Network Theorems
Linearity Property:
Linearity is the property of an element describing a linear relationship between cause and effect. Although the property applies to many circuit elements, we shall limit its applicability to resistors in this chapter. The property is a combination of both the homogeneity (scaling) property and
the additivity property. The homogeneity property requires that if the input (also called the excitation) is multiplied by a constant, then the output (also called the response) is multiplied by the same constant. For a resistor, for example, Ohm’s law relates the input i to the output v,
If the current is increased by a constant k, then the voltage increases
correspondingly by k, that is,
The additivity property requires that the response to a sum of inputsis the sum of the responses to each input applied separately. Using the voltage-current relationship of a resistor, if
v1 = i1R
and
v2 = i2R
then applying (i1 + i2) gives
A linear circuit is one whose output is linearly related (or directly proportional) to its input.
Superposition Principle:
If a circuit has two or more independent sources, one way to determine the value of a specific variable (voltage or current) is to use nodal or mesh analysis Another way is to determine the contribution of each independent source to the variable and then add them up. The latter
approach is known as the superposition.
The idea of superposition rests on the linearity property
"The superposition principle states that the voltage across (or current through) an bear a linear relationship to one another.Element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone."
- Superposition is not limited to circuit analysis but is applicable in many fields where cause and effect
we must keep two things in mind:
1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.Other terms such as killed, made inactive, deadened,or set equal to zero are often used to conveythe same idea.
2. Dependent sources are left intact because they are controlled by circuit variables.
Steps to Apply Super position Principle :
1. Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis.
2. Repeat step 1 for each of the other independent sources.
3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
Keep in mind that superposition is based on linearity. For this reason, it is not applicable to the effect on power due to each source,because the power absorbed by a resistor depends on the square of the voltage or current. If the power value is needed, the current through (or voltage across) the element must be calculated first using superposition.
Thevenin's Theorem:
It often occurs in practice that a particular element in a circuit is variable (usually called the load) while other elements are fixed. As a typical example, a household outlet terminal may be connected to different appliances constituting a variable load. Each time the variable element is changed, the entire circuit has to be analyzed all over again. To avoid this problem, Thevenin’s theorem provides a technique by which the fixed part of the circuit is replaced by an equivalent circuit
"Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when
the independent sources are turned off."
Norton's Theorem:
In 1926, about 43 years after Thevenin published his theorem, E. L. Norton, an American engineer at Bell Telephone Laboratories, proposed a similar theorem
"Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the
independent sources are turned off."
Since VTh, IN, and RTh are related according to Eq. (4.11), to determine the Thevenin or Norton equivalent circuit requires that we find:
• The open-circuit voltage voc across terminals a and b.
• The short-circuit current isc at terminals a and b.
• The equivalent or input resistance Rin at terminals a and b when
all independent sources are turned off.
We Can Also Find:
Maximum Power Transfer Theorem:
In many practical situations, a circuit is designed to provide power to a load. While for electric utilities, minimizing power losses in the process of transmission and distribution is critical for efficiency and economic reasons, there are other applications in areas such as communications
where it is desirable to maximize the power delivered to a load. We now address the problem of delivering the maximum power to a load when given a system with known internal losses. It should be noted that this will result in significant internal losses greater than or equal to the power
delivered to the load.
The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load
"Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (RL = RTh)."
Maximum Power Transferred is :