Two-port network

(PD) Image: John R. Brews
Two-port network with symbol definitions. A Thevenin voltage source with Thevenin impedance Z_Th drives port 1, and impedance Z_L loads port 2

A two-port network is an electrical circuit with two pairs of terminals. As shown in the figure, two terminals constitute a port only if they satisfy the essential requirement known as the port condition, namely, the same current must enter and leave a port. Examples include small-signal models for transistors (such as the hybrid-pi model), filters and matching networks. The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz.

A two-port network makes possible the replacement of either a complete circuit or part of it by a "black box" with a only four distinct parameters, enabling us to separate its behavior from that of its internal constituents, thus simplifying analysis. Any linear circuit with four terminals can be transformed into a two-port network provided that it does not contain an independent source and satisfies the port conditions.

The parameters used to describe a two-port network are the following: z, y, h, g. Each choice corresponds to a different choice for which pair of variables from port 1 and port 2 are chosen to be independent, externally applied sources and which two will be the dependent resultant variables determine by the two-port parameters (see the figure). The port currents and voltages are denoted as follows:

${V_{1}}$ = Port 1 voltage

${I_{1}}$ = Port 1 current

${V_{2}}$ = Port 2 voltage

${I_{2}}$ = Port 2 current

The relations between inputs and outputs usually are expressed in matrix notation.

These variables are most useful at low to moderate frequencies. At high frequencies (for example, microwave frequencies) power and energy are more useful variables, and the two-port approach based on current and voltages that is discussed here is replaced by an approach based upon scattering parameters.

Though some authors use the terms two-port network and four-terminal network interchangeably, the latter represents a more general concept. Not all four-terminal networks are two-port networks. A pair of terminals can be called a port only if the current entering one is equal to the current leaving the other (the port condition). Only those four-terminal networks in which the four terminals can be paired into two ports can be called two-ports.

Impedance parameters (z-parameters)

(PD) Image: John R. Brews
Z-equivalent two port showing independent variables I₁ and I₂.

The figure shows the two-port driven by two external current sources, making the input currents I₁ and I₂ the independent variables controlled from outside the two-port. The port voltages are determined in terms of these input currents by the z-parameters defined by:

$\left[{\begin{array}{c}V_{1}\\V_{2}\end{array}}\right]=\left[{\begin{array}{cc}z_{11}&z_{12}\\z_{21}&z_{22}\end{array}}\right]\left[{\begin{array}{c}I_{1}\\I_{2}\end{array}}\right]$.

$z_{11}={V_{1} \over I_{1}}{\bigg |}_{I_{2}=0}\qquad z_{12}={V_{1} \over I_{2}}{\bigg |}_{I_{1}=0}$

$z_{21}={V_{2} \over I_{1}}{\bigg |}_{I_{2}=0}\qquad z_{22}={V_{2} \over I_{2}}{\bigg |}_{I_{1}=0}$

Notice that all the series connected elements represented by z-parameters have dimensions of ohms, as do the dependent source parameters.

Example: bipolar current mirror with emitter degeneration

(PD) Image: John R. Brews
Bipolar current mirror: I₁ is the reference current and I₂ is the output current.

(PD) Image: John R. Brews
Small-signal circuit for bipolar current mirror: I₁ is the amplitude of the small-signal reference current and I₂ is the amplitude of the small-signal output current.

The figure at left shows a bipolar current mirror with emitter resistors to increase its output resistance. Transistor Q₁ is diode connected, which is to say its collector-base voltage is zero. The figure at right shows the small-signal circuit equivalent to the transistor circuit. Transistor Q₁ is represented by its emitter resistance r_E ≈ V_T / I_E (V_T = thermal voltage, I_E = Q-point emitter current), a simplification made possible because the dependent current source in the hybrid-pi model for Q₁ draws the same current as a resistor 1 / g_m connected across r_π. The second transistor Q₂ is represented by its hybrid-pi model. Table 1 below shows the z-parameter expressions that make the z-equivalent two-port electrically equivalent to the small-signal circuit for the mirror.

Table 1	Expression	Approximation
$Z_{21}={\begin{matrix}{V_{\mathrm {2} } \over I_{\mathrm {1} }}\end{matrix}}{\Big |}_{I_{2}=0}$	$-(\beta r_{O}-R_{E})$ ${\begin{matrix}{\frac {r_{E}+R_{E}}{r_{\pi }+r_{E}+2R_{E}}}\end{matrix}}$	$-\beta r_{o}$${\begin{matrix}{\frac {r_{E}+R_{E}}{r_{\pi }+2R_{E}}}\end{matrix}}$
$Z_{11}={\begin{matrix}{\frac {V_{1}}{I_{1}}}\end{matrix}}{\Big |}_{I_{2}=0}$	$(r_{E}+R_{E})$ $//$ $(r_{\pi }+R_{E})$	${\displaystyle }$
$Z_{22}={\begin{matrix}{\frac {V_{2}}{I_{2}}}\end{matrix}}{\Big |}_{I_{1}=0}$	$\ ($$1+\beta$ ${\begin{matrix}{\frac {R_{E}}{r_{\pi }+r_{E}+2R_{E}}}\end{matrix}})$ $r_{O}$ $+{\begin{matrix}{\frac {r_{\pi }+r_{E}+R_{E}}{r_{\pi }+r_{E}+2R_{E}}}\end{matrix}}$$R_{E}$	$\ ($$1+\beta$${\begin{matrix}{\frac {R_{E}}{r_{\pi }+2R_{E}}}\end{matrix}})$ $r_{O}$
$Z_{12}={\begin{matrix}{V_{\mathrm {1} } \over I_{\mathrm {2} }}\end{matrix}}{\Big |}_{I_{1}=0}$	$R_{E}$ ${\begin{matrix}{\frac {r_{E}+R_{E}}{r_{\pi }+r_{E}+2R_{E}}}\end{matrix}}$	$R_{E}$ ${\begin{matrix}{\frac {r_{E}+R_{E}}{r_{\pi }+2R_{E}}}\end{matrix}}$

In this table, the bipolar transistor emitter resistance denoted r_E is defined as:

$r_{E}={\frac {r_{\pi }}{\beta +1}}\ ,$

and the notation '//' indicates the resistance of resistors in parallel:

$R_{1}{\text{//}}R_{2}=\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)^{-1}\ .$

The negative feedback introduced by resistors R_E can be seen in these parameters. For example, when used as an active load in a differential amplifier, I₁ ≈ -I₂, making the output impedance of the mirror approximately R₂₂ -R₂₁ ≈ 2 β r_OR_E /( r_π+2R_E ) compared to only r_O without feedback (that is with R_E = 0 Ω) . At the same time, the impedance on the reference side of the mirror is approximately R₁₁ -R₁₂ ≈ ${\begin{matrix}{\frac {r_{\pi }}{r_{\pi }+2R_{E}}}\end{matrix}}$ $(r_{E}+R_{E})$, only a moderate value, but still larger than r_E with no feedback. In the differential amplifier application, a large output resistance increases the difference-mode gain, a good thing, and a small mirror input resistance is desirable to avoid Miller effect.

Admittance parameters (y-parameters)

(PD) Image: John R. Brews
Y-equivalent two port showing independent variables V₁ and V₂.

The figure shows the two-port driven by two external voltage sources, making the input voltages V₁ and V₂ the independent variables controlled from outside the two-port. The port currents are determined in terms of these input voltages by the y-parameters defined by:

$\left[{\begin{array}{c}I_{1}\\I_{2}\end{array}}\right]=\left[{\begin{array}{cc}y_{11}&y_{12}\\y_{21}&y_{22}\end{array}}\right]\left[{\begin{array}{c}V_{1}\\V_{2}\end{array}}\right]$.

where

$y_{11}={I_{1} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad y_{12}={I_{1} \over V_{2}}{\bigg |}_{V_{1}=0}$

$y_{21}={I_{2} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad y_{22}={I_{2} \over V_{2}}{\bigg |}_{V_{1}=0}$

The network is said to be reciprocal if $y_{12}=y_{21}$. Notice that all the shunt-connected elements are represented by y-parameters with dimensions of siemens, as are the dependent source parameters.

Hybrid parameters (h-parameters)

(PD) Image: John R. Brews
H-equivalent two-port showing independent variables I₁ and V₂.

The figure shows the two-port driven by two external sources, a current source at port 1 and a voltage source at port 2, making the input current I₁ and input voltage V₂ the independent variables controlled from outside the two-port. The voltage at port 1, V₁, and the current at port 2, I₂, are determined in terms of these inputs by the h-parameters defined by:

${V_{1} \choose I_{2}}={\begin{pmatrix}h_{11}&h_{12}\\h_{21}&h_{22}\end{pmatrix}}{I_{1} \choose V_{2}}$.

where

$h_{11}={V_{1} \over I_{1}}{\bigg |}_{V_{2}=0}\qquad h_{12}={V_{1} \over V_{2}}{\bigg |}_{I_{1}=0}$

$h_{21}={I_{2} \over I_{1}}{\bigg |}_{V_{2}=0}\qquad h_{22}={I_{2} \over V_{2}}{\bigg |}_{I_{1}=0}$

Often this circuit is selected when a current amplifier is described, because the port 1 input is the independent input current and port 2 output is the dependent current.

Notice that off-diagonal h-parameters are dimensionless, while the series-connected diagonal element has dimensions of ohms, while the shunt-connected diagonal element has dimensions of siemens.

Example: common base amplifier

(PD) Image: John R. Brews
Common base circuit with active load and current drive.

(PD) Image: John R. Brews
Common-base amplifier with AC current source I₁ as signal input and unspecified load supporting voltage V₂ and a dependent current I₂.

The common base amplifier is shown at left. The current source in the collector lead is an ideal DC source, and as such will not allow passage of any varying signal through itself. Its purpose in the circuit is to provide a DC bias current that sets the operating point (the quiescent transistor state about which the transistor varies in processing the signal). A signal current is applied to the emitter, and an output voltage is required at the collector. The output node is shown as an open circuit (no load) at the left, but to make use of the amplifier, a load can be attached, which then will draw a signal current from the collector node.

The small-signal circuit at the right assumes the voltage at the collector is a specified variable, and the output current is to be determined. The input current at the emitter also is a specified variable, and the signal voltage V_E developed at the emitter is a dependent variable.

Tabulated formulas in Table 2 make the h-equivalent circuit of the transistor agree with the small-signal low-frequency circuit found using the transistor hybrid-pi model at the right. Notation: r_π = base resistance of transistor, r_O = output resistance, and g_m = transconductance. The negative sign for h₂₁ reflects the convention that I₁, I₂ are positive when directed into the two-port. Parameter h₂₁ represents the current gain of the circuit viewed as a current amplifier. A non-zero value for h₁₂ means the output voltage affects the input voltage, that is, this amplifier is bilateral. If h₁₂ = 0, the amplifier is unilateral.

Table 2	Expression	Approximation
$h_{21}={\begin{matrix}{I_{\mathrm {2} } \over I_{\mathrm {1} }}\end{matrix}}{\Big |}_{V_{2}=0}$	${\begin{matrix}-{\frac {{\frac {\beta }{\beta +1}}r_{O}+r_{E}}{r_{O}+r_{E}}}\end{matrix}}$	${\begin{matrix}-{\frac {\beta }{\beta +1}}\end{matrix}}$
$h_{11}={\begin{matrix}{\frac {V_{1}}{I_{1}}}\end{matrix}}{\Big |}_{V_{2}=0}$	$r_{E}//r_{O}$	$r_{E}$
$h_{22}={\begin{matrix}{\frac {I_{2}}{V_{2}}}\end{matrix}}{\Big |}_{I_{1}=0}$	${\begin{matrix}{\frac {1}{(\beta +1)(r_{O}+r_{E})}}\end{matrix}}$	${\begin{matrix}{\frac {1}{(\beta +1)r_{O}}}\end{matrix}}$
$h_{12}={\begin{matrix}{V_{\mathrm {1} } \over V_{\mathrm {2} }}\end{matrix}}{\Big |}_{I_{1}=0}$	$\ {\begin{matrix}{\frac {r_{E}}{r_{E}+r_{O}}}\end{matrix}}\$	$\ {\begin{matrix}{\frac {r_{E}}{r_{O}}}\end{matrix}}\$ << 1

The transistor β is related to the transistor transconductance g_m as:

$\beta =g_{m}r_{\pi }\ .$

Inverse hybrid parameters (g-parameters)

(PD) Image: John R. Brews
G-equivalent two-port showing independent variables V₁ and I₂.

The figure shows the two-port driven by two external sources, a voltage source at port 1 and a current source at port 2, making the input voltage V₁ and input current I₂ the independent variables controlled from outside the two-port. The current at port 1, I₁, and the voltage at port 2, V₂, are determined in terms of these inputs by the g-parameters defined by:

${I_{1} \choose V_{2}}={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}{V_{1} \choose I_{2}}$.

where

$g_{11}={I_{1} \over V_{1}}{\bigg |}_{I_{2}=0}\qquad g_{12}={I_{1} \over I_{2}}{\bigg |}_{V_{1}=0}$

$g_{21}={V_{2} \over V_{1}}{\bigg |}_{I_{2}=0}\qquad g_{22}={V_{2} \over I_{2}}{\bigg |}_{V_{1}=0}$

Often this circuit is selected to describe a voltage amplifier, as the port 1 input is an independent voltage, and the port 2 output is a dependent voltage. Notice that off-diagonal g-parameters are dimensionless, while the series-connected diagonal element has dimensions of ohms, while the shunt-connected diagonal element has dimensions of siemens.

Example: common base amplifier

(PD) Image: John R. Brews
Bipolar transistor with base grounded and signal applied to emitter.

(PD) Image: John R. Brews
Common-base amplifier with AC voltage source V₁ as signal input and unspecified load delivering current I₂ at a dependent voltage V₂.

The common base amplifier is shown again at the left. This time, a signal voltage is applied to the emitter, and an output current is taken from the collector.

The small-signal circuit at the right assumes the current at the collector is a specified variable, and the output voltage is to be determined. The input voltage at the emitter also is a specified variable, and the signal current driven into the emitter is a dependent variable.

The tabulated formulas in Table 3 make the g-equivalent two-port for the amplifier agree with the small-signal circuit found using the small-signal low-frequency hybrid-pi model for the transistor. Notation: r_π = base resistance of transistor, r_O = output resistance, and g_m = transconductance. Parameter g₂₁ represents the voltage gain of this circuit viewed as a voltage amplifier. The negative sign for g₁₂ reflects the convention that I₁, I₂ are positive when directed into the two-port. A non-zero value for g₁₂ means the output current affects the input current, that is, this amplifier is bilateral. If g₁₂ = 0, the amplifier is unilateral.

Table 3	Expression
$g_{21}={\begin{matrix}{V_{\mathrm {2} } \over V_{\mathrm {1} }}\end{matrix}}{\Big |}_{I_{2}=0}$	$g_{m}r_{O}+1$
$g_{11}={\begin{matrix}{\frac {I_{1}}{V_{1}}}\end{matrix}}{\Big |}_{I_{2}=0}$	${\begin{matrix}{\frac {1}{r_{\pi }}}\end{matrix}}$
$g_{22}={\begin{matrix}{\frac {V_{2}}{I_{2}}}{\Big |}_{V_{1}=0}\end{matrix}}$	$r_{O}$
$g_{12}={\begin{matrix}{I_{\mathrm {1} } \over I_{\mathrm {2} }}\end{matrix}}{\Big |}_{V_{1}=0}$	$-1$