Sinusoidal Steady-State Analysis

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Introduction

In this chapter, we want to see how nodal analysis, mesh analysis, Thevenin's theorem, Norton's theorem, superposition, and source transformations are applied in analyzing ac circuits. Since these techniques were already introduced for dc circuits, our major effect here will be to illustrate with examples.
     Analyzing ac circuits usually requires three steps.




Steps to Analyze AC circuits:

1. Transform the circuits to the phasor or frequency domain 
2. Solve the problem using circuit techniques (nodal analysis, mesh analysis, superposition, etc. )
3. Transform the resulting phasor to the time domain.

Step 1 is not necessary if the problem is specified in the frequency domain. In step 2, analysis is performed in the same manner as dc circuit analysis except that complex numbers are involved. Having done and learned Chapter 9, we are adept at handling step 3.

10.2 Nodal Analysis

     Nodal analysis is more commonly used than mesh or loop analysis for analysing

networks.  It can be used to determine the unknown node voltages of both planar

and non-planar  circuits.   Nodal equations are usually formed by applying

Kirchhoff’s Current Law to the nodes with unknown voltages, whereas equations

based on Kirchoff’s Voltage Law are used to form the mesh equations. In order to

apply nodal analysis to a circuit, the first step is to select a reference node or

datum node and then assign a voltage at each of the other nodes with respect to

the reference node.  In a circuit with dc sources, the node that has the lowest

voltage is usually selected as the reference node and then the other node voltages

would be positive.  Often the node that has the maximum elements connected to

it in a circuit tends to be the reference node.  Many electronic circuits have a

metallic chassis and the reference node, usually  the negative terminal of the dc

source present in the circuit, happens to be connected to the chassis.  It is  common

practice to connect the chassis to earth terminal of the utility supply.  Then the

reference node is at earth or ground potential and hence the reference terminal is

referred to as the ground terminal, even if it is not earthed.   In a power system, the

casing of power appliances is usually earthed and the neutral of the utility supply

remains connected to earth at the source.  The reference node in such a power

system is then at zero potential. 

In nodal analysis, the voltage at the reference node is assumed to be zero.  The

voltages at other nodes are expressed with reference to the datum node.  Since the

unknown node voltages are determined by nodal analysis, it is logical to write the

KCL equations at nodes.  The procedure can be summarized as follows.

• Select a reference node and treat it to be at zero or ground potential.
• Label the nodes with unknown voltages.
• At each of these nodes, mark currentsin the elements asflowing away from the node.
• Form KCL equations and solve the set of simultaneous equations for theunknown voltages.

     The voltage across an element can be expressed as the difference of  voltages at

nodes to which it is connected.  Use the passive sign convention and mark  the

currents to be flowing away from the node at which the KCL is applied.  Then the

mutual conductance term would have a negative sign whereas the self-conductance

at a node would be a positive value.  This aspect should become clear after a few

examples.  It is worth stressing that labelling the nodes and assigning current

direction properly are really important.  The practice of this technique would

enable one to write the KCL equations and the matrix equations by inspection.

The basis of nodal analysis is Kirchhoff's current law. Since KCL is valid for phasors, as demonstrated in Section 9.6 we can analyze ac circuits by nodal analysis. 

     This chapter has shown nodal analysis can be applied to simple circuits and   to

circuits with dependent sources. The next chapter describes the sources or signals

used for exciting an electrical circuit and their properties

10. 3 Mesh Analysis

   

      Kirchhoff's voltage law (KVL) forms the basis of mesh analysis. The validity of KVL for ac circuits was shown in Section 9.6 and is illustrated in the following examples. Keep in mind that the very nature of mesh analysis is that it is to be applied to planar circuits. 

    The two general analytical techniques used in network analysis are mesh or loop

analysis and nodal analysis.  This chapter describes mesh analysis, whereas the

next chapter is on nodal analysis. 

        

     Mesh analysis is a systematic technique to evaluate all voltages and currents in a

circuit.  It is based on Kirchoff’s Voltage Law and Ohm’s Law. This technique can

be applied to ac circuits also.  The way to form meshes or loops is described first,

followed by an example. Then a set of simultaneous equations are formed, with the

number  of these equations equaling the  number  of unknown variables to be

determined.  These unknown variables are the loop currents or the mesh currents.

The simultaneous equations can be compactly described by matrix equations, and

the solution to matrix equations can be obtained using Cramer's rule. Gaussian

elimination method can also be used to obtain the solution.  Solution by hand can

be laborious if the number of unknowns exceeds three. Some examples are

presented, to illustrate the technique of mesh analysis.  Since it is essential to be

familiar with passive sign convention, it is explained first.

    This chapter has introduced mesh analysis to resistive circuits with dependent and

independent sources.  Given a circuit with dependent sources, it is difficult to

specify a particular technique for forming the necessary equations. The number

of independent equations to be formed should equal the number of unknowns.  In

that case, the square matrix would have a non-zero determinant and then the

solution is unique.  The next chapter describes how nodal analysis can be applied

to such circuits.

10. 4  Superpositon Theorem

    Since ac circuits are linear, the superposition theorem applies to ac circuits the same way it applies to dc circuits. The theorem becomes important if the circuit has sources operating at different frequencies. In this case, since the impedances depend on frequencies, we must have different frequency domain circuit for each frequency. The total response must be obtained by adding individual responses in the time domain. It is incorrect to try to add the responses int the phasor or frequency domain.Why? Because the exponential factor e^jwt is implicit in sinusoidal analysis, and that factor would change for every angular frequency w. It would therefore not make sense to add response at different operating at different frequencies, one must add the response due to the individual frequencies in the time domain.

Importance of Superposition Theorem

     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. Thistheorem 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.

Properties of Superposition Theorem

    There are two guiding properties of superposition theorem.  The first is the

property of homogeneity or proportionality, and the second is the property of

additivity.     

Limitations of Superposition Theorem

    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 leadto simplification of analysis.  It is not the best technique to determine all currents

and voltages in a circuit, driven by multiple of sources.    

Relationship with Mesh and Nodal Analysis

    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 nonlinear  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.   

    The analysis of linear circuits is founded on the superposition theorem.  Eventhough 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.

10.5 Thevenin's Theorem

   Thevenin’s theorem is a popular theorem, used often for analysis of electronic

circuits.  Its theoretical value is due to the insight it offers about the circuit. This

theorem states that a linear circuit containing one or more sources and other linear

elements can be represented by a voltage source and a resistance. Using this

theorem, a model of the circuit can be developed based on its output characteristic.

Let us try to  find  out what Thevenin’s  theorem is  by using an investigative

approach.

   Thevenin's theorem are applied to ac ciruits in the same way as they are to dc circuits. The only additional effort arises from the need manipulate complex numbers. the frequency domain version of a Thevenin equivalent circuit, where a linear is circuit is replaced by a voltage source in series with an impedance. The Thevenin theorem equivalent circuit must be detemined at eacg frequency. This leads to entirely different equivalent circuits, one for each frequency, not one equivalent circuit with equivalent sources and equivalent impedance.

10.6 Superposition Theorem

   

    The circuits in this set of problems consist of independent sources, resistors and a meter. In 

particular, these circuits do not contain dependent sources. Each of these circuits has a seriesparallel structure that makes it possible to simplify the circuit by repeatedly   

•  Performing source transformations. 

•  Replacing series or parallel resistors by an equivalent resistor.  

•  Replacing series voltage sources by an equivalent voltage source. 

•  Replacing parallel current sources by an equivalent source source. 

    Each simplification is done in such a way that the voltage or current measured by the meter is not 

disturbed. Generally, that requires beginning the simplification at the opposite end of the circuit 

from the meter and then working toward the meter.  

   Eventually, the circuit is small enough to be easily solved using Ohm’s and Kirchhoff’s Laws. 

10.7 Summary

1. We apply nodal and mesh analysis to ac circuits by applying KCL and KVL to the phasor form of the circuits.
2. In solving for the steady state response of a circuit that has independent sources with different frequencies, each independent source must be considered separately. The most natural approach to analyzing such circuits is t The concept of source transformation is also applicable in the frequency domain. o apply the superposition theorem. A separate phasor circuit for each frequency must be solved independently. and corresponding response should be obtained in the time domain. The overall response is the sum of the time domain responses of all the individual phasor circuit.
3. The Thevenin equivalent of an ac circuit consists of a voltage source Vth in series with the Thevenin impedance Zth.
4. The concept of source transformation is also applicable in the frequency domain.