Two-port networks
Introduced| English | Meaning |
|---|---|
| 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 | Connecting stages one after another, so the output of one feeds the input of the next. |
| 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.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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 —
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$\det(M_{1}M_{2}) = \det M_{1} \cdot \det M_{2}$, 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
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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 $z_{12} = z_{21}$.