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
Impedance Combinations
Series
If two or more impedances $Z_1$, $Z_2$, ... $Z_n$ are connected in series they are equivalent with a impedance with $$Z_{eff}=Z_1+Z_2+...+Z_n$$ That means we can replace one of the $n$ impedances with an equivalent impedance with resistance $Z_{eff}$ and replace the other impedances with short-circuits (wires).
For instance, considering the circuit in
, impedances $Z_1$ and $Z_2$ are connected in series. Therefore, we can keep one the impedances, say $Z_1$, replace its value with $Z_{eff}=Z_1+Z_2$, and replace impedance $Z_2$ with a wire.Parallel
If two or more impedances $Z_1$, $Z_2$, ... $Z_n$ are connected in parallel they are equivalent with a impedance with $$Z_{eff}=\frac{1}{\frac{1}{Z_1}+\frac{1}{Z_2}+...+\frac{1}{Z_n}}$$ That means we can replace one of the $n$ impedances with an equivalent impedance with resistance $Z_{eff}$ and remove all the other impedances. When we have only two impedances connected in parallel, we can replace them with a single impedances with resistance $$Z_{eff}=\frac{Z_1 Z_2}{Z_1+Z_2}$$
For instance, considering the circuit in
, impedances $Z_3$, $Z_4$, and $Z_5$ are connected in parallel. Therefore, we can keep one the impedances, say $Z_3$, replace its value with $Z_{eff}=\frac{1}{\frac{1}{Z_3}+\frac{1}{Z_4}+\frac{1}{Z_5}}$, and remove impedance $Z_4$ from the circuit. Similarly, we could keep $Z_4$ and remove $Z_3$ and $Z_5$, or keeep $Z_5$ and remove $Z_3$ and $Z_4$.