In the circuit of Fig. 13.93,
- (a) find the coupling coefficient,
- (b) calculate vo,
- (c) determine the energy stored in the coupled inductors at t = 2 s.
(a)
Calculate the coupling coefficient of the circuit in Figure 13.93.
Answer to Problem 24P
The coupling coefficient is
Explanation of Solution
Given data:
Refer to Figure 13.93 in the textbook for the circuit with coupled coils.
The value of
Calculation:
Consider the expression for the coefficient of coupling in the coupled coils.
Substitute 1 H for M, 4 H for
Conclusion:
Thus, the coupling coefficient is
(b)
Calculate the voltage
Answer to Problem 24P
The value of voltage
Explanation of Solution
Given data:
From Figure 13.93, the value of
Calculation:
Write the expression for the inductive reactance.
Write the expression for the capacitive reactance.
Substitute 4 H for
Substitute 2 H for
Substitute 1 H for
Substitute
Calculate load impedance
Modify the Figure 13.93 by transforming the time-domain circuit with coupled-coils to frequency domain of the circuit with coupled-coils. The frequency domain equivalent circuit is shown in Figure 1.
From Figure 1, consider that the loops 1 and 2 contain the currents
Apply Kirchhoff's voltage law to the loop 1 in Figure 1.
Apply Kirchhoff's voltage law to the loop 2 in Figure 1.
Write equations (3) and (4) in matrix form as follows.
Write the MATLAB code to solve the equation (5).
A = [(1+j*8) j*2;j*4 (0.5+j*7.5)];
B = [6; 0];
I = inv(A)*B
The output in command window:
I =
0.13036 - 0.84468i
-0.09912 + 0.44389i
From the MATLAB output, the currents
And
Write the expression for the voltage
Substitute
Convert the phasor form to time domain form.
Conclusion:
Thus, the value of voltage
(c)
Calculate the stored energy in the coupled coils at
Answer to Problem 24P
The energy stored in the coupled coils is
Explanation of Solution
Calculation:
From part (b), write the currents
Substitute 2 s for t in Equation (6).
Substitute 2 s for t in Equation (7).
Write the expression for the total energy stored in the coupled coils.
Substitute 4 H for
Conclusion:
Thus, the energy stored in the coupled coils is
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