INTRODUCTION TO ELECTRONICS: The Superposition Theorem

Most of the circuits we’ve encountered thus far contain a single voltage ( V ) source that provides current ( I ) to the system. Suppose, however, that a current determination must be made for the following dual-voltage circuit type:

The presence of two voltage sources eliminates any series-parallel relationships that would exist between the circuit’s resistors ( R ). For this reason, an alternative circuit-analysis approach is used. We must first realize that both voltage sources give rise to the I₂ current across the R₂ resistor. The superposition theorem helps us deal with this conundrum via a “ division-of-labor “ approach: Each circuit loop is analyzed separately. By using techniques employed by Thevenin’s theorem, voltage sources can be shorted, thus leaving a more manageable circuit type in place:

The R₁ resistor is now in series with the R₂ and R₃ resistors, which are now parallel to one another:

R_T(S1) = R₁ + R₂∥R₃

Ohm’s law is now used to determine the total current ( I_T ) that leaves then eventually re-enters the source:

I_T(S1) = ( V_S1 / R_T(S1) )

The next step requires usage of the current-divider rule. The current across R₂ is a fraction of I_T, and it is directly proportional to R₃; if resistance across R₃ is high, current across R₃ will be low, and thus, current across R₂ will be high ( and vice-versa ): Furthermore, it is of crucial importance to realize that the current across R₂ is a current component created by V_s1 alone!!! The current contribution from V_s2 will be determined in a separate step that, fortunately for us, is a repeat of the aforementioned steps that have been taken:

I_2(S1) = [ R₃ / ( R₂ + R₃ ) ]( I_T(S1) )

We must now place a short across V_s1 and evaluate the circuit from the V_s2 vantage point:

From the vantage point of V_s2, the R₃ resistor is in series with the R₁ and R₂ resistors that are now parallel to one another:

R_T(S2) = R₃ + R₁∥R₂ 

I_T(S2) = ( V_S2 / R_T(S2) )

Next, the V_s2 current contribution is determined using the current-divider formula:

I_2(S2) = [ R₁ / ( R₁ + R₂ ) ]( I_T(S2) )

The I_2(S1) and I_2(S2) current components may now be added together to obtain I₂: