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
Resistor Combinations
Series
If two or more resistors $R_1$, $R_2$, ... $R_n$ are connected in series they are equivalent with a resistor with $$R_{eff}=R_1+R_2+...+R_n$$ That means we can replace one of the $n$ resistors with an equivalent resistor with resistance $R_{eff}$ and replace the other resistors with short-circuits (wires).
For instance, considering the circuit in
, resistors $R_1$ and $R_2$ are connected in series. Therefore, we can keep one the resistors, say $R_1$, replace its value with $R_{eff}=R_1+R_2$, and replace resistor $R_2$ with a wire.Parallel
If two or more resistors $R_1$, $R_2$, ... $R_n$ are connected in parallel they are equivalent with a resistor with $$R_{eff}=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}}$$ That means we can replace one of the $n$ resistors with an equivalent resistor with resistance $R_{eff}$ and remove all the other resistors. When we have only two resistors connected in parallel, we can replace them with a single resistors with resistance $$R_{eff}=\frac{R_1 R_2}{R_1+R_2}$$
For instance, considering the circuit in
, resistors $R_3$, $R_4$, and $R_5$ are connected in parallel. Therefore, we can keep one the resistors, say $R_3$, replace its value with $R_{eff}=\frac{1}{\frac{1}{R_3}+\frac{1}{R_4}+\frac{1}{R_5}}$, and remove resistor $R_4$ from the circuit. Similarly, we could keep $R_4$ and remove $R_3$ and $R_5$, or keep $R_5$ and remove $R_3$ and $R_4$.Inductor Combinations
Series
If two or more inductors $L_1$, $L_2$, ... $L_n$ are connected in series they are equivalent with a inductor with $$L_{eff}=L_1+L_2+...+L_n$$ That means we can replace one of the $n$ inductors with an equivalent inductor with inductance $L_{eff}$ and replace the other inductors with short-circuits (wires).
For instance, considering the circuit in
, inductors $L_1$ and $L_2$ are connected in series. Therefore, we can keep one the inductors, say $L_1$, replace its value with $L_{eff}=L_1+L_2$, and replace inductor $L_2$ with a wire.Parallel
If two or more inductors $L_1$, $L_2$, ... $L_n$ are connected in parallel they are equivalent with a inductor with $$L_{eff}=\frac{1}{\frac{1}{L_1}+\frac{1}{L_2}+...+\frac{1}{L_n}}$$ That means we can replace one of the $n$ inductors with an equivalent inductor with inductance $L_{eff}$ and remove all the other inductors. When we have only two inductors connected in parallel, we can replace them with a single inductors with inductance $$L_{eff}=\frac{L_1 L_2}{L_1+L_2}$$
For instance, considering the circuit in
, inductors $L_3$, $L_4$, and $L_5$ are connected in parallel. Therefore, we can keep one the inductors, say $L_3$, replace its value with $L_{eff}=\frac{1}{\frac{1}{L_3}+\frac{1}{L_4}+\frac{1}{L_5}}$, and remove inductor $L_4$ from the circuit. Similarly, we could keep $L_4$ and remove $L_3$ and $L_5$, or keep $L_5$ and remove $L_3$ and $L_4$.Capacitor Combinations
Series
If two or more capacitors $C_1$, $C_2$, ... $C_n$ are connected in series they are equivalent with a capacitor with $$C_{eff}=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}+...+\frac{1}{C_n}}$$ That means we can replace one of the $n$ capacitors with an equivalent capacitor with capacitance $C_{eff}$ and replace the other capacitors with short-circuits (wires). When we have only two capacitors connected in parallel, we can replace them with a single capacitors with capacitance $$C_{eff}=\frac{C_1 C_2}{C_1+C_2}$$
For instance, considering the circuit in
, capacitors $C_1$ and $C_2$ are connected in series. Therefore, we can keep one the capacitors, say $C_1$, replace its value with $C_{eff}=\frac{C_1 C_2}{C_1+C_2}$, and replace capacitor $C_2$ with a wire.Parallel
If two or more capacitors $C_1$, $C_2$, ... $C_n$ are connected in parallel they are equivalent with a capacitor with $$C_{eff}=C_1+C_2+...+C_n$$ That means we can replace one of the $n$ capacitors with an equivalent capacitor with capacitance $C_{eff}$ and remove all the other capacitors.
For instance, considering the circuit in
, capacitors $C_3$, $C_4$, and $C_5$ are connected in parallel. Therefore, we can keep one the capacitors, say $C_3$, replace its value with $C_{eff}=C_3+C_4+C_5$, and remove capacitors $C_4$ and $C_5$ from the circuit. Similarly, we could keep $C_4$ and remove $C_3$ and $C_5$, or keep $C_5$ and remove $C_3$ and $C_4$.-
Resistance simplifications (analytical)
2000
2001
2002
-
Resistance simplifications (numerical)
2500
2501
2502
2503
-
Inductance simplifications (analytical)
2100
2101
2102
-
Inductance simplifications (numerical)
2600
2601
2602
2603
-
Capacitance simplifications (analytical)
2200
2201
2202
-
Capacitance simplifications (numerical)
2700
2701
2702
2703