Calculate the coupling coefficient of the circuit that make the
Answer to Problem 29P
The required coupling coefficient is
Explanation of Solution
Given data:
Refer to Figure 13.98 in the textbook for the circuit with coupled coils.
In Figure 13.98, consider that the primary and secondary loops contain the currents
The value of
The value of
Calculation:
Calculate the inductors in frequency domain.
Write the expression for the inductive reactance.
Substitute 30 mH for
Substitute 50 mH for
Consider that the value of X.
Substitute
Consider that the second side reflect on the primary side. Consider the expression for the input impedance.
Write the expression for the current
Substitute 330 V for V and Equation (3) in (4).
Consider the expression for the power dissipated in the
Substitute 1.288 kW for p.
Substitute 16 A for
Square on both sides of the equation.
Simplify the equation as follows.
By solving the above equation, there are two positive values of X and they are,
Consider that the value of X as 38.128.
Substitute 38.128 for X in Equation (2).
Consider the expression for the coefficient of coupling in the coupled coils.
Substitute 38.128 mH for M, 30 mH for
Modify the Figure 13.98 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 (6) and (7) in matrix form as follows.
Write the MATLAB code to solve the equation (8).
A = [(10+j*30) j*(-38.128); j*(-38.128) (20+j*50)];
B = [330; 0];
I = inv(A)*B
The output in command window:
I =
15.535 - 3.829i
11.219 + 1.568i
From the MATLAB output, the currents
And
Write the currents
Substitute 1.5 ms for t in Equation (9).
Substitute 1.5 ms for t in Equation (10).
Write the expression for the total energy stored in the coupled coils.
Substitute 30 mH for
Conclusion:
Thus, the required coupling coefficient is
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Chapter 13 Solutions
Fundamentals of Electric Circuits
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