z22, h12 = z12/z22, h21 = −z21/z22, h22 = 1/z22.
  • ABCD from Z: A = z11/z21, B = ΔZ/z21, C = 1/z21, D = z22/z21.
  • Z from ABCD: z11 = A/C, z12 = (AD − BC)/C, z21 = 1/C, z22 = D/C.
  • Terminated: Zin = z11 − z12·z21/(z22 + ZL).

Worked examples

Example 1 (standard): parameters of a T network. Given: a T network with series arms 2 Ω (port 1 side) and 3 Ω (port 2 side) and a 5 Ω shunt arm. Find the Z, Y and ABCD parameters.

  1. Z parameters: z11 = 2 + 5 = 7 Ω, z22 = 3 + 5 = 8 Ω, z12 = z21 = 5 Ω.
  2. Determinant: ΔZ = 7·8 − 5·5 = 31 Ω².
  3. Y parameters: y11 = 8/31 = 0.258 S, y22 = 7/31 = 0.226 S, y12 = y21 = −5/31 = −0.161 S.
  4. ABCD: A = 7/5 = 1.4, B = 31/5 = 6.2 Ω, C = 1/5 = 0.2 S, D = 8/5 = 1.6.
  5. Reciprocity check: AD − BC = 2.24 − 1.24 = 1 ✓. The network is not symmetric, since A ≠ D.

Example 2 (GATE level): a terminated two-port. Given: an active two-port with z11 = 10 Ω, z12 = 4 Ω, z21 = 40 Ω and z22 = 20 Ω. It is driven by Vs = 10 V with Rs = 2 Ω and loaded by RL = 30 Ω. Find Zin, V2 and the gain V2/Vs.

  1. Input impedance: Zin = 10 − (4·40)/(20 + 30) = 10 − 3.2 = 6.8 Ω.
  2. Input current: I1 = 10/(2 + 6.8) = 1.136 A, so V1 = 6.8 × 1.136 = 7.73 V.
  3. Output side: V2 = z21·I1 + z22·I2 with V2 = −30·I2. So −30·I2 = 40·I1 + 20·I2, which gives I2 = −40·I1/50 = −0.909 A.
  4. Output voltage: V2 = −30 × (−0.909) = 27.27 V.
  5. Gains: V2/Vs = 2.73 and V2/V1 = 3.53. Because z12 ≠ z21, the network is non-reciprocal (active).

Answer: Zin = 6.8 Ω, V2 = 27.3 V, V2/Vs = 2.73.

Common mistakes

  • Using the wrong termination: Z parameters need open circuits, while Y and the h21 and h11 terms need short circuits.
  • Losing the sign of I2 in ABCD. With I2 entering port 2, B = −V1/I2 and D = −I1/I2.
  • Assuming reciprocity means symmetry, or the reverse.
  • Inverting Z to get Y element by element (y11 = 1/z11). Invert the whole matrix.
  • Forgetting that h12 and h21 are dimensionless, while h11 is in Ω and h22 in S.
  • Applying the parameter equations to a network that contains independent sources.

For GATE IN

Common question types: find one parameter set for a T, π, lattice or dependent-source network; convert between Z, Y, h and ABCD; test reciprocity and symmetry; input impedance or voltage gain of a terminated two-port; identify which parameters do not exist for a given network. Practise the measurement definitions (open or short at the right port), and memorise the conversion table from Z.

Quick check

  1. What termination is used to measure y21?
  2. A two-port has A = 2, B = 3 Ω, C = 1 S and D = 2. Is it reciprocal? Is it symmetric?
  3. What is the unit of h22?
  4. For a T network with arms 4 Ω, 4 Ω and shunt 6 Ω, what is z12?

Answers: 1. Port 2 short-circuited (V2 = 0). 2. AD − BC = 4 − 3 = 1, so it is reciprocal, and A = D, so it is symmetric. 3. Siemens. 4. 6 Ω.

Interview questions

All Electrical Circuits interview questions

Try answering each one aloud before you open it.

  1. 1.What is a two-port network, and why is it important in electrical engineering?Concept

    A two-port network is an electrical circuit or device with two pairs of terminals, one pair for input and one pair for output. It is important because it simplifies the analysis of complex circuits by allowing engineers to model and analyze the behavior of the network using parameters like impedance (Z), admittance (Y), hybrid (h), and transmission (ABCD) parameters.

  2. 2.Explain the Z-parameters in a two-port network.Concept

    Z-parameters, or impedance parameters, are used to describe the relationship between the voltages and currents at the input and output ports of a two-port network. They are defined by the equations V1 = Z11I1 + Z12I2 and V2 = Z21I1 + Z22I2, where V1 and V2 are the voltages at the input and output, and I1 and I2 are the currents at the input and output.

  3. 3.What are Y-parameters, and how do they differ from Z-parameters?Concept

    Y-parameters, or admittance parameters, describe the relationship between the currents and voltages at the input and output ports of a two-port network. They are defined by the equations I1 = Y11V1 + Y12V2 and I2 = Y21V1 + Y22V2. Unlike Z-parameters, which relate voltages to currents, Y-parameters relate currents to voltages, making them useful for parallel circuit analysis.

  4. 4.Describe the h-parameters in a two-port network.Concept

    h-parameters, or hybrid parameters, are a set of four parameters that describe the input impedance, reverse voltage gain, forward current gain, and output admittance of a two-port network. They are defined by the equations V1 = h11I1 + h12V2 and I2 = h21I1 + h22V2. These parameters are particularly useful for analyzing transistor circuits.

  5. 5.What are ABCD parameters, and where are they typically used?Concept

    ABCD (transmission) parameters relate the input quantities to the output quantities: V1 = A·V2 − B·I2 and I1 = C·V2 − D·I2, with I2 taken as entering port 2. With I2 taken as leaving port 2, the signs become +. A and D are dimensionless, B is in ohms and C in siemens. They are used for transmission lines and cascaded networks, because the ABCD matrix of a cascade is the product of the individual matrices. A reciprocal network has AD − BC = 1, and a symmetric one has A = D.

  6. 6.Why are Z-parameters preferred for series-connected two-ports, while Y-parameters are preferred for parallel-connected ones?Application

    When two two-ports are connected in series at both ports, they carry the same port currents and their port voltages add, so the overall Z matrix is the sum of the individual Z matrices. When they are connected in parallel, they share the port voltages and their currents add, so the Y matrices add. The choice therefore follows the interconnection, provided the port condition still holds after connection; Brune's test or an isolating transformer ensures that. Similarly, h-parameters add for series–parallel connections and ABCD matrices multiply for cascades.

  7. 7.What happens to the Z-parameters of a two-port network if the network is reciprocal?Application

    Reciprocity gives z12 = z21: the open-circuit transfer impedance is the same in both directions, so a current injected at port 1 produces the same open-circuit voltage at port 2 as the reverse experiment. All passive RLC networks and transformers are reciprocal. This is not the same as symmetry, which needs z11 = z22. Equivalent conditions are y12 = y21, h12 = −h21 and AD − BC = 1.

  8. 8.How can you convert Z-parameters to Y-parameters in a two-port network?Application

    To convert Z-parameters to Y-parameters, you need to find the inverse of the Z-parameter matrix. The Y-parameters are given by Y11 = Z22/ΔZ, Y12 = -Z12/ΔZ, Y21 = -Z21/ΔZ, and Y22 = Z11/ΔZ, where ΔZ = Z11Z22 - Z12Z21 is the determinant of the Z-parameter matrix.

  9. 9.A two-port has Y11 = 0.5 S, Y12 = −0.1 S, Y21 = −0.1 S and Y22 = 0.4 S. Find the input impedance when the output is open-circuited.Numerical

    With port 2 open, I2 = 0, so 0 = Y21·V1 + Y22·V2 and V2 = −Y21·V1/Y22 = 0.25·V1. Then I1 = Y11·V1 + Y12·V2 = (0.5 − 0.025)·V1 = 0.475·V1, so Zin = 1/0.475 = 2.105 Ω. This is z11 = Y22/ΔY = 0.4/0.19. Note that 1/Y11 = 2 Ω is the input impedance with the output short-circuited, not open-circuited.

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