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Superposition Theorem: Decomposition and Synthesis of Multi-Source Linear Circuits

Linear circuit with voltage source \(U_s\) and current source \(I_s\) | Individual contributions | Superposition verification \(U_{out} = U_{out1} + U_{out2}\)

Circuit Parameters

24 V
5 V50 V
0.10 A
0.01 A0.50 A
100 Ω
10 Ω1000 Ω
200 Ω
10 Ω1000 Ω

Analysis View

Output Voltage \(U_{out}\) (Voltage between a and b)

\(U_{out1}\) (\(U_s\) contribution) 0.00 V
\(U_{out2}\) (\(I_s\) contribution) 0.00 V
\(U_{out}\) (Total) 0.00 V
\(U_out\)(total) = \(U_out1\) + \(U_out2\) = 0.00 V
\(U_s\) contribution \(U_{out1}\)
\(I_s\) contribution \(U_{out2}\)
Total \(U_{out}\)
Superposition Theorem Verification: Total Response = Sum of Individual Independent Source Contributions

\(U_s\) acting alone

U_s acting alone

\(I_s\) acting alone

I_s acting alone

Complete Circuit (Superposition)

Complete circuit

Output voltage \(U_{out}\) (Voltage between a and b)

Theoretical Basis

In a linear circuit, the response (voltage or current) at any point equals the sum of the responses produced by each independent source acting alone.
When \(U_s\) acts alone, \(I_s\) is open-circuited; when \(I_s\) acts alone, \(U_s\) is short-circuited.
\(U_{out1} = U_s \cdot \frac{R_2}{R1 + R_2}\)
\(U_{out2} = I_s \cdot (R_1 \parallel R_2)\)

Superposition Properties

Zeroing a voltage source -> replace with a short circuit (wire).
Zeroing a current source -> replace with an open circuit (disconnection).
The superposition theorem applies only to linear circuits (linear resistors, linear dependent sources, independent sources).
Power cannot be directly superposed (since power is a quadratic function of voltage/current).