The superposition theorem is a derived result of the

superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...

suited to the

network analysis Network analysis can refer to: * Network theory, the analysis of relations through mathematical graphs ** Social network analysis, network theory applied to social relations * Network analysis (electrical circuits) See also *Network planning and d ...

of electrical circuits. The superposition theorem states that for a

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction ...

(notably including the subcategory of time-invariant linear systems) the response (

voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...

or current) in any branch of a bilateral

linear circuit A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit ''F(x)'' when a linear combination of signals ''ax1(t) + bx2(t)'' is applied to it is equal to the linear combination o ...

having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal

impedances In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the compl ...

. 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 A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circu ...

(thereby eliminating difference of potential i.e. ''V''=0; internal impedance of ideal voltage source is zero (

)). * 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 In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of te ...

or Thevenin equivalent. The theorem is applicable to linear networks (time varying or time invariant) consisting of independent sources, linear

dependent source In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current elsewhere in the network.I. D. Mayergoyz, Wes Lawson ''Basic electric circuit theory: a one-semester tex ...

s, linear passive elements (

resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...

inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...

s) and linear

transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...

s. Superposition works for voltage and current but not power. In other words, the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents. However, if the linear network is operating in steady-state and each external independent source has a different frequency, then superposition can be applied to compute the average power or active power. If at least two independent sources have the same frequency (for example in power systems, where many generators operate at 50 Hz or 60 Hz), then superposition can't be used to determine average power.

Gas pressure analogy

The electric circuit superposition theorem is analogous to Dalton's law of

partial pressure In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal g ...

which can be stated as the total pressure exerted by an ideal gas mixture in a given volume is the algebraic sum of all the pressures exerted by each gas if it were alone in that volume.

References

* ''Electronic Devices and Circuit Theory'' (9th ed.) by Boylestad and Nashelsky * ''Basic Circuit Theory'' by C. A. Desoer and E. H. Kuh * ''Edward Hughes revised by John.K, Keith.B etal (2008) Electrical and Electronic Technology (10th ed.) Pearson page 75-77

External links

On the Application of Superposition to Dependent Sources in Circuit Analysis
nbsp;– proves superposition of dependent sources is valid. Circuit theorems {{electric-stub de:Superposition (Physik)#Elektrotechnik