Two-port networks
Introduit| English | Sens |
|---|---|
| two-port network | A circuit treated as a box with one input port and one output port, described by its parameters. |
| parameter set | parameter set |
| cascade | en cascade |
| reciprocal/rɪˈsɪprəkl/ | reciprocal |
A box with two pairs of wires
- An amplifier module, a matching network and a length of cable each have an input pair and an output pair of wires. The designer joining them never needs the parts inside.
- What matters is how the box behaves at its terminals. Four numbers capture it, and chaining boxes becomes matrix multiplication.
Ports and conventions
- A two-port network 二端口网络 has port 1 ($V_{1}$, $I_{1}$) and port 2 ($V_{2}$, $I_{2}$). The outside world sees only these four quantities.
- A linear two-port links them with two equations. Each parameter set 参数组 picks two of the four as inputs and gives the other two.
To use a two-port in a design, you need to know the components inside it.
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The parameter set describes everything the outside world can see at the two ports. Two boxes with the same parameters behave identically in any circuit.
z-parameters: measured with ports open
- z-parameters give voltages from currents, both currents flowing into the ports: $V_{1} = z_{11}I_{1} + z_{12}I_{2}$ and $V_{2} = z_{21}I_{1} + z_{22}I_{2}$.
- Each is one open-circuit measurement. Open port 2 ($I_{2} = 0$) and $z_{11} = \dfrac{V_{1}}{I_{1}}$, $z_{21} = \dfrac{V_{2}}{I_{1}}$.
With port 2 open (I2 = 0), the input voltage is
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An open output removes the $I_{2}$ term, leaving $V_{1} = z_{11}I_{1}$: one parameter from one open-circuit test.
A T-network has 5 Ω in each series arm and 20 Ω in the shunt arm. What is z11, in ohms?
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With port 2 open, the output series arm carries no current, so port 1 sees $5 + 20 = 25\,\Omega$. By symmetry $z_{22} = 25\,\Omega$, and $z_{12} = z_{21} = 20\,\Omega$, the shared shunt arm.
ABCD parameters chain together
- ABCD parameters give the input from the output: $V_{1} = AV_{2} + B(-I_{2})$ and $I_{1} = CV_{2} + D(-I_{2})$. Here $-I_{2}$ is the current out of port 2, into the next stage.
- A cascade 级联 feeds one block's output into the next, so its ABCD matrix is the product of theirs, taken in order.
Cascading shunt-Y then series-Z gives the ABCD matrix
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The product $(1, 0; Y, 1)(1, Z; 0, 1) = (1, Z; Y, 1 + YZ)$. Order matters: series-Z first would give $(1 + YZ, Z; Y, 1)$.
Worked example: an L-section
- Shunt $Y$ then series $Z$: $\begin{pmatrix} 1 & 0 \\ Y & 1 \end{pmatrix}\begin{pmatrix} 1 & Z \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & Z \\ Y & 1 + YZ \end{pmatrix}$.
- With $Z = j10\,\Omega$ and $Y = -j0.1\,\text{S}$: $1 + YZ = 1 + 1 = 2$, so $A = 1$, $B = j10$, $C = -j0.1$, $D = 2$.
Check the determinant: $AD - BC = 2 - (j10)(-j0.1) = 2 - 1 = 1$. Every network of R, L and C (a reciprocal 互易 network) has $AD - BC = 1$.
A cascade of reciprocal sections, each with AD − BC = 1, has overall AD − BC equal to —
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$\det(M_{1}M_{2}) = \det M_{1} \cdot \det M_{2}$, so each section contributes a factor of 1. Any cascade of reciprocal sections keeps $AD - BC = 1$, a quick check on every calculation.
Worked example: back to z-parameters
- Convert with $z_{11} = \dfrac{A}{C}$, $z_{12} = \dfrac{AD - BC}{C}$, $z_{21} = \dfrac{1}{C}$, $z_{22} = \dfrac{D}{C}$. With $C = -j0.1$: $z_{11} = z_{12} = z_{21} = j10\,\Omega$ and $z_{22} = j20\,\Omega$.
- $z_{12} = z_{21}$ is reciprocity, visible in the numbers. With port 2 open, the series arm carries no current, so the input sees only the shunt arm: $\dfrac{1}{Y} = j10\,\Omega$.
For a reciprocal two-port in z-parameters, always
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Reciprocity is the off-diagonal symmetry $z_{12} = z_{21}$; the L-section has both equal to $j10\,\Omega$. $z_{11} = z_{22}$ is a different property, physical symmetry, which the L-section lacks.
Three common slips
- The sign of $I_{2}$. ABCD uses the current out of port 2; z-parameters use the current in.
- Multiplying in the wrong order. Matrix products do not commute: the first block's matrix goes on the left.
- Assuming reciprocity with transistors inside. Dependent sources break $z_{12} = z_{21}$ and $AD - BC = 1$.
Swapping the order of two different sections in a cascade leaves the overall ABCD matrix unchanged.
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Matrix multiplication does not commute: shunt-then-series gives $(1, Z; Y, 1 + YZ)$, series-then-shunt gives $(1 + YZ, Z; Y, 1)$.
Recap
- A two-port is four port quantities linked by a parameter set.
- z-parameters come from open-circuit tests; ABCD matrices multiply along a cascade.
- Reciprocal networks: $AD - BC = 1$ and $z_{12} = z_{21}$.