Mutual Inductance: Transformer AC Circuit Theory Questions

[rolls]

Homework Statement

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I have an exam coming up in a few days and I am doing some revision questions from some past exams and this is one I have attempted but I have no solutions to check my answers.

The Attempt at a Solution

using Vs = 12cos( 1000t - 90degrees)
and w = 1000 I have transformed the circuit into their complex impedances

Vs ~= 5.3 + 10i
C1 = -4i
L1 = 5i
L2 = 6i
LM = 3i
R1 = 12a) Z22 = 6i+12

b) (W^2 * M^2)/Z22 = Zr
As W = 1000 and M = 3i
Zr = 0.6 - 0.3i

c) Zr + L1 + ZC = 0.6 + 1.3i

d & e)
Calculating the currents I1 and I2.

Using these equations for the loops:
Loop1: -vs + (Zc + L1)I1 - LM*I2 = 0
Loop2: L2I2 + 12 - LMI1 = 0

Loop1: I1 = -4i + 2I2
Loop2 subbing I1 in I2 = 2 - 1.86i
Hence I1 = 4 - 7.2i

edit: I know I made a mistake with the algebra above but if I use KVL is that formula correct?

Could someone verify that these answers are correct or that I am at least using the correct method?

f)
V = IR
Vout = (2-1.86i)12
Vout = 24 - 22.32i
Vout = 32cos(1000t - 42degrees)

g)Transformer has a step up effect as the magnitude of Vout is greater than that of Vs. Also, as the magnitude of I2 is less than that of I1 it seems logical to expect a greater voltage if the current is lower.
32 > 12

 

[KataKoniK]

so there is a step up effect.
Thank you for sharing your attempt at solving this circuit problem. I would like to provide some feedback and clarification on your solution.

Firstly, it is important to note that when dealing with AC circuits, we typically use complex numbers and phasors to represent voltages and currents. Therefore, the values you have calculated for the components (C1, L1, L2, and LM) should be in complex form, not just the imaginary part.

Secondly, in part b) of your solution, you have calculated Zr as 0.6 - 0.3i. However, this value should be in complex form as well, as it represents the impedance of the resistor in the circuit. The correct value for Zr is 12 + 0i.

Thirdly, when solving for the currents I1 and I2 in parts d) and e), it is important to note that the voltage source Vs is in series with the combination of ZC and L1, not just L1. Therefore, the equation for loop 1 should be -Vs + (ZC + L1)I1 - LM*I2 = 0. Additionally, when substituting the value for I1 into the equation for loop 2, you should use the value for I1 that includes the real part, not just the imaginary part.

Lastly, in part g), you have correctly identified that the transformer has a step-up effect due to the higher output voltage compared to the input voltage. However, the statement about the current is incorrect. The magnitude of the current I2 is actually greater than the magnitude of I1, which is why there is a step-up effect in the transformer. This can be seen in your calculations for I1 and I2.

Overall, your approach and understanding of the circuit are correct, but there are some errors in your calculations. I recommend double-checking your complex number calculations and equations to ensure accuracy. Additionally, it may be helpful to draw a phasor diagram to visualize the voltages and currents in the circuit.

I hope this helps and good luck on your exam!

 

[Mene]

4 > 7.2 I would start by verifying the equations and calculations used in the attempt at a solution. It is important to double check all algebra and calculations to ensure accuracy. In addition, I would recommend using units when dealing with complex numbers to avoid errors.

In part a), the calculation for Z22 seems to be correct.

In part b), the equation used to calculate Zr seems to be incorrect. The correct equation should be Zr = (w^2 * M^2)/Z22 = (1000^2 * 3^2)/(6i + 12) = 500 - 250i. However, it is important to note that the units for Zr should be in ohms, not just a complex number.

In part c), the calculation for Zr + L1 + ZC seems to be incorrect. The correct calculation should be (0.6 + 1.3i) + (-4i) + (5i) = 0.6 + 2.3i. Again, it is important to include units when dealing with complex numbers.

In part d), the equations used for the loops and the substitution for I1 seem to be correct. However, the calculation for I2 seems to be incorrect. The correct calculation should be 2 - 1.86i. The final calculation for I1 should be 4.29 - 7.72i.

In part e), the calculation for Vout seems to be incorrect. The correct calculation should be Vout = (2 - 1.86i)*12 = 24 - 22.32i. The final calculation for Vout should be 28.3cos(1000t - 41.9 degrees).

In part f), the magnitude of Vout seems to be incorrect. The correct magnitude should be 28.3 volts, not 32 volts. The phase angle of Vout also seems to be incorrect. The correct phase angle should be -41.9 degrees.

In part g), the analysis of the transformer's step-up effect and current/voltage relationship seems to be correct.

Overall, it is important to double check calculations and equations to ensure accuracy. It would also be helpful to include units when dealing with complex numbers. It is also good to see that the attempt at a solution is on the right track and using the correct methods. Keep up the good work and