This page contains some worked examples to illustrate how ac circuits can be analyzed.
Problem:
For the bridged_T network in Fig. 51, the load is connected across its output terminals, A and B. Find current I4 through load impedance Z4. Equation (14.1) specifies the values of elements in the circuit.
SOLUTION USING THEVENIN'S THEOREM:
The bridged_T network in Fig. 51 is re-presented in Fig. 52, showing how Thevenin's voltage and the Norton's current can be obtained. We can form node equations at node, A and B to get the Thevenin's voltage. The node voltages are VA0 and VC0. The equations obtained are presented below.
Equation (14.2) is the KCL equation for the node with voltage VC0. Equation (14.3) is the KCL equation for the node with voltage VA0. The two equations can be presented by a matrix equation, as shown by equation (14.4). The solution yield Thevenin's voltage, as displayed by equation (14.5). From the circuit in Fig. 52, we get Thevenin's impedance as follows.
We get equation (14.6) for the circuit in Fig. 52, when the load terminals are short-circuited. The short-circuit current IN can be obtained, as shown by equation (14.7). Once the Thevenin's voltage and the Norton's current are known, we can get the Thevenin's impedance, as shown by equation (14.8). The load current can be obtained, as shown by equation (14.9).
SOLUTION USING STAR TO DELTA TRANSFORMATION
We can perform star to delta transformation and then obtain the current I4 through the load impedance Z4. We can obtain the Thevenin's equivalent circuit for the circuit in Fig. 53 by using star-to-delta transformation. The star-connected network connected to nodes a, b and c can be replaced by an equivalent delta connected-network, as shown by the circuit in Fig. 54.. From the circuits shown in Fig. 54, we can get Thevenin's voltage and Thevenin's impedance, as shown below.
The impedances of the equivalent delta-connected network are presented by equations (14.10), (14.11) and (14.12). The Thevenin's voltage is expressed by equation (14.13).
Equation (14.14) shows how the Norton's current can be obtained. Once the Thevenin's voltage and the Norton's current are known, we can get the Thevenin's impedance, as shown by equation (14.15). The load current can be obtained, as shown by equation (14.16).
SOLUTION OBTAINED BY USING BRIDGE CONFIGURATION
The bridged-T network with the load can be represented by a bridge network, as shown in Fig. 55. For this network, we can obtain the node voltages, V3 and V4. Then the the current I4 through load impedance Z4 can be obtained, as shown below.
Equations (14.17) and 914.18) are the KCL equations formed at nodes with voltages, V3 and V4. The matrix equation, containing the admittance matrix and the column vectors, is presented by equation (14.19). The value of voltage V4 and current I4 are expressed by equations (14.20) and (14.21).
Problem:
Obtain the relationship between L, C, R1 and R2 if the current Im marked in Fig. 56 is zero at all frequencies.
Solution:
There are many bridge circuits, each with some advantages. The bridge circuits are useful for measurements of inductances and capacitances. The detector is located where resistor Rm is shown. We can take Rm as the resistance of the detector. Under balanced conditions, current Im is zero. then the detector has no deflection. It means that when current Im is zero, we have equation (14.22 ). When two phasors are equal, the real parts of the two phasors are equal and the imaginary parts of the phasors are equal. Hence equate the real parts of both voltages and the imaginary parts of both voltages. We get the following equations:
Equation (14.25) is obtained by equating the real parts of phasors VA and VB. Equation (14.26) is obtained by equating the imaginary parts of phasors VA and VB. Equations (14.25) and (14.26) can be equated with each other, since both these equations have the same expression on the left side. Then we get equation (14.27). The condition for current Im to be zero at all frequencies is that the ratio L/C should be equal to the product of R1 and R2.
Figure 57 shows an Anderson bridge used to measure inductance Lx and its series resistance Rx with high accuracy.
a. Transform the delta-connected network ABD into its equivalent star-connected network and hence determine the balance equations for Lx and Rx.
b. If R2 = 1 kW, R3 = 1 kW, R4 = 0.5 kW, R5 = 0.2 kW and C = 2 mF, determine the values of Lx and Rx at balance.
Solution:
Replace the delta-connected to nodes A, B and D, by a star-connected network, as shown in Fig. 58.
From the circuit in Fig. 58, we get the following equations.
After replacing the delta-connected by a star-connected network, the bridge circuit in Fig. 58 appears as shown in Fig. 59.
From Fig. 59, we get that
Substituting expressions for ZB and ZD, we get that
From the values given, Rx = 500 W, and Lx = 1.6 H.
Find the VA delivered by the two sources present in the circuit shown in Fig. 60. Note that the voltage source VA and the current source are IB specified in terms of their rms values and angles.
Solution:
For the circuit in Fig. 60, we get that
On solving equation (14.34), we get that
Note that the power delivered is negative.
Show that an n-mesh, passive, three-terminal network shown in Fig. 60 may be replaced with a delta-connection of three impedances. Comment on the result.
Solution:
Let voltage V1 be applied between terminals a and c and voltage V2 be applied between terminals b and c . The n-mesh equations formed for the network is shown below.
From equation (14.40), we can write that
where Dz is the determinant of the loop impedance matrix and D11 , D12 , D21 and D22 are the cofactors of z11, z12 , z21 and z22 respectively. Let the equation above be represented in the form of 2×2 matrix equation. Then
From Fig. 62, we have that
Comparing equations (14.42) and (14.43), we get that
It is possible that the real part of ZA, ZB and ZC may turn out to be a negative value and then it is not physically realizable.
This chapter has presented analysis of ac circuits using network theorems and the general techniques of analysis. Some examples have been presented to highlight the techniques used for solution. It is possible to obtain the solution to a problem in more than one way. Since sinusoidal signals are commonly used, a special technique to analyze ac circuits has been outlined in this chapter and the previous chapter. A network may be excited by a signal, other than a dc or an ac signal. In such a case, we need to have background knowledge of linear differential equations with constant coefficients. We deal with this topic in the next chapter.