Linear Circuit Analysis


Current Source Combinations

Series

Current sources should never be combined in series. What would happen if we connect two current sources with different currents in series?

Parallel

If two or more current sources $I_1$, $I_2$, ... $I_n$ are connected in parallel they can be replaced with a single current source with $$I_{eff}=\pm I_1 \pm I_2\pm...\pm I_n$$ where the terms in the right-hand side are taken with $+$ sign if the corresponding current source $I_i$ points towards the same node as $I_{eff}$ and with $-$ sign if $I_i$ points to the other node. Since all the current sources are connected in parallel, when we replace $I_i$ with $I_{eff}$, the other current sources are removed (i.e. replaced with open circuits).

For instance, consider the circuit in Fig. 1, in which current sources $I_1$, $I_2$, and $I_3$ are connected in parallel. In this circuit, we can keep one current source, say $I_1$, replace its value with $$\begin{equation}I_{1,eff}=I_1-I_2+I_3\end{equation}$$ and remove current sources $I_2$ and $I_3$ from the circuit. Notice that $I_1$ and $I_3$ appear with positive sign in the above equation because both $I_1$ and $I_3$ point towards node $v_3$, just like $I_{1,eff}$. $I_2$ appears with negative sign because it points towards node $v_4$ while $I_{1,eff}$ points towards node $v_3$. If we kept current source $I_2$, we had to replace its value with $$\begin{equation}I_{2,eff}=-I_1+I_2-I_3\end{equation}$$ $I_1$ and $I_3$ appear with positive sign in the above equation because $I_2$ points towards node $v_4$, just like $I_{2,eff}$. $I_1$ and $I_3$ appear with negative sign because they point towards node $v_3$ while $I_{2,eff}$ points towards node $v_4$. Also, if we kept current source $I_3$, we had to replace its value with $$\begin{equation}I_{3,eff}=I_1-I_2+I_3\end{equation}$$

I R1 R2 I1 I2 I3 v1 v2 v4 v3 R R1 R2 I1,eff v3 v4 v2 v1 R R1 R2 I2,eff v3 v4 v2 v1 R R1 R2 I3,eff v3 v4 v2 v1
Fig. 1. When combining multiple current sources that are connected in parallel, we keep one current source and remove the others. The 4 diagrams shown in this figure are equivalent with each other.
See also