Linear Circuit Analysis


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 Fig. 1, 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.

R R1 R2 R3 R4 R5 R Reff R3 R4 R5 R Reff R3 R4 R5
Fig. 1. When combining multiple resistors that are connected in series, we keep one resistor and short-circuit the others. The 3 diagrams are equivalent with each other.
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 Fig. 2, 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 $R_4$.

R R1 R2 R3 R4 R5 R R1 R2 Reff R R1 R2 Reff R R1 R2 Reff
Fig. 2. When combining multiple resistors that are connected in parallel, we keep one resistor and remove the others. The 4 diagrams are equivalent with each other.

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 Fig. 3, 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.

L L1 L2 L3 L4 L5 R Leff L3 L4 L5 R Leff L3 L4 L5
Fig. 3. When combining multiple inductors that are connected in series, we keep one inductor and short-circuit the others. The 3 diagrams are equivalent with each other.
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 Fig. 4, 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 $L_4$.

L L1 L2 L3 L4 L5 R L1 L2 Leff R L1 L2 Leff R L1 L2 Leff
Fig. 4. When combining multiple inductors that are connected in parallel, we keep one inductor and remove the others. The 4 diagrams are equivalent with each other.

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 Fig. 5, 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.

C C1 C2 C3 C4 C5 R Ceff C3 C4 C5 R Ceff C3 C4 C5
Fig. 5. When combining multiple capacitors that are connected in series, we keep one capacitor and short-circuit the others. The 3 diagrams are equivalent with each other.
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 Fig. 6, 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 $C_4$.

C C1 C2 C3 C4 C5 R C1 C2 Ceff R C1 C2 Ceff R C1 C2 Ceff
Fig. 6. When combining multiple capacitors that are connected in parallel, we keep one capacitor and remove the others. The 4 diagrams are equivalent with each other.