INTRODUCTION TO ELECTRONICS: The Superposition Theorem

Q: What is the total current ( I_T ) and voltage ( V₃ ) across resistor R₃?

A: In order to begin evaluating the circuit from the vantage point of V_s1, we place a short across V_s2 :

Within this circuit, negatively charged electrons move upward and across the R₃ resistor towards the positively charged source terminal. The authors of electronics textbooks randomly choose whether to use conventional vs. physical charge flow within circuit diagrams, so it is a good idea to be familiar with both designations. From the vantage point of V_s1, the R₁ resistor is in series with the R₂ and R₃ resistors situated parallel to one other:

R_T(S1) = R₁ + R₂∥R₃

R_T(S1) = R₁ + [ R₂R₃ / ( R₂ + R₃ ) ]

R_T(S1) = 1.0 kΩ + [ ( 1.0 kΩ )( 2.2 kΩ )  / ( 3.2 kΩ ) ]

R_T(S1) = 1.0 kΩ + 688 kΩ

R_T(S1) = 1.69 kΩ

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I_T(S1) = ( V_s1 / R_T(S1) )

I_T(S1) = ( 20 V / 1.69 kΩ )

I_T(S1) = 11.8 mA

The current across R₃ is a fraction of the total current upstream of it. We use the current divider formula to determine its value:

I_3(S1) = [ R₂ / ( R₂ + R₃ ) ]( I_T(S1) )

I_3(S1) = [ 1.0 kΩ / ( 3.2 kΩ ) ]( 11.8 mA )

I_3(S1) = ( 0.313 )( 11.8 mA )

I_3(S1) = 3.69 mA

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The same procedures are used to evaluate the circuit from the vantage point of V_s2 :

The current now moves in the opposite direction across the R₃ resistor. This must be taken into account when the final current tabulation is made. From the vantage point of V_s2, the resistance between the source terminals is provided by R₂ in series with resistors R₁ and R₃ situated parallel to one another:

R_T(S2) = R₂ + R₁∥R₃

R_T(S2) = R₂ + [ R₁R₃ / ( R₁ + R₃ ) ]

R_T(S2) = 1.0 kΩ + [ ( 1.0 kΩ )( 2.2 kΩ )  / ( 3.2 kΩ ) ]

R_T(S2) = 1.0 kΩ + 688 kΩ

R_T(S2) = 1.69 kΩ

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I_T(S2) = ( V_s2 / R_T(S2) )

I_T(S2) = ( 15 V / 1.69 kΩ )

I_T(S2) = 8.88 mA

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I_3(S2) = [ R₂ / ( R₂ + R₃ ) ]( I_T(S1) )

I_3(S2) = [ 1.0 kΩ / ( 3.2 kΩ ) ]( 8.88 mA )

I_3(S2) = ( 0.313 )( 8.88 mA )

I_3(S2) = 2.78 mA

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I_3(total) = I_3(S1) – I_3(S2) 

I_3(total) = 3.69 mA – 2.78 mA

I_3(total) = 0.91 mA = 910µA

Thus, the net current flows upward across the R₃ resistor.

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V_R3 = ( I_3(total) )( R₃ )

V_R3 = ( 910µA )( 2.2 kΩ )

V_R3 = ( 910 x 10^-6 A )( 2.2 kΩ )

V_R3 = 2 V