User:John R. Brews/WP Import

(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 in order to describe a two-port network are the following: z, y, h, g, T. They are usually expressed in matrix notation and they establish relations between the port currents and voltages (see the figure):

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

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

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

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

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 z-parameters have dimensions of ohms.

Example: bipolar current mirror with emitter degeneration

Figure 3: Bipolar current mirror: i₁ is the reference current and i₂ is the output current; lower case symbols indicate these are total currents that include the DC components

Figure 4: Small-signal bipolar current mirror: I₁ is the amplitude of the small-signal reference current and I₂ is the amplitude of the small-signal output current

Figure 3 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. Figure 4 shows the small-signal circuit equivalent to Figure 3. 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 circuit of Figure 2 electrically equivalent to the small-signal circuit of Figure 4.

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

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Y-parameters have dimensions of siemens.

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 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 and the voltage at port 2 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 wanted at the output. The resistors shown in the diagram can be general impedances instead.

Notice that off-diagonal h-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another.

Example: common base amplifier

Figure 7: Common-base amplifier with AC current source I₁ as signal input and unspecified load supporting voltage V₂ and a dependent current I₂.

Note: Tabulated formulas in Table 2 make the h-equivalent circuit of the transistor from Figure 6 agree with its small-signal low-frequency hybrid-pi model in Figure 7. 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. 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

Inverse hybrid parameters (g-parameters)

Figure 8: G-equivalent two-port showing independent variables V₁ and I₂; g₁₁ is reciprocated to make a resistor

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

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 when a voltage amplifier is wanted at the output. Notice that off-diagonal g-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another. The resistors shown in the diagram can be general impedances instead.

Example: common base amplifier

Figure 9: Common-base amplifier with AC voltage source V₁ as signal input and unspecified load delivering current I₂ at a dependent voltage V₂.

Note: Tabulated formulas in Table 3 make the g-equivalent circuit of the transistor from Figure 8 agree with its small-signal low-frequency hybrid-pi model in Figure 9. Notation: r_π = base resistance of transistor, r_O = output resistance, and g_m = transconductance. 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
$g_{21}={\begin{matrix}{V_{\mathrm {2} } \over V_{\mathrm {1} }}\end{matrix}}{\Big |}_{I_{2}=0}$	$g_{m}r_{O}+1$
$g_{11}={\begin{matrix}{\frac {I_{1}}{V_{1}}}\end{matrix}}{\Big |}_{I_{2}=0}$	${\begin{matrix}{\frac {1}{r_{\pi }}}\end{matrix}}$
$g_{22}={\begin{matrix}{\frac {V_{2}}{I_{2}}}{\Big |}_{V_{1}=0}\end{matrix}}$	$r_{O}$
$g_{12}={\begin{matrix}{I_{\mathrm {1} } \over I_{\mathrm {2} }}\end{matrix}}{\Big |}_{V_{1}=0}$	$-1$

ABCD-parameters

The ABCD-parameters are known variously as chain, cascade, or transmission parameters.

${V_{1} \choose I_{1}}={\begin{pmatrix}A&B\\C&D\end{pmatrix}}{V_{2} \choose I_{2}}$.

where

$A={V_{1} \over V_{2}}{\bigg |}_{I_{2}=0}\qquad B={V_{1} \over I_{2}}{\bigg |}_{V_{2}=0}$

$C={I_{1} \over V_{2}}{\bigg |}_{I_{2}=0}\qquad D={I_{1} \over I_{2}}{\bigg |}_{V_{2}=0}$

Note that we have inserted negative signs in front of the fractions in the definitions of parameters B and D. The reason for adpoting this convention (as opposed to the convention adopted above for the other sets of parameters) is that it allows us to represent the transmission matrix of cascades of two or more two-port networks as simple matrix multiplications of the matrices of the individual networks. This convention is equivalent to reversing the direction of I₂ so that it points in the same direction as the input current to the next stage in the cascaded network.

This technique is exactly analogous to the use of ABCD matrices for ray tracing in the science of optics. See also ray transfer matrix.

Table of transmission parameters

The table below lists transmission parameters for some simple network elements.

Element	Matrix	Remarks
Series resistor	${\begin{pmatrix}1&-R\\0&1\end{pmatrix}}$	R = resistance
Shunt resistor	${\begin{pmatrix}1&0\\-1/R&1\end{pmatrix}}$	R = resistance
Series conductor	${\begin{pmatrix}1&-1/G\\0&1\end{pmatrix}}$	G = conductance
Shunt conductor	${\begin{pmatrix}1&0\\-G&1\end{pmatrix}}$	G = conductance
Series inductor	${\begin{pmatrix}1&-Ls\\0&1\end{pmatrix}}$	L = inductance s = complex angular frequency
Shunt capacitor	${\begin{pmatrix}1&0\\-Cs&1\end{pmatrix}}$	C = capacitance s = complex angular frequency

Combinations of two-port networks

Series connection of two 2-port networks: Z = Z1 + Z2
Parallel connection of two 2-port networks: Y = Y1 + Y2

Example: Cascading two networks

Suppose we have a two-port network consisting of a series resistor R followed by a shunt capacitor C. We can model the entire network as a cascade of two simpler networks:

$\mathbf {T} _{1}={\begin{pmatrix}1&-R\\0&1\end{pmatrix}}$

$\mathbf {T} _{2}={\begin{pmatrix}1&0\\-Cs&1\end{pmatrix}}$

The transmission matrix for the entire network T is simply the matrix multiplication of the transmission matrices for the two network elements:

$\mathbf {T} =\mathbf {T} _{2}\cdot \mathbf {T} _{1}$

$={\begin{pmatrix}1&0\\-Cs&1\end{pmatrix}}\cdot {\begin{pmatrix}1&-R\\0&1\end{pmatrix}}$

$={\begin{pmatrix}1&-R\\-Cs&1+RCs\end{pmatrix}}$

Thus:

${\begin{pmatrix}V_{2}\\I_{2}\end{pmatrix}}={\begin{pmatrix}1&-R\\-Cs&1+RCs\end{pmatrix}}{\begin{pmatrix}V_{1}\\I_{1}\end{pmatrix}}$

Notes regarding definition of transmission parameters

1) It should be noted that all these examples are specific to the definition of transmission parameters given here. Other definitions exist in the literature, such as:

${V_{1} \choose I_{1}}={\begin{pmatrix}A&B\\C&D\end{pmatrix}}{V_{2} \choose -I_{2}}$

2) The format used above for cascading (ABCD) examples cause the "components" to be used backwards compared to standard electronics schematic conventions. This can be fixed by taking the transpose of the above formulas, or by making the $V_{1},I_{1}$ the left hand side (dependent variables). Another advantage of the $V_{1},I_{1}$ form is that the output can be terminated (via a transfer matrix representation of the load) and then $I_{2}$ can be set to zero; allowing the voltage transfer function, 1/A to be read directly.

3) In all cases the ABCD matrix terms and current definitions should allow cascading. 4

Networks with more than 2 ports

While two port networks are very common (e.g. amplifiers and filters), other electrical networks such as directional couplers and isolators have more than 2 ports. The following representations can be extended to networks with an arbitrary number of ports:

• Admittance (Y) Parameters
• Impedance (Z) Parameters
• Scattering (S) Parameters

They are extended by adding appropriate terms to the matrix representing the other ports. So 3 port impedance parameters result in the following relationship:

$\left[{\begin{array}{c}V_{1}\\V_{2}\\V_{3}\end{array}}\right]=\left[{\begin{array}{ccc}Z_{11}&Z_{12}&Z_{13}\\Z_{21}&Z_{22}&Z_{23}\\Z_{31}&Z_{32}&Z_{33}\end{array}}\right]\left[{\begin{array}{c}I_{1}\\I_{2}\\I_{3}\end{array}}\right]$.

It should be noted that the following representations cannot be extended to more than two ports:

• Hybrid (h) parameters
• Inverse hybrid (g) parameters
• Transmission (ABCD) parameters
• Scattering Transmission (T) parameters