Study Notes - Two Port Network - Network Theory (Electric Circuits) - Electrical

Two Port Network

A port is a pair of terminals through which a current may enter or leave a network. A single two-terminal element (resistor, capacitor, inductor) is a one-port network. A two-port network has two such pairs of terminals: one pair designated as the input port and the other as the output port. Two-port models are widely used to characterise amplifiers, transistors, transformers and many linear network blocks in communications, control and power systems.

• The current entering one terminal of a port must leave through the other terminal; therefore the net current entering a port is zero.
• To characterise a linear two-port network we relate the four terminal quantities V1, V2, I1 and I2. The expressions that relate these voltages and currents are called two-port parameters.
• The term driving-point is used when the dependent and independent variables are at the same port. The term transfer is used when dependent and independent variables are at different ports.

Z-parameters (Open-circuit impedance parameters)

Choose port currents I1 and I2 as independent variables and express port voltages in terms of these currents. The Z-parameters (open-circuit impedance parameters) are defined by

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

The Z-parameters are obtained by open-circuiting the other port when measuring a parameter:

• Z11 = V1 / I1 with I2 = 0 (output open circuit) - driving-point input impedance.
• Z12 = V1 / I2 with I1 = 0 (input open circuit) - reverse transfer impedance.
• Z21 = V2 / I1 with I2 = 0 (output open circuit) - forward transfer impedance.
• Z22 = V2 / I2 with I1 = 0 (input open circuit) - driving-point output impedance.

Y-parameters (Short-circuit admittance parameters)

Choose port voltages V1 and V2 as independent variables and express port currents in terms of these voltages. The Y-parameters (short-circuit admittance parameters) are defined by

I₁ = Y₁₁V₁ + Y₁₂V₂

I₂ = Y₂₁V₁ + Y₂₂V₂

The Y-parameters are obtained by short-circuiting the other port when measuring a parameter:

• Y11 = I1 / V1 with V2 = 0 - short-circuit driving-point input admittance.
• Y12 = I1 / V2 with V1 = 0 - short-circuit reverse transfer admittance.
• Y21 = I2 / V1 with V2 = 0 - short-circuit forward transfer admittance.
• Y22 = I2 / V2 with V1 = 0 - short-circuit driving-point output admittance.

h-parameters (Hybrid parameters)

The h-parameters are hybrid because they mix voltages and currents as independent and dependent variables. Choose I1 and V2 as independent variables and write V1 and I2 as dependent:

V₁ = h₁₁I₁ + h₁₂V₂

I₂ = h₂₁I₁ + h₂₂V₂

• h11 = V1 / I1 with V2 = 0 - input impedance with output shorted (short-circuit input impedance).
• h12 = V1 / V2 with I1 = 0 - reverse voltage gain with input open (reverse open-circuit voltage ratio).
• h21 = I2 / I1 with V2 = 0 - forward short-circuit current gain (dimensionless).
• h22 = I2 / V2 with I1 = 0 - output admittance with input open (open-circuit output admittance).

g-parameters (Inverse hybrid parameters)

The g-parameters (inverse hybrid) choose V1 and I2 as independent variables and express I1 and V2 as dependent:

I₁ = g₁₁V₁ + g₁₂I₂

V₂ = g₂₁V₁ + g₂₂I₂

• g₁₂ and g₂₁ are dimensionless.
• g₁₁ has dimensions of admittance and g₂₂ has dimensions of impedance (reciprocals of each other in appropriate units).

ABCD (Transmission or T) parameters

Transmission parameters relate sending-end (port-1) variables to receiving-end (port-2) variables. They are convenient for cascaded networks and power transmission problems. Using the sign convention where the load current enters port-2, define

V₁ = A V₂ + B (-I₂)

I₁ = C V₂ + D (-I₂)

The negative sign with I₂ is used so that currents at both ports are treated as entering the two-port. The parameters have these typical interpretations:

• A is the reverse voltage ratio with open output.
• B is the reverse transfer impedance with shorted output.
• C is the reverse transfer admittance with open output.
• D is the reverse current ratio with shorted output.

Important network properties

• Symmetry: The two-port is symmetric if the ports are interchangeable without changing the behaviour. In parameter terms this requires Z11 = Z22 and Z12 = Z21 (equivalently Y11 = Y22 and Y12 = Y21). For ABCD parameters a symmetric network has A = D.
• Reciprocity: A linear two-port containing only passive reciprocal elements or bilateral dependent sources is reciprocal. Reciprocity implies Z12 = Z21 and Y12 = Y21. For ABCD parameters reciprocity leads to AD - BC = 1 (with the sign and convention used above).

Conversions between parameter sets

Two-port parameter sets are related by matrix algebra. The most common conversion is between Z and Y parameters. For a 2×2 Z matrix:

[Z] = [ [Z₁₁, Z₁₂], [Z₂₁, Z₂₂] ]

The Y matrix is the inverse of Z:

[Y] = [Z]⁻¹

Using the 2×2 inversion formula:

det(Z) = Z₁₁Z₂₂ - Z₁₂Z₂₁

Y₁₁ = Z₂₂ / det(Z)

Y₁₂ = -Z₁₂ / det(Z)

Y₂₁ = -Z₂₁ / det(Z)

Y₂₂ = Z₁₁ / det(Z)

Similarly, Z = Y⁻¹ and each other parameter set can be obtained from another by algebraic relations or by matrix inversion and multiplication. Other useful conversion formulae (obtained by algebraic manipulation) are often presented in textbooks and lab manuals.

Interconnection of two-port networks

Two-port networks may be combined to build larger networks. The common interconnections and their equivalent parameter rules are:

• Series connection (ports in series): When two two-ports are connected in series (same current through corresponding terminals), their Z-parameter matrices add elementwise. If [Z]₁, [Z]₂, ... , [Z]_nare the Z matrices of series connected stages, then the equivalent Z matrix is

  [Z]_eq = [Z]₁ + [Z]₂ + ... + [Z]_n

  
• Parallel connection: When two two-ports are connected in parallel (corresponding terminals joined so same voltages appear across them), their Y-parameter matrices add elementwise. If [Y]₁, [Y]₂, ... , [Y]_nare the Y matrices, then the equivalent Y matrix is

  [Y]_eq = [Y]₁ + [Y]₂ + ... + [Y]_n

  
• Cascade (series-feed) connection: When two two-ports are cascaded (output of one connected to input of next, matched port conventions), their ABCD (transmission) matrices multiply. If [A]₁, [A]₂, ... , [A]_nare the ABCD matrices, then the equivalent ABCD matrix is

  [A]_eq = [A]₁ × [A]₂ × ... × [A]_n

  

Notes on practical use and applications

• Choice of parameter set depends on the intended operation: use Z for series combinations and open-circuit measurements; Y for parallel combinations and short-circuit measurements; ABCD for cascaded stages and transmission lines; h and g are convenient for transistor amplifier descriptions because their independent variables match typical circuit connections.
• Reciprocity and symmetry conditions simplify parameter matrices and reduce the number of independent elements to be measured.
• Two-port models allow modular analysis: measure or compute a block's parameters once and reuse them when cascading or connecting blocks to form larger systems.

Summary

Two-port networks provide compact linear models that relate port voltages and currents using different parameter sets: Z, Y, h, g and ABCD. Each set is obtained by choosing particular independent variables and is most useful for specific interconnection types: Z for series, Y for parallel, ABCD for cascade. Conversions between parameter sets use matrix inversion and algebraic relations. Symmetry and reciprocity are important properties that reduce measurement effort and have clear algebraic tests (for example Z12 = Z21 for reciprocity; AD - BC = 1 for reciprocity in ABCD form under standard conventions).