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.
See also