Two-Port Networks

Taking the currents as independent gives

Each parameter is isolated by setting one current to zero — that is, by open-circuiting the corresponding port. Hence the alternative name open-circuit parameters.

All four have units of ohms. Note the index convention: the first subscript identifies where the voltage is measured, the second where the current is applied.

A caution. The \(z\) parameters do not exist for every network. A network whose \(z\) matrix is singular has no \(y\) parameters, and vice versa — an ideal transformer, for instance, has neither. This is one reason several parameter sets are needed.

Taking the voltages as independent gives the dual set:

Each parameter is found by setting a voltage to zero — by short-circuiting a port — hence short-circuit parameters:

All four are in siemens. The \(y\) matrix is the inverse of the \(z\) matrix — not the reciprocal of each element, a distinction Section 19.6 makes precise.

Where \(z\) parameters suit a T network, \(y\) parameters suit a Pi network, whose parameters can be written down by inspection:

where \(Y_{2}\) is the series branch and \(Y_{1}\), \(Y_{3}\) the two shunts. Note the minus sign on the transfer terms: it arises because a positive \(\mathbf{V}_{2}\) drives current out of port 1 through the shared branch. Off-diagonal \(y\) parameters are negative for any passive network.

Neither \(z\) nor \(y\) parameters suit a transistor. A transistor's input behaves like a low impedance and its output like a high one, so open-circuiting the input or short-circuiting the output is awkward in practice. The hybrid parameters mix the two descriptions to match:

The four have different units — which is exactly why they are called hybrid:

The subscript letters denote input, reverse, forward, output, and the trailing \(e\) denotes the common-emitter configuration. \(h_{fe}\) is the current gain \(\beta\) familiar from transistor work — its appearance here as a two-port parameter is the reason the whole scheme is standard in electronics.

The inverse hybrid or \(g\) parameters swap the roles, expressing \(\mathbf{I}_{1}\) and \(\mathbf{V}_{2}\) in terms of \(\mathbf{V}_{1}\) and \(\mathbf{I}_{2}\). The \(g\) matrix is the inverse of the \(h\) matrix.

The transmission or ABCD parameters express the input quantities in terms of the output ones — the natural direction for a signal travelling through a chain of networks:

Note the minus signs. They arise because the convention here takes \(\mathbf{I}_{2}\) as leaving the network, matching the physical picture of current flowing onward into the next stage. This is the one place in the chapter where the entering-current convention is set aside, and it must be remembered when converting.

These parameters are indispensable for transmission lines and for any cascade, because — as Section 19.8 shows — cascaded networks multiply their \(ABCD\) matrices. No other parameter set has that property.

Since all four sets describe the same network, any one determines the rest. The derivation is straightforward algebra: solve one pair of equations for the desired variables and read off the coefficients.

The \(z\)-to-\(y\) relationship is the cleanest, because the two matrices are simply inverses:

with \(\Delta_{z} = z_{11}z_{22}-z_{12}z_{21}\). It follows immediately that \(\Delta_{y} = 1/\Delta_{z}\).

Note carefully: \(y_{11} \neq 1/z_{11}\). The matrices are inverses, not the individual elements. This is the single most common error in the whole chapter.

There is no need to memorise this table. Two things are worth remembering instead: \(\left[y\right] = \left[z\right]^{-1}\), and that any conversion can be re-derived in a minute by solving the defining equations. The determinants \(\Delta_{z}\), \(\Delta_{y}\) and \(\Delta_{h}\) appear throughout, so compute them first.

Two structural properties reduce the four parameters to three or two, and both have simple physical meanings.

Reciprocity. A network is reciprocal if interchanging an ideal voltage source and an ideal ammeter between the two ports leaves the ammeter reading unchanged. Every network built solely from resistors, inductors, capacitors and mutual inductance is reciprocal; only networks containing dependent sources — that is, active devices — can fail to be.

The condition takes a different algebraic form in each parameter set, which is a persistent source of confusion:

Symmetry. A network is symmetric if its two ports may be interchanged without any observable difference — electrically, the network looks the same from either end. A symmetric network is necessarily reciprocal, but the converse does not hold: the T network of Example 19.1 is reciprocal (\(z_{12} = z_{21} = 40\)) yet not symmetric (\(z_{11} = 60 \neq 70 = z_{22}\)), because its series arms differ.

A symmetric network has only two independent parameters instead of four, and its \(ABCD\) matrix satisfies both \(A = D\) and \(AD-BC = 1\). Symmetric sections are the building blocks of classical filter design, precisely because identical sections can be cascaded without impedance mismatches at the joins.

The practical power of the two-port description appears when networks are combined. Each connection type has a parameter set for which the combination rule is trivial — and that is why several sets exist.

Series connection. Both inputs are in series and both outputs are in series, so the currents are shared and the voltages add. Since \(z\) parameters give voltage in terms of current, the matrices simply add:

Parallel connection. The voltages are shared and the currents add, so by the dual argument:

Cascade connection. The output of the first network becomes the input of the second, which is precisely the relation \(ABCD\) parameters express. The matrices multiply:

The order matters, since matrix multiplication does not commute. \(\left[T_{1}\right]\) is written first because the signal encounters \(N_{1}\) first. Reversing the order describes a physically different circuit.

A caution on series and parallel connections. The addition rules assume the port condition still holds after connection — that the current entering each port's upper terminal still equals that leaving its lower one. Interconnecting two-ports can violate this, in which case the rules fail. The Brune test checks for it, and cascade connection is free of the difficulty entirely, which is one more reason the \(ABCD\) set is the most used in practice.

The dominant application of two-port theory is the small-signal analysis of transistor amplifiers. A manufacturer publishes the four \(h\) parameters, and every important amplifier property follows from them without any knowledge of the device's internal physics.

With a load \(R_{L}\) connected at the output, the standard results are

where \(\Delta_{h} = h_{ie}h_{oe}-h_{re}h_{fe}\). The negative voltage gain signals the phase inversion characteristic of a common-emitter stage.

• A port is a terminal pair satisfying the port condition; a two-port is fully described by four parameters, whatever it contains.

• \(z\) parameters are found by open-circuit tests, \(y\) parameters by short-circuit tests.

• \(h\) parameters mix the two and suit transistors, where \(h_{fe}\) is the familiar current gain.

• \(ABCD\) parameters express input in terms of output, with \(\mathbf{I}_{2}\) taken as leaving the network.

• \(\left[y\right] = \left[z\right]^{-1}\) — the matrices are inverses, not the individual elements.

• Reciprocity holds for any network of R, L, C and mutual inductance, and is expressed differently in each parameter set.

• Symmetry means the ports may be interchanged; it implies reciprocity but not conversely.

• Series connections add \(z\), parallel connections add \(y\), cascades multiply \(T\) — and cascade order matters.

• Transistor gains and impedances follow from the four \(h\) parameters alone.

• The condition \(AD-BC = 1\) is the most convenient arithmetic check in the whole chapter.

• Taking \(y_{11} = 1/z_{11}\). The matrices are inverses; the elements are not reciprocals. In Example 19.5 the two differ by a factor of 1.6.

• Getting the current convention wrong. Both currents enter the network for \(z\), \(y\) and \(h\) parameters — but \(\mathbf{I}_{2}\) is taken as leaving for \(ABCD\), which is where the minus signs come from.

• Using the wrong reciprocity condition. It is \(z_{12} = z_{21}\) but \(h_{12} = -h_{21}\) and \(AD-BC = 1\). Applying one set's condition to another is a frequent error.

• Confusing reciprocity with symmetry. Every symmetric network is reciprocal; most reciprocal networks are not symmetric. Example 19.1's T network is the case in point.

• Reversing the cascade order. \(\left[T_{1}\right]\left[T_{2}\right]\) and \(\left[T_{2}\right]\left[T_{1}\right]\) describe different circuits.

• Adding \(z\) matrices for a parallel connection. Series adds \(z\), parallel adds \(y\). Only cascade uses \(T\).

• Forgetting the minus sign in a Pi network's \(y_{12}\). Off-diagonal \(y\) parameters are negative for passive networks.

• Assuming every network has every parameter set. An ideal transformer has neither \(z\) nor \(y\) parameters, though it has \(ABCD\) parameters.

• Including independent sources inside the two-port. The definitions assume none; dependent sources are permitted and are what make active two-ports possible.

• Skipping the \(AD-BC = 1\) check. It costs one line and catches nearly every arithmetic slip.

Work these before opening the answers. Verify reciprocity wherever it applies.

1. P19.1 Find the \(z\) parameters of a T network with series arms of 10 \(\Omega\) and 10 \(\Omega\) and a shunt of 25 \(\Omega\). Is it symmetric?

   Show answer

   \[z_{11} = 10+25 = 35~\Omega, \quad z_{12} = z_{21} = 25~\Omega, \quad z_{22} = 10+25 = 35~\Omega\]

   Since \(z_{11} = z_{22}\) the network is symmetric, and being made of resistors it is also reciprocal. Only two parameters are independent.

2. P19.2 A two-port has \(\left[z\right] = \begin{bmatrix}10&6\\6&8\end{bmatrix}~\Omega\). Find its \(y\) parameters.

   Show answer

   \[\Delta_{z} = 80-36 = 44\]

   \[y_{11} = \frac{8}{44} = 0.1818, \quad y_{12} = y_{21} = \frac{-6}{44} = -0.1364, \quad y_{22} = \frac{10}{44} = 0.2273~\mathrm{S}\]

   Check: \(\Delta_{y} = (0.1818)(0.2273)-(0.1364)^{2} = 0.02273 = 1/44\) \(\checkmark\)

3. P19.3 Find the \(h\) parameters of the network in P19.2.

   Show answer

   \[h_{11} = \frac{\Delta_{z}}{z_{22}} = \frac{44}{8} = 5.500~\Omega, \qquad h_{12} = \frac{6}{8} = 0.7500\]

   \[h_{21} = -\frac{6}{8} = -0.7500, \qquad h_{22} = \frac{1}{8} = 0.1250~\mathrm{S}\]

   Reciprocity: \(h_{12} = -h_{21}\) \(\checkmark\)

4. P19.4 Find the \(ABCD\) parameters of the network in P19.2 and verify reciprocity.

   Show answer

   \[A = \frac{10}{6} = 1.667, \quad B = \frac{44}{6} = 7.333~\Omega, \quad C = \frac{1}{6} = 0.1667~\mathrm{S}, \quad D = \frac{8}{6} = 1.333\]

   \[AD-BC = \left(1.667\right)\left(1.333\right)-\left(7.333\right)\left(0.1667\right) = 2.222-1.222 = 1.000 \;\checkmark\]

5. P19.5 Write the \(ABCD\) matrix of a single series impedance \(Z\), and of a single shunt admittance \(Y\).

   Show answer

   \[\text{Series } Z: \begin{bmatrix}1&Z\\0&1\end{bmatrix}, \qquad \text{Shunt } Y: \begin{bmatrix}1&0\\Y&1\end{bmatrix}\]

   Both have determinant 1, so both are reciprocal, as any passive element must be.

6. P19.6 Use P19.5 to find the \(ABCD\) matrix of an L-section: a 4 \(\Omega\) series resistor followed by a 0.5 S shunt.

   Show answer

   \[\left[T\right] = \begin{bmatrix}1&4\\0&1\end{bmatrix}\begin{bmatrix}1&0\\0.5&1\end{bmatrix} = \begin{bmatrix}1+2&4\\0.5&1\end{bmatrix} = \begin{bmatrix}3&4\\0.5&1\end{bmatrix}\]

   Check: \(AD-BC = 3-2 = 1\) \(\checkmark\)

7. P19.7 Two two-ports with \(\left[z_{1}\right] = \begin{bmatrix}4&2\\2&5\end{bmatrix}\) and \(\left[z_{2}\right] = \begin{bmatrix}6&3\\3&7\end{bmatrix}\) are connected in series. Find the overall \(z\) matrix.

   Show answer

   \[\left[z\right] = \begin{bmatrix}10&5\\5&12\end{bmatrix}~\Omega\]

   Still reciprocal, since \(z_{12} = z_{21}\) for both components and addition preserves it.

8. P19.8 A transistor has \(h_{ie} = 2~\mathrm{k}\Omega\), \(h_{re} = 0\), \(h_{fe} = 100\), \(h_{oe} = 20~\mu\mathrm{S}\), with \(R_{L} = 2~\mathrm{k}\Omega\). Find \(A_{v}\), \(A_{i}\) and \(Z_{\text{in}}\).

   Show answer

   With \(h_{re} = 0\), \(\Delta_{h} = h_{ie}h_{oe} = (2000)(20\times10^{-6}) = 0.04\).

   \[A_{v} = \frac{-\left(100\right)\left(2000\right)}{2000+\left(0.04\right)\left(2000\right)} = \frac{-200\,000}{2080} = -96.15\]

   \[A_{i} = \frac{100}{1+0.04} = 96.15, \qquad Z_{\text{in}} = h_{ie} = 2000~\Omega\]

   With no reverse feedback the input impedance is exactly \(h_{ie}\).

9. P19.9 A network has \(\left[h\right]\) with \(h_{12} = 0.4\) and \(h_{21} = 0.4\). Is it reciprocal? What does this imply?

   Show answer

   The reciprocity condition for \(h\) parameters is \(h_{12} = -h_{21}\). Here \(0.4 \neq -0.4\), so the network is not reciprocal. It must therefore contain a dependent source — it is an active network, and cannot be built from R, L, C and mutual inductance alone.

10. P19.10 Explain why the \(ABCD\) parameters of a cascade of \(n\) reciprocal networks always satisfy \(AD-BC = 1\).

    Show answer

    The overall matrix is the product \(\left[T_{1}\right]\left[T_{2}\right]\cdots\left[T_{n}\right]\), and the determinant of a product equals the product of the determinants. Each factor is reciprocal and so has determinant 1, giving \(1\times1\times\cdots\times1 = 1\). Physically: cascading passive networks cannot create an active one.