Calculating Power and Impedance in Mutually Coupled Coils

[greg997]

The problem I have is the following:
Two Coils have a resistance of 25 ohms and an inductance of 20 mH. Coils are mutually coupled with a coefficient of coupling of 0.6.

(a) Calculate the voltage input to the first coil to develop a power of 1.6W
at a frequency of 1 kHz. with a load of 55<-40 connected to the second coil.

This one I calculated as Vin =29.25<27.5 Volt. The magnitude seems correct according to multisim. But I am not sure about the angle. Anyone knows how to check that in Multisim?(b) Calculate the power developed in the load by referring the secondary
impedance to the primary.

This is the main problem I have,because I am not sure I understand the question.
The secondary Z2 consists of R+jXl of the winding and R-jXc of the load.
I have found the Zeffective and primary current . Ths is what I did in point a) of the question. I got the primary current, but what part of the total impedance on the primary side is the load impedance?
What should I do?
Is it like I have Iprimary and Isecondary, and then using their ratio refer just load impedance to primary and using the primary current get the power? Is the primary current going to be the same as in point a)? Is the question b) correct? Perhaps it should say "secondary impedance of the load"?
thank you

 

[rude man]

First off, (a) is not clearly stated. What power? Power to the load only, power everywhere ... ? I'm guessig power to the load only, based on the wording of part (b).

Second, your answer to (a) should not include a phase angle. Define the input voltage phase to be zero.

Forgetting multisim which I don't know:

(a) Go with the equivalent circuit of a transformer with finite primary & secondary resistances, k < 1, and turns ratio of 1. Your ckt should therefore consist of the winding resistances, leakage inductances since k < 1, and of course the load. Determine what the output voltage must be, then work the equations to get input voltage.

(b) is just a check of (a). Use the input voltage you got from (a) and solve for load power. Use the same equiv. ckt as in (a).

 

[gneill]

You can use the coupling constant k and the inductances L1, L2, of the coils to determine the mutual inductance M between the coils.

What's the relationship between k, M, L1, L2?

With the mutual inductance in hand you should be able to write the loop equations for the primary and secondary circuits. Since the frequency of operation is a fixed given it would probably make life easier to work with impedances for all the parts. Don't be afraid to simplify appropriate constant expressions down to single numerical (complex) values and assign them names; this will help to keep the algebra from taking over the planet (or at least your scratchpad)

My take on part (a) is that they're looking for an input voltage that will cause that voltage supply to provide 1.6W of real power driving the primary. If that is indeed the case, then if you solve for the current in the primary then P = VI*, where I* is the conjugate current phasor (just negate the imaginary term of the current). The real portion of P is the real power dissipated.

 

[greg997]

Can either of you give me more hint on how to start with point b)

I mean the initial calculations and perhaps some circuit diagram. I still don't know how to reflect that load onto primary. Is the power in the load going to be the same on the primary as on secondary side, if the k<1? If it was k=1 then I assume the power would be the same.
I would appreciate that help.

 

[gneill]

Can either of you give me more hint on how to start with point b)

I mean the initial calculations and perhaps some circuit diagram. I still don't know how to reflect that load onto primary. Is the power in the load going to be the same on the primary as on secondary side, if the k<1? If it was k=1 then I assume the power would be the same.
I would appreciate that help.

I understand how secondary circuit impedances are reflected into the primary for ideal transformers where the coefficient of coupling k is unity, but I've not come across a discussion for cases where k ≠ 1. No doubt it's out there somewhere

However, I think that if you follow my advice in post #3 and solve the equations for the current in the primary, you should be able to pick out the "reflected" load in the equation for the primary current.

 

[rude man]

Can either of you give me more hint on how to start with point b)

I mean the initial calculations and perhaps some circuit diagram. I still don't know how to reflect that load onto primary. Is the power in the load going to be the same on the primary as on secondary side, if the k<1? If it was k=1 then I assume the power would be the same.
I would appreciate that help.

To make sense of the term 'reflected' you need to set up an equivalent circuit including an ideal transformer. For an ideal transformer, an impedance Z (this includes secondary leakage inductance and resistance!) hanging on its secondary winding is reflected back to the primary by ZN² where N is the primary-to-secondary turns ratio. For example, a load resistor R across the secondary looks like NR² at the primary, with the transformer now removed.

Like gneill I don't know any other way to 'reflect' a secondary across a k<1 transformer, even if otherwise ideal, without going to an equivalent circuit invoking an ideal transformer. On top of that, the 'transformer' here has finite winding resistances.

BTW don't be confused when we talk about a 'transformer', because the two coils in your problem constitute just that.

I could give you a textbook reference for an equivalent circuit but that book would be outdated by now. I can only hint that such a circuit would show primary and secondary resistances 1:1, and primary and secondary leakage inductances thanks to k being < 1. I no longer use my PSPICE software (I am retired!) so you'll need to scrounge around for the equiv. ckt.

One more hint: N = 1 here. This can be assumed without loss of generality.