Linear Circuit Analysis


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 Fig. 1, 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.

Z Z1 Z2 Z3 Z4 Z5 R Zeff Z3 Z4 Z5 R Zeff Z3 Z4 Z5
Fig. 1. When combining multiple impedances that are connected in series, we keep one impedance and short-circuit the others. The 3 diagrams are equivalent with each other.
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 Fig. 2, 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 $Z_4$.

Z Z1 Z2 Z3 Z4 Z5 R Z1 Z2 Zeff R Z1 Z2 Zeff R Z1 Z2 Zeff
Fig. 2. When combining multiple impedances that are connected in parallel, we keep one impedance and remove the others. The 4 diagrams are equivalent with each other.
See also