Two Port Network - 1

Table of Contents
1. Network parameters
2. Z-parameters
3. Y-parameters
4. Transmission parameters (ABCD)
5. Inverse transmission parameters (A' B' C' D')
6. Notes and practical remarks
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Two Port network

A port is a pair of terminals through which current may enter or leave a network. Two-terminal elements such as resistors, capacitors and inductors form one-port networks. A two-port network is an electrical network having two distinct ports (two terminal pairs) that serve as access points for an input and an output. The current entering one terminal of a port leaves the other terminal of that same port. Three-terminal devices (for example, transistors) or larger networks can be represented as two-port networks by appropriate connections.

When the network is linear, it is possible to fully characterise its behaviour by a small set of parameters obtained from voltage and current measurements at the ports. These parameters permit prediction of how the two-port will interact with other circuits.

Network parameters

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

Z-parameters

The Z-parameters relate port voltages to port currents by linear equations in the form

V₁ = Z₁₁ I₁ + Z₁₂ I₂

V₂ = Z₂₁ I₁ + Z₂₂ I₂

In matrix notation,

[V] = [Z][I], where [V] =  [V₁ V₂]ᵀ, [I] =  [I₁ I₂]ᵀ and [Z] =  [[Z₁₁ Z₁₂] [Z₂₁ Z₂₂]].

The Z-parameters are obtained using open-circuit conditions at the other port:

• Z₁₁ is the input driving-point impedance at port 1 with port 2 open (I₂ = 0)
• Z₂₂ is the output driving-point impedance at port 2 with port 1 open (I₁ = 0)
• Z₁₂ is the reverse transfer impedance: the ratio V₁/I₂ with I₁ = 0 (port 1 open)
• Z₂₁ is the forward transfer impedance: the ratio V₂/I₁ with I₂ = 0 (port 2 open)

...the input driving-point impedance with the output port open-circuited

...the reverse transfer impedance with the input port open-circuited

...the forward transfer impedance with the output port open-circuited

...the output driving-point impedance with the input port open-circuited

The Z-parameter equations can be rearranged to show an equivalent circuit representation in terms of series impedances and current-controlled voltage sources (CCVS). Rewriting the first equation gives an alternative grouping useful for drawing the equivalent circuit:

V₁ = (Z₁₁ - Z₁₂) I₁ + Z₁₂ (I₁ + I₂)

V₂ = (Z₂₁ - Z₁₂) I₁ + (Z₂₂ - Z₁₂) I₂ + Z₁₂ (I₁ + I₂)

• The terms Z₁₂I₂ and Z₂₁I₁ act as CCVS elements in the equivalent circuit representation.

Equivalent circuit for the rearranged Z-equations (one possible representation).

Another equivalent circuit showing series and controlled sources.

Condition of reciprocity and symmetry (Z-parameters)

• Reciprocal network: A two-port is reciprocal when the transfer impedances are equal: Z₁₂ = Z₂₁. In passive reciprocal networks this follows from the symmetry of the network.
• Symmetrical network: A two-port is symmetrical when its input and output driving-point impedances are equal: Z₁₁ = Z₂₂.

Illustration related to reciprocity and symmetry conditions (Z₁₁ = Z₂₂ for symmetry).

Y-parameters

The Y-parameters (admittance parameters) relate port currents to port voltages by

I₁ = Y₁₁ V₁ + Y₁₂ V₂

I₂ = Y₂₁ V₁ + Y₂₂ V₂

In matrix form, [I] = [Y][V].

The Y-parameters are measured with the other port short-circuited (V = 0):

• Y₁₁ is the input driving-point admittance at port 1 with port 2 shorted (V₂ = 0)
• Y₂₂ is the output driving-point admittance at port 2 with port 1 shorted (V₁ = 0)
• Y₁₂ is the reverse transfer admittance: I₁/V₂ with V₁ = 0 (port 1 shorted)
• Y₂₁ is the forward transfer admittance: I₂/V₁ with V₂ = 0 (port 2 shorted)

...input driving-point admittance (Y₁₁), measured with the output port short-circuited

...reverse transfer admittance (Y₁₂)

...forward transfer admittance (Y₂₁)

...output driving-point admittance (Y₂₂)

Equivalent circuit for Y-parameters

An equivalent pi or tee network may be drawn to represent the Y-parameter relations by splitting Y₁₁, Y₂₂ and mutual admittance terms.

• Rewriting the Y-equations to show admittance branches and mutual branches gives a representation useful for current-source and admittance-network thinking.

For example, the currents may be regrouped as

I₁ = (Y₁₁ + Y₁₂) V₁ - Y₁₂ (V₁ - V₂)

I₂ = (Y₂₂ + Y₁₂) V₂ - Y₁₂ (V₂ - V₁)

Equivalent circuit for these Y-parameter groupings (showing shunt admittances and mutual admittance branch).

• Reciprocity: For reciprocal networks Y₁₂ = Y₂₁.
• Symmetry: For symmetrical networks Y₁₁ = Y₂₂.

Transmission parameters (ABCD)

Transmission (ABCD) parameters relate the port variables so that the input (port 1) quantities are expressed in terms of the output (port 2) quantities. One commonly used sign convention writes

V₁ = A V₂ + B (-I₂)

I₁ = C V₂ + D (-I₂)

or in matrix form, [V₁ I₁]ᵀ = [ [A B] [C D] ] [V₂ -I₂]ᵀ.

The transmission parameters are determined under particular terminal conditions:

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

...A is the reverse voltage ratio with the receiving end open-circuited

...C is the transfer admittance with the receiving end open-circuited

...B is the transfer impedance with the receiving end short-circuited

...D is the reverse current ratio with the receiving end short-circuited

Condition for reciprocity and symmetry (ABCD)

• Reciprocity: A two-port described by ABCD parameters is reciprocal if the determinant of the A-matrix equals 1, that is AD - BC = 1.
• Symmetry: The network is symmetrical if A = D.

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

The inverse transmission parameters express the output port variables in terms of the input port variables. Using the same sign convention for port currents,

V₂ = A' V₁ + B' (-I₁)

I₂ = C' V₁ + D' (-I₁)

The inverse parameters are obtained similarly by applying the appropriate open- and short-circuit conditions at the sending end:

...A' is the forward voltage ratio with the sending end open-circuited

...C' is the transfer admittance with the sending end open-circuited

...B' is the transfer impedance with the sending end short-circuited

...D' is the forward current ratio with the sending end short-circuited

Condition of reciprocity and symmetry (inverse parameters)

• Reciprocity: For the inverse transmission set the corresponding determinant condition AD - BC = 1 carries through to the inverse representation appropriately.
• Symmetry: For inverse parameters the network is symmetrical when A' = D'.

Notes and practical remarks

• Choice of parameter set depends on the application: Z and Y are convenient for series/shunt connections, ABCD parameters are convenient for cascaded two-ports (they multiply), while h-parameters are often used for transistor models at low frequencies.
• Measurement conditions must be respected: Z-parameters use open-circuit at the opposite port, Y-parameters use short-circuit, and ABCD definitions depend on the particular open/short conditions indicated in their definitions.
• Reciprocity is a property of reciprocal (non-gyrator) networks; many passive linear networks built from resistors, capacitors, inductors and mutual inductances are reciprocal. Active devices or those with magnetic bias (gyrators) can break reciprocity.
• Symmetry is a special case of reciprocity where the two ports are identical in their driving-point behaviour.

Summary

This chapter introduced two-port networks and the common parameter sets used to represent them: Z, Y, ABCD and inverse ABCD. Each parameter set has a clear physical meaning tied to specific measurement conditions (open or short at the other port). Conditions for reciprocity and symmetry are stated for each parameter set. Equivalent circuit forms illustrate how controlled sources and series/shunt elements represent mutual interaction between ports.