| bonfire | 17-04-2011 12:23 AM |
Superposition theorem
1 Attachment(s) 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: - 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)).
- 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) http://fourier.eng.hmc.edu/e84/lectures/ch2/img77.png and output (response, effect) http://fourier.eng.hmc.edu/e84/lectures/ch2/img78.png can be described mathematically as a function http://fourier.eng.hmc.edu/e84/lectures/ch2/img79.png. Superposition principle applies if this function is linear:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img80.png
Superposition principle has two aspects:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img81.png
The interpretation is that when the effect is directly proportional to the cause in a system http://fourier.eng.hmc.edu/e84/lectures/ch2/img82.png, then we can consider several causes (http://fourier.eng.hmc.edu/e84/lectures/ch2/img83.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img84.png) individually and then combine the individual effects (http://fourier.eng.hmc.edu/e84/lectures/ch2/img85.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img86.png). 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., http://fourier.eng.hmc.edu/e84/lectures/ch2/img87.png or http://fourier.eng.hmc.edu/e84/lectures/ch2/img88.png).
http://fourier.eng.hmc.edu/e84/lectu...rposition0.gif
Superposition of voltage sources:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img89.png
where http://fourier.eng.hmc.edu/e84/lectures/ch2/img90.png (http://fourier.eng.hmc.edu/e84/lectures/ch2/img64.png short circuit) and http://fourier.eng.hmc.edu/e84/lectures/ch2/img91.png (http://fourier.eng.hmc.edu/e84/lectures/ch2/img63.png short circuit). However, note that superposition principle does not apply to power:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img92.png
Superposition of current sources:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img93.png
where http://fourier.eng.hmc.edu/e84/lectures/ch2/img94.png (http://fourier.eng.hmc.edu/e84/lectures/ch2/img54.png open circuit) and http://fourier.eng.hmc.edu/e84/lectures/ch2/img95.png (http://fourier.eng.hmc.edu/e84/lectures/ch2/img53.png open circuit). Again, superposition principle does not apply to power:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img96.png
Example 1: The previous example can also be solved by superposition theorem.
http://fourier.eng.hmc.edu/e84/lectu...ffexample1.gif
First turn the voltage source of 20V off (short-circuit with 0V), and get
http://fourier.eng.hmc.edu/e84/lectures/ch2/img97.png
Second turn the voltage source of 32V off and get
http://fourier.eng.hmc.edu/e84/lectures/ch2/img98.png
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:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img99.png
Example 2: Find voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png and current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png.
http://fourier.eng.hmc.edu/e84/lectu...erposition.gif
First, we solve this problem using node-voltage method. Assume the currents http://fourier.eng.hmc.edu/e84/lectures/ch2/img53.png (left branch), http://fourier.eng.hmc.edu/e84/lectures/ch2/img101.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img54.png (right branch) all leave the top node, where the voltage is http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png (with respect to the bottom treated as ground). By KCL, we have
http://fourier.eng.hmc.edu/e84/lectures/ch2/img102.png
Solving for http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png, we get:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img103.png
and
http://fourier.eng.hmc.edu/e84/lectures/ch2/img104.png
Next, using superposition theorem, we get
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