Linear Circuit Analysis
1. Introduction
2. Basic Concepts
- Currents and voltages
- Linear circuits
- Linear components
- Loops and nodes
- Series and parallel
- R, L & C combinations
- V & I combinations
- Power and energy
3. Simple Circuits
- Ohm's law
- Kirchhoff's current law
- Kirchhoff's voltage law
- Single loop circuits
- Single node-pair circuits
- Voltage division
- Current division
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
- Superposition
- Source transformation
- The $V_{test}/I_{test}$ method
- Norton equivalent
- Thévenin equivalent
- Max power transfer
6. AC Analysis
7. Operational Amplifiers
8. Laplace Transforms
9. Time-Dependent Circuits
- Introduction
- First-order transients
- Nodal analysis
- Mesh analysis
- Laplace transforms
- Additional techniques
10. Two-port networks
Introduction
A two-port network is an electric network with two ports, in which each port is connected to an external circuit (see Fig. 1). Because of Kirchhoff's current law, the current entering one terminal of each port must equal the current emerging from the other terminal of the same port (this requirement is known as the port condition). Notice that the port condition is not necessarily a restriction that we impose only on the architecture of the two-port network itself, but also on the configuration of the external circuits that can apply voltages and inject currents in the terminals of the network.
It is customary to label the voltages and currents of two-port networks as shown in Fig. 1. One of the ports (usually the port on the left) is usually called the input port
and is denoted with subscript 1, while the other port is the output port and is denoted with subscript 2.
An important result related to two-port networks is that currents $I_1$ and $I_2$ depend only on the potential difference at the input and output terminals (i.e. they depend only on $V_1$ and $V_2$ and not on the exact values of the potentials of each terminal). This results can be easily proved using mesh analysis and invoking the port condition. If addition, if the two-port network is linear (which means is made using linear components such as R, L, C, transformers, OpAmps, ideal dependent and independent sources, etc.), we can express the terminal currents as linear combination of the input and output voltages $$\begin{equation}I_1=y_{11}V_1+y_{12}V_2 + i_{10}\end{equation}$$ $$\begin{equation}I_1=y_{21}V_1+y_{22}V_2 + i_{20}\end{equation}$$ where $y_{11}$,...,y_{22}, $i_{10}$, and $i_{20}$ are constant coefficients.
If the two-port network does not contain any independent sources (but can still contains R, L, C and dependent sources), the terminal currents can be expressed as $$\begin{equation}I_1=y_{11}V_1+y_{12}V_2\end{equation}$$ $$\begin{equation}I_2=y_{21}V_1+y_{22}V_2\end{equation}$$ where $y_{ij}$, $i_{10}$ are now called the admittance parameters (or y-parameters) of the linear two-port. In this webbook we are interested only in linear two-ports that do not contain independent sources and in which the input and output terminals satisfy the above relationships. We will also assume that the previous system can be inverted.
Summary of two-port parameters
Any two-port can be completely described using the y-parameters introduced in the previous section. These parameters give terminal currents if the terminal voltages are given. In general, if we know any two quantities (say $I_1$ and $V_1$), the other two quantities (in this case $I_2$ and $V_2$) can be computed by inverting the above equations. Since we have 4 quantities ($I_1$, $I_2$, $V_1$ and $V_2$) and 2 quantities need to be specified and we can compute the other 2 quantities, we can have $\frac{4!}{2! 2!}=6$ cases in total, as shown in the Table 1. If you want to see how to compute one set of parameters to another set of parameters, check the Conversion formulas page.
| Name | Input quantities | Output quantities | Parameters | |
|---|---|---|---|---|
| 1 | Admittance parameters | $\begin{bmatrix}V_1\\V_2\end{bmatrix}$ | $\begin{bmatrix}I_1\\I_2\end{bmatrix} = \begin{bmatrix}y_{11} & y_{12}\\y_{21} & {y_22}\end{bmatrix} \begin{bmatrix}V_1\\V_2\end{bmatrix}$ | $y_{11}$, $y_{12}$, $y_{21}$, $y_{22}$ |
| 2 | Impedance parameters | $\begin{bmatrix}I_1\\I_2\end{bmatrix}$ | $\begin{bmatrix}V_1\\V_2\end{bmatrix} = \begin{bmatrix}z_{11} & z_{12}\\z_{21} & {z_22}\end{bmatrix} \begin{bmatrix}I_1\\I_2\end{bmatrix}$ | $z_{11}$, $z_{12}$, $z_{21}$, $z_{22}$ |
| 3 | Hybrid parameters | $\begin{bmatrix}V_1\\I_2\end{bmatrix}$ | $\begin{bmatrix}V_1\\I_2\end{bmatrix} = \begin{bmatrix}h_{11} & h_{12}\\h_{21} & {h_22}\end{bmatrix} \begin{bmatrix}I_1\\V_2\end{bmatrix}$ | $h_{11}$, $h_{12}$, $h_{21}$, $h_{22}$ |
| 4 | Inverse hybrid parameters | $\begin{bmatrix}I_1\\V_2\end{bmatrix}$ | $\begin{bmatrix}I_1\\V_2\end{bmatrix} = \begin{bmatrix}g_{11} & g_{12}\\g_{21} & {g_12}\end{bmatrix} \begin{bmatrix}V_1\\I_2\end{bmatrix}$ | $g_{11}$, $g_{12}$, $g_{21}$, $g_{22}$ |
| 5 | Transmission parameters | $\begin{bmatrix}I_2\\V_2\end{bmatrix}$ | $\begin{bmatrix}V_1\\I_1\end{bmatrix} = \begin{bmatrix}A & B\\C & D\end{bmatrix} \begin{bmatrix}V_2\\-I_2\end{bmatrix}$ | $A$, $B$, $C$, $D$ or $t_{11}$, $t_{12}$, $t_{21}$, $t_{22}$ |
| 6 | Inverse transmission parameters or A'B'C'D' parameters |
$\begin{bmatrix}I_1\\V_1\end{bmatrix}$ | $\begin{bmatrix}V_2\\I_2\end{bmatrix} = \begin{bmatrix}A'' & B'\\C'' & D'\end{bmatrix} \begin{bmatrix}V_1\\-I_1\end{bmatrix}$ | $A'$, $B'$, $C'$, $D'$ or $t'_{11}$, $t'_{12}$, $t'_{21}$, $t'_{22}$ |
Definitions
Table 2 summarizes each of the 6 parameters introduced in the previous section.
| Admittance parameters | Impedance parameters | Hybrid parameters | Inverse hybrid parameters | Transmission parameters | Inverse transmission parameters |
|---|---|---|---|---|---|
| $$y_{11}=\left.{\frac{I_1}{V_1}} \right|_{V_2=0}$$ $$y_{12}=\left.{\frac{I_1}{V_2}} \right|_{V_1=0}$$ $$y_{21}=\left.{\frac{I_2}{V_1}} \right|_{V_2=0}$$ $$y_{22}=\left.{\frac{I_2}{V_2}} \right|_{V_1=0}$$ | $$z_{11}=\left.{\frac{V_1}{I_1}} \right|_{I_2=0}$$ $$z_{12}=\left.{\frac{V_1}{I_2}} \right|_{I_1=0}$$ $$z_{21}=\left.{\frac{V_2}{I_1}} \right|_{I_2=0}$$ $$z_{22}=\left.{\frac{V_2}{I_2}} \right|_{I_1=0}$$ | $$h_{11}=\left.{\frac{V_1}{I_1}} \right|_{V_2=0}$$ $$h_{12}=\left.{\frac{V_1}{V_2}} \right|_{I_1=0}$$ $$h_{21}=\left.{\frac{I_2}{I_1}} \right|_{V_2=0}$$ $$h_{22}=\left.{\frac{I_2}{V_2}} \right|_{I_1=0}$$ | $$g_{11}=\left.{\frac{I_1}{V_1}} \right|_{I_2=0}$$ $$g_{12}=\left.{\frac{I_1}{I_2}} \right|_{V_1=0}$$ $$g_{21}=\left.{\frac{V_2}{V_1}} \right|_{I_2=0}$$ $$g_{22}=\left.{\frac{V_2}{I_2}} \right|_{V_1=0}$$ | $$A=\left.{\frac{V_1}{V_2}} \right|_{I_2=0}$$ $$B=-\left.{\frac{V_1}{I_2}} \right|_{V_2=0}$$ $$C=\left.{\frac{I_1}{V_2}} \right|_{I_2=0}$$ $$D=-\left.{\frac{I_1}{I_2}} \right|_{V_2=0}$$ | $$A'=\left.{\frac{V_2}{V_1}} \right|_{I_1=0}$$ $$B'=-\left.{\frac{V_2}{I_1}} \right|_{V_1=0}$$ $$C'=\left.{\frac{I_2}{V_1}} \right|_{I_1=0}$$ $$D'=-\left.{\frac{I_2}{I_1}} \right|_{V_1=0}$$ |
Applications
Two-port linear networks have applications in filters, mathicng netwworks, amplifiers, transmission lines, antennas, power systems, small-signal analysis of electronic circuits (containing diodes, transistors, etc.), and communication systems. In these applications, two-port networks can used to isolate portions of larger circuits and the two-port network is regarded as a "black box" with its properties specified in matrix form.
Sample Solved Problems
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DC two-port networks (numerically and analytically)
2 loops and 3 R
2 loops and 3 R (analytical)
3 loops, 6 R)
5 loops, 3 loops, 6 R (analytical)
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AC two-port networks (numerically)
2 loops (1 R, 1 L, 1 C) or (2 R, 0/1 L, 1/0 C)
2 loops (2 R, 1 L, 1 C) or (3 R, 1 L, 1 C)
3 loops, 4 R, 1 L, 1 C
3 loops, 4 R, 1 L, 1 C, 1 dependent source
5 loops, 4 R, 1 L, 1 C, 1 dependent source