PROPERTIES OF SUPERPOSITION THEOREM

LIMITATIONS OF SUPERPOSITION THEOREM

RELATIONSHIP WITH MESH AND NODAL ANALYSIS

PROPORTIONALITY IN ELEMENTS

ADDITIVITY PROPERTY

APPLICATION OF SUPERPOSITION THEOREM

AN EXAMPLE FOR USING SUPERPOSITION THEOREM

ANOTHER WORKED EXAMPLE

SUMMARY

Network theorems provide insight into the behaviour and properties of electrical circuits. Superposition theorem is of theoretical importance, because it is fundamental to linear circuit analysis. A circuit is linear only when it behaves in accordance with superposition theorem. This theorem states that the linear responses in a circuit can be obtained as the algebraic sum of responses, due to each of the independent sources acting alone. This theorem defines the behaviour of a linear circuit. Within the context of linear circuit analysis, this theorem provides the basis for all other theorems. Given a linear circuit, it is easy to see how mesh analysis and nodal analysis make use of the principle of superposition.

There are two guiding properties of superposition theorem.

- The first is the property of homogeneity or proportionality.
- The second is the property of additivity.

As stated earlier, the linear responses in a circuit can be obtained using this theorem as the algebraic sum of responses, due to each of the independent sources acting alone. Current and voltage associated with an element are linear responses. On the other hand, power in an element is not a linear response. It is a non-linear function, varying proportionately either with the square of voltage across the element or with the square of current through the element. Hence it is not possible to apply superposition theorem directly to determine power associated with an element. In addition, application of superposition theorem does not normally lead to simplification of analysis. It is not the best technique to determine all currents and voltages in a circuit, driven by multiple of sources.

Superposition theorem is valid for linear circuits and analysis of linear circuits is relatively easy. On the other hand, the principle of superposition is not valid for non-linear circuits. And the analysis of non-linear circuits is quite complex complex and difficult. It is possible to apply mesh and nodal analysis to non-linear circuits. However, within the context of linear circuits, mesh or nodal analysis of a circuit illustrates how the principle of superposition is ever so pervasive in defining the behaviour of linear circuits.

A linear circuit consists of linear elements. The passive elements, the dependent sources and the independent sources used in a linear circuit are linear. Let us how linearity is defined for each type of element.

In the case of a resistor, the voltage across a resistor varies proportionally with its current. The ratio of voltage to current is resistance. Power in a resistor varies with the square of its voltage or its current.

In the case of an inductor, the linearity is between the flux linkage and the current. The ratio of flux linkage to current is inductance. The product of inductance and the rate of change of current is the inductor voltage. In an ideal inductor, the core of inductor does not get saturated. In an inductor with a magnetic core, saturation of flux occurs at some level, depending on the material used for the core.

In the case of a capacitor, the linearity is between the charge stored and the capacitor voltage. The ratio of charge to voltage is capacitance. The product of capacitance and the rate of change of voltage is the capacitor current. In an ideal capacitor, the capacitor voltage can be very high. In a real capacitor, the dielectric within the capacitor breaks down a magnetic core at some level, depending on the dielectric.

In the case of dependent source, the output variable varies proportionately with the controlling variable. In the case of ideal independent source, linearity implies that a voltage can supply any current at constant voltage, and that a current source can sustain any load voltage while supplying constant current.

The principle of additivity can be explained with the help of a sketch. When there are two sources in a circuit, the contribution due to each source can be found out separately and the response due to both sources is the algebraic sum of contributions due to each source acting alone. This aspect is illustrated by the circuit shown above. The current through the resistor is the sum of currents due to source *V*_{1} and source *V*_{2}, acting alone. When the contribution due to source *V*_{1} is to be calculated, the contribution due to source *V*_{2} should be zero. Hence source *V*_{2} can be replaced by a short circuit. When source *V*_{2} is replaced by a short circuit, its contribution has to be nil. Similarly, when the contribution due to source *V*_{2} is to be calculated, the contribution due to source *V*_{1} should be zero. Hence source *V*_{1} can be replaced by a short circuit. The relevant circuits are shown above.

If the direct application of superposition theorem is not easy, the question arises when it is suitable to use superposition theorem. It is best to use superposition theorem to find a particular current or voltage in a circuit, when the circuit has multiple independent sources. This theorem states that the linear responses in a circuit can be obtained as the algebraic sum of responses, due to each of the independent sources acting alone. A voltage source that makes no contribution is replaced by a short-circuit, whereas a current source that makes no contribution is replaced by an open-circuit. The internal resistance of the source is left in the circuit, as it is where it is. The worked examples are used to show how to apply superposition theorem to circuits.

A simple circuit is used to illustrate how the principle of superposition can be used to obtain the current through the resistor in the circuit shown in Fig. 1.

When superposition theorem is used, the response due to only one independent source is obtained at a time. The other sources are replaced, by either open-circuits or short-circuits, as the case may be. In this circuit, there are two sources, voltage *V*_{1} and voltage *V*_{2}. When response due to source *V*_{1} is calculated, source *V*_{2} is replaced by a short-circuit. Let the current through the resistor be* I*_{1}, as shown in Figure 2. When response due to source *V*_{2} is calculated, source *V*_{1} is replaced by a short-circuit. Let the current through the resistor be *I*_{2}.

Without using superposition theorem, it can be stated that the voltage across the resistor is as shown by equation (1). Then we can express the current through the resistor by equation (2), and it can be split into two fractions, as shown by equation (3). We see next how we can apply superposition theorem.

The expressions for currents *I*_{1} and *I*_{2} are marked in Figure 2. Using superposition theorem, the current through the resistor is obtained as the sum of currents *I*_{1} and *I*_{2}. Equation (4) is the same as the previous equation obtained for current through the resistor.
It is seen that the total current in the resistor is expressed as the sum of two responses, due to each source acting alone. That is, when the response due to voltage source *V*_{1} is to be calculated, the contribution of voltage source *V*_{2} should be zero and then source *V*_{2} is replaced by a short circuit. A voltage source with zero volts allows current to pass through it, and does not contribute anything to a circuit. That is, when contribution of an ideal voltage source is to be zero, it can be replaced by a short circuit. Similarly, when the response due to source *V*_{2} is to be calculated, the contribution of source *V*_{1} should be zero and it is replaced by a short circuit.
Equation (4) is a linear equation. In this equation, *V*_{1} and *V*_{2} are the independent variables. For example, if value of *V*_{1} changes from 6 Volts to 12 Volts, its contribution to current through resistor doubles. Hence the response due to an independent variable varies proportionately.

That is, the ratio of independent variable to the response is a constant. In this case, the Ohms Law applies, as shown by equation (5). Equation (5) expresses the property of proportionality of a linear relationship. The other property of a linear relationship is the additive property, expressed by equation (4). Here the total response is expressed as the algebraic sum of responses, due to each independent source acting alone. Equation (4) is formed based on the principle of superposition. The current through resistor R in the circuit shown in Fig. 1 can be determined using the superposition theorem, and the power absorbed by this resistor is determined in the end, after finding the resultant current in the resistor due to both the sources.

The values of components present in the circuit in Fig. 1 are presented by equation (6).

Current through the resistor is obtained, as shown by equation (7).

Once the current through the resistor is known, the power absorbed by it can be found out, as shown by equation (8). Next, it is shown why superposition theorem cannot be applied directly to determine the power absorbed by the resistor R present in the circuit in Fig. 1.

Based on the circuits presented in Fig.2, the power absorbed by the resistor, due each of the sources acting alone, can be obtained. Equation (9) states the power absorbed by the resistor when *V*_{1} = 12 Volts, and *V*_{2} = 0 V.

Equation (10) states the power absorbed by the resistor when *V*_{1} = 0 Volts, and *V*_{2} = 6 V.

The power absorbed by the resistor due to both sources is presented by equation (8), which is 18 Watts. On the other hand, the sum of P1 and P2 is
90 Watts. In this circuit, the power associated with source *V*_{1} is - 36 Watts, since it is delivering power. The resistor absorbs 18 Watts, and the remaining power is absorbed source *V*_{2}.

A circuit has been presented in Fig. 3. Determine current *I*_{2}. The values of components have been specified.
Solution:
It is possible to apply the superposition theorem directly. But in this case, the mesh equations are formed, and then the components in the solution can be split into two parts, to show how the principle of superposition is in operation, even if it is not applied directly.

Given the circuit in Fig. 3, the mesh equation that can be formed is of the type shown by equation (12). The values of the elements in the loop resistance matrix can be obtained by inspection of the circuit.

The elements *a*_{11}, *a*_{22}, and *a*_{11} along the main diagonal reflect the self-resistance of the loops, whereas the other elemenst are the mutual resistances, common to two loops. Note that all the loops are marked in the clockwise direction, and the current through any element is either the loop current or the difference of two loop currents. In this context, the mutual resistances have negative values. It can also be seen that the loop-resistance matrix is symmetric.

Let the determinant if the loop resistance matrix be calculated, as shown by equation (14). Then we can obtain the value of current *I*_{2} using Cramer's rule, as shown by equation (15). The value of current *I*_{2} can be expressed in terms of cofactors of *V*_{1} and *V*_{2}, as shown by equation (16).

Contribution to current *I*_{2} due to *V*_{1} and *V*_{2} can be calculated separately, as shown above, and the value of *I*_{2} is obtained to be 2 A. It can be seen that the use of mesh analysis reflects the application of superposition theorem.

The analysis of linear circuits is founded on the superposition theorem. Even though the direct use of superposition theorem is not always easy, it remains the guiding principle for the behaviour of linear circuits. Next we take up Thevenin's theorem.