# Substitution Theorem

#### What is a Substitution Theorem?

The substitution theorem states the following:
If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.

#### How did substitution theorem works?

Consider the circuit in Fig. 1, in which the voltage across and current through the branch a-b are determined. Through the use of the substitution theorem, a number of equivalent a-b branches are shown in Fig. 2.
Note that for each equivalent, the terminal voltage and current are the same.
Fig. 1: Demonstrating the effect of the substitution theorem.
Fig. 2: Equivalent branches for the branch $a$-$b$ in Fig. 1.

#### How to apply substitution theorem?

Understand that this theorem cannot be used to solve networks with two or more sources that are not in series or parallel. For it to be applied, a potential difference or current value must be known or found using one of the techniques discussed earlier. One application of the theorem is shown in Fig. 3. Note that in the figure the known potential difference V was replaced by a voltage source, permitting the isolation of the portion of the network including $R_3$, $R_4$, and $R_5$. Recall that this was basically the approach used in the analysis of the ladder network as we worked our way back toward the terminal resistance $R_5$.
The current source equivalence of the above is shown in Fig. 4, where a known current is replaced by an ideal current source, permitting the isolation of $R_4$ and $R_5$.
Fig. 3: Demonstrating the effect of knowing a voltage at some point in a complex network.
Fig. 4: Demonstrating the effect of knowing a current at some point in a complex network.