How do I solve this circuit using the impedance form?

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Discussion Overview

The discussion revolves around solving a circuit problem involving mutual inductance using impedance form. Participants explore different methods to approach the problem, including the application of Kirchhoff's Voltage Law (KVL) and the incorporation of mutual inductance into the circuit analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving a circuit problem and mentions two methods found online: using an energy equation and applying KVL for each coil.
  • Another participant suggests adjusting the circuit diagram to include the effects of mutual inductance as a current-controlled dependent voltage source in series with the inductors.
  • A participant proposes converting the circuit into a T-Circuit, but this is challenged by another who clarifies the role of mutual inductance in the circuit.
  • Equations involving the currents and mutual inductance are presented, but the participant is uncertain about rearranging them correctly.
  • There is a suggestion to use impedance directly instead of differential equations, indicating a shift in approach to solving the problem.
  • Participants discuss the relationship between voltage, current, and impedance, with a focus on the formula V=I*Z.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to solve the circuit problem. There are multiple competing views on how to incorporate mutual inductance and whether to use differential equations or impedance directly.

Contextual Notes

Some participants express uncertainty about the correct application of equations and the implications of mutual inductance in their circuit analysis. There are unresolved mathematical steps and assumptions regarding the circuit configuration.

berryberry
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I am currently revising over some past papers and this question has come up which I am having problems with. My notes don't show how to solve this and I don't have any worked examples.

So far from looking on the internet, I have seen two methods. The first is using an equation for energy and the second is to use KVL for each coil and then substituting the equations.

I've tried to understand using both of these methods but I still can't apply to this question.

http://img714.imageshack.us/img714/3693/elecprob.jpg

Uploaded with ImageShack.us

If anyone could give me pointers on what I need to be doing, or a brief example of how it's done, it would be greatly appreciated as I'm having a hard time getting my head around it.

Thanks.
 
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How might you adjust the circuit diagram to incorporate the effects of the mutual inductance (the j3Ω indicated above the inductors) into each loop?
 
Would that be to convert it into a T-Circuit?
 
berryberry said:
Would that be to convert it into a T-Circuit?

Nope. The mutual inductance acts in each loop as though it were a current-controlled dependent voltage source in series with that loops inductor, with the control currents being the currents in the other loop.

For example, if M is the mutual inductance, i1 the current in the first loop, and i2 the current in the second loop, then in the first loop you could insert a voltage source in series with the inductor with a value M*i2. The same can be done for the second loop with the control current being i1.

This manipulation of the circuit diagram should allow you to write the KVL loop equations for each loop in a straightforward fashion.
 
If I'm understanding you correctly, that it is to use these equations (with the inclusion of R*I for the resistors)...
http://img189.imageshack.us/img189/750/unled1dg.jpg

Uploaded with ImageShack.us

If so what I have done so far is...

100/_0 = 10I1 + j10 dI1/dt + j3 dI2/dt
50/_45 = 5I2 +j5 dI2/dt + j3 dI1/dt

Although I'm not quite sure on how I would rearrange this, if this is the corrent way forward.
 
Last edited by a moderator:
berryberry said:
If I'm understanding you correctly, that it is to use these equations (with the inclusion of R*I for the resistors)...
http://img189.imageshack.us/img189/750/unled1dg.jpg

Uploaded with ImageShack.us

If so what I have done so far is...

100/_0 = 10I1 + j10 dI1/dt + j3 dI2/dt
50/_45 = 5I2 +j5 dI2/dt + j3 dI1/dt

Although I'm not quite sure on how I would rearrange this, if this is the corrent way forward.

Yes, you're on the right track. Since you're dealing with sinusoidal sources of the same frequency and you're given the impedances of the reactive components (the inductances), you can dispense with the differential form and use the impedances directly.

You'll have two equations in two unknowns (the currents).
 
Last edited by a moderator:
gneill said:
(the inductances), you can dispense with the differential form and use the impedances directly.

I'm not sure what you mean by this.

I assume replacing di/dt for just I?
 
berryberry said:
I'm not sure what you mean by this.

I assume replacing di/dt for just I?

Yup.
 
Impedance form: V=I*Z where Z=j*omega*M
 

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