Superposition theorem


The superposition theorem for electrical circuits states that the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances.
To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:

1. Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential. i.e. V=0, internal impedance of ideal voltage source is ZERO (short circuit)).
2. Replacing all other independent current sources with an open circuit (thereby eliminating current. i.e. I=0, internal impedance of ideal current source is infinite (open circuit).

This procedure is followed for each source in turn, then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.
The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Norton equivalent or Thevenin equivalent.
Applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, linear passive elements Resistors, Inductors, Capacitors and linear transformers.



Superposition Theorem: Detail

• A generic system with input (stimulus, cause) and output (response, effect) can be described mathematically as a function . Superposition principle applies if this function is linear:
  
  
  
  Superposition principle has two aspects:
  
  
  The interpretation is that when the effect is directly proportional to the cause in a system , then we can consider several causes ( and ) individually and then combine the individual effects ( and ). An electrical system of linear components (with linear voltage-current relation) is a linear system (if only voltage and current are of interest). When there exist multiple energy sources, the currents and voltages in the circuit can be found as the algebraic sum of the corresponding values obtained by assuming only one source at a time, with all other sources turned off (voltage sources treated as short circuit, current sources treated as open circuit).
  As superposition principle only applies to linear functions, it cannot be applied to nonlinear functions such as power (e.g., or ).
  
  Superposition of voltage sources:
  
  
  
  where ( short circuit) and ( short circuit). However, note that superposition principle does not apply to power:
  
  
  Superposition of current sources:
  
  
  where ( open circuit) and ( open circuit). Again, superposition principle does not apply to power:
  
  
  Example 1: The previous example can also be solved by superposition theorem.
  
  First turn the voltage source of 20V off (short-circuit with 0V), and get
  
  
  
  Second turn the voltage source of 32V off and get
  
  
  The overall currents can then be found as the algebraic sums of the corresponding values obtained with one voltage source turned on at a time:
  
  
  Example 2: Find voltage and current .
  
  First, we solve this problem using node-voltage method. Assume the currents (left branch), and (right branch) all leave the top node, where the voltage is (with respect to the bottom treated as ground). By KCL, we have
  
  
  
  Solving for , we get:
  
  
  and
  
  
  Next, using superposition theorem, we get
  • Find and with the current source off (open circuit with zero current):
    
  • Find and with the voltage source off (short circuit with zero voltage):
    
    
    Both and have a negative sign as their direction and polarity are opposite to those of the assumed current and voltage.
  • Find the sum of the two:
    

Attached Files

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