Two Port Network - 2

Table of Contents
1. Two Port network
2. Network parameters - overview
3. Z-parameters (open-circuit impedance)
4. Y-parameters (admittance, short-circuit)
5. ABCD parameters (Transmission parameters)
6. Inverse transmission parameters (A' B' C' D')
7. h-parameters (Hybrid parameters)
8. g-parameters (Inverse hybrid parameters)
9. Interrelations between parameter sets
10. Conditions for reciprocity and symmetry - summary
11. Interconnection of two-port networks
12. Barlett's bisection theorem
13. Symmetrical lattice network
14. Image impedance
15. Useful reminders and practical notes
16. Examples and practice (suggested)
17. Extended reference images and circuit diagrams
18. Additional diagrams and derivations
19. Complete interconnection examples and diagrams
20. Barlett theorem diagrams and lattice examples
21. Image impedance and further illustrations
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Two Port network

Introduction

A two-port network is an electrical network with two distinct pairs of terminals (ports) that provide access for connection to external circuits. Each port is a pair of terminals through which a current may enter or leave the network. Two-terminal elements (resistors, inductors, capacitors) form one-port networks; three-terminal devices (for example transistors) can be modelled or configured as two-port networks for analysis. If the network is linear, its behaviour can be fully characterised by a set of two-port parameters that relate voltages and currents at the two ports. These parameters allow prediction of how the network will interact with other networks.

Network parameters - overview

• Z-parameters (open-circuit impedance parameters)
• Y-parameters (short-circuit admittance parameters)
• ABCD parameters (transmission parameters)
• A' B' C' D' parameters (inverse transmission parameters)
• h-parameters (hybrid parameters)
• g-parameters (inverse hybrid parameters)

Z-parameters (open-circuit impedance)

Definition in matrix form:

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

Meaning of each parameter (measured with the other port open-circuited):

• Z₁₁ - input driving-point impedance with output open (V₂=0, I₂=0): Z₁₁ = V₁/I₁ |_I2=0
• Z₁₂ - reverse transfer impedance (ratio V₁/I₂ with I₁=0)
• Z₂₁ - forward transfer impedance (ratio V₂/I₁ with I₂=0)
• Z₂₂ - output driving-point impedance with input open (V₁=0, I₁=0): Z₂₂ = V₂/I₂ |_I1=0

The two controlled sources Z₁₂I₂ and Z₂₁I₁ are current-controlled voltage sources (CCVS) in the equivalent circuit representation.

Reciprocity and symmetry (Z-parameters)

• Reciprocal network: Z₁₂ = Z₂₁.
• Symmetrical network: Z₁₁ = Z₂₂ (and for a symmetrical reciprocal network also Z₁₂ = Z₂₁).

Y-parameters (admittance, short-circuit)

Definition in matrix form:

I₁ = Y₁₁V₁ + Y₁₂V₂

I₂ = Y₂₁V₁ + Y₂₂V₂

Meaning of each parameter (measured with the other port short-circuited):

• Y₁₁ - input driving-point admittance with output short (V₂=0): Y₁₁ = I₁/V₁ |_V2=0
• Y₁₂ - reverse transfer admittance (I₁/V₂ with V₁=0)
• Y₂₁ - forward transfer admittance (I₂/V₁ with V₂=0)
• Y₂₂ - output driving-point admittance with input short (V₁=0): Y₂₂ = I₂/V₂ |_V1=0

Reciprocity and symmetry (Y-parameters)

• Reciprocal network: Y₁₂ = Y₂₁.
• Symmetrical network: Y₁₁ = Y₂₂.

ABCD parameters (Transmission parameters)

Transmission (or chain) parameters relate the sending-end variables to the receiving-end variables. One common form is:

V₁ = A V₂ + B I₂

I₁ = C V₂ + D I₂

Definitions (special cases):

• A = V₁/V₂ with I₂=0 (receiving end open).
• B = V₁/I₂ with V₂=0 (receiving end short).
• C = I₁/V₂ with I₂=0.
• D = I₁/I₂ with V₂=0.

Reciprocity and symmetry (ABCD)

• Reciprocal passive network: AD - BC = 1.
• Symmetrical network: A = D.

Inverse transmission parameters (A' B' C' D')

The inverse transmission parameters relate receiving-end variables to sending-end variables:

V₂ = A' V₁ + B' I₁

I₂ = C' V₁ + D' I₁

Reciprocity and symmetry (inverse transmission)

• Reciprocal network: corresponding determinant condition holds (same AD - BC = 1 relation transforms accordingly).
• Symmetry: A' = D' for a symmetrical network.

h-parameters (Hybrid parameters)

Hybrid parameters are commonly used to model transistors because they mix voltage and current variables so that some parameters are measured with an open input while others with a shorted output. The matrix form used here is:

V₁ = h₁₁I₁ + h₁₂V₂

I₂ = h₂₁I₁ + h₂₂V₂

Interpretation:

• h₁₁ - input impedance with output short-circuited: h₁₁ = V₁/I₁ |_V2=0.
• h₁₂ - reverse voltage gain with input open-circuited: h₁₂ = V₁/V₂ |_I1=0.
• h₂₁ - forward current gain with output short-circuited: h₂₁ = I₂/I₁ |_V2=0.
• h₂₂ - output admittance with input open-circuited: h₂₂ = I₂/V₂ |_I1=0.

Condition of reciprocity (h-parameters)

h₁₂ = - h₂₁

g-parameters (Inverse hybrid parameters)

Inverse hybrid (g) parameters use V and I at port 1 and I and V at port 2 in the alternative hybrid arrangement:

I₁ = g₁₁V₁ + g₁₂I₂

V₂ = g₂₁V₁ + g₂₂I₂

Interpretation:

• g₁₁ - input admittance with output open-circuited.
• g₁₂ - effect of output current on input current (measured with V₁=0).
• g₂₁ - forward voltage gain with output open-circuited.
• g₂₂ - output impedance with input short-circuited.

Condition of reciprocity (g-parameters)

g₁₂ = - g₂₁

Interrelations between parameter sets

Relationship between Z and Y

The Y-parameter matrix is the inverse of the Z-parameter matrix. For a 2×2 network, if

Z = [ [Z₁₁, Z₁₂], [Z₂₁, Z₂₂] ]

then

Y = Z^-1 = 1/Δ_Z [ [Z₂₂, -Z₁₂], [-Z₂₁, Z₁₁] ]

where Δ_Z = Z₁₁Z₂₂ - Z₁₂Z₂₁.

ABCD parameters in terms of Z-parameters

Starting from Z relations:

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

One convenient set of relations (assuming I₂ is the current leaving port 2) is:

A = Z₁₁/Z₂₁,

B = (Z₁₁Z₂₂ - Z₁₂Z₂₁)/Z₂₁,

C = 1/Z₂₁,

D = Z₂₂/Z₂₁.

Z-parameters in terms of h-parameters

From the h-parameter equations

V₁ = h₁₁I₁ + h₁₂V₂

I₂ = h₂₁I₁ + h₂₂V₂

solve (ii) for V₂ and substitute into (i) to get Z parameters. The resulting relations are:

Z₁₁ = h₁₁ - (h₁₂ h₂₁)/h₂₂

Z₁₂ = h₁₂/h₂₂

Z₂₁ = - h₂₁/h₂₂

Z₂₂ = 1/h₂₂

• Find Y parameters in terms of ABCD parameters by inverting the appropriate relations.
• Find h-parameters in terms of ABCD parameters using direct algebraic substitution.

Conditions for reciprocity and symmetry - summary

Interconnection of two-port networks

1. Series connection

When two two-port networks are connected in series (port-to-port series connection, where corresponding port terminals are series connected), the overall Z-parameters are the sum of the Z-parameters of individual networks:

Z_total = Z_a + Z_b

Currents at each corresponding port are same: I_1a = I_1b = I₁, I_2a = I_2b = I₂. Voltages add: V₁ = V_1a + V_1b, V₂ = V_2a + V_2b.

2. Parallel connection

For parallel connection of two networks across both ports, the overall Y-parameters are the sum of the Y-parameters of the individual networks:

Y_total = Y_a + Y_b

Currents add at each port and voltages are the same across each port: I₁ = I_1a + I_1b, I₂ = I_2a + I_2b, V_1a = V_1b = V₁, V_2a = V_2b = V₂.

3. Series-parallel and parallel-series combinations

Combining networks in mixed series/parallel configurations requires converting to a common parameter set (Z or Y or h) and applying the appropriate add rule. For example, if series connection appears at input side and parallel at output side, write relations accordingly and combine.

4. Cascade (or cascade-connected) networks

When two two-ports are connected in cascade (output of first connected to input of second, with port polarities observed), the overall ABCD matrix is the matrix product of the individual ABCD matrices in the connection order:

[A B; C D]_total = [A B; C D]_a × [A B; C D]_b

For cascade connection pay attention to current direction: typical convention uses I measured into port 1 and out of port 2; sign conventions must be consistent when multiplying matrices.

Barlett's bisection theorem

• This theorem applies to symmetrical networks. A symmetrical reciprocal network can be split into two identical "halfsections".
• Let Z_OC be the open-circuited driving-point impedance of the half-section and Z_SCH the short-circuited driving-point impedance of the half-section. Then the two-port Z-parameters of the full symmetrical reciprocal network are given by:

In particular, for many symmetrical networks

Z₁₁ = (Z_OC + Z_SCH)/2

Z₁₂ = (Z_OC - Z_SCH)/2

These relations are useful when designing filters and ladder networks where half-sections are used.

Symmetrical lattice network

A symmetrical lattice (bridge) network is a special two-port formed from two series arms and two cross arms, used frequently in equaliser and filter design. For a symmetrical lattice built from a half-section:

Z_b ≈ Z_OCH

Z_a ≈ Z_SCH

Image impedance

• If Z_i1 is the impedance seen at port 1 when an impedance Z_i2 is connected at port 2, and Z_i2 is the impedance seen at port 2 when Z_i1 is connected at port 1, then Z_i1 and Z_i2 are called the image impedances at port 1 and port 2 respectively.

Useful reminders and practical notes

• Choose the parameter set (Z, Y, ABCD, h, g) that makes interconnection and measurement easiest for a given problem. For example, h-parameters are convenient for transistor small-signal models; ABCD parameters are convenient for cascaded transmission/line sections; Z and Y are natural for series/parallel connections respectively.
• When converting between parameter sets, check determinant conditions (e.g., Δ_Z ≠ 0 when inverting Z to obtain Y).
• Always use consistent sign conventions for currents and voltages. Many formulae assume currents are positive into port 1 and out of port 2 unless a different convention is stated.
• Reciprocity conditions depend on the network being passive and reciprocal (no ideal gyrators or active nonreciprocal elements). Symmetry requires identical behaviour when ports are interchanged.

Examples and practice (suggested)

• Convert a given 2×2 Z-matrix to Y-matrix by finding the determinant Δ_Z and applying the inverse formula.
• Given h-parameters of a transistor small-signal model, compute the equivalent Z-parameters using the relations linking h to Z.
• For two cascade sections with ABCD matrices given, compute the overall ABCD by matrix multiplication and then find the input impedance for a known load.

Extended reference images and circuit diagrams

Additional diagrams and derivations

Complete interconnection examples and diagrams

Barlett theorem diagrams and lattice examples

Image impedance and further illustrations