Transformer & Mesh Current Problem

In summary: SPICE.In summary, the question is asking what transformator in the middle should be used when writing the mesh equations for an upper mesh. The answer is to use the transformer winding that is in phase with the input voltage.
  • #1
Cetullah
31
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Hello people, I ve recently started to solve questions about magnetic coupled circuits, but I am stuck at a question that I ve seen for first time. I want to write the mesh equations but I simply don't know what should I do with the transformator in the middle when writing the equation for upper mesh. I seek your guidance, thank you.

The question is in the attachment.
 

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  • #2
Cetullah said:
Hello people, I ve recently started to solve questions about magnetic coupled circuits, but I am stuck at a question that I ve seen for first time. I want to write the mesh equations but I simply don't know what should I do with the transformator in the middle when writing the equation for upper mesh. I seek your guidance, thank you.

The question is in the attachment.

Do you know the AC transfer function for a transformer? Usually the turns ratio will be listed, but I'm guessing that is just a 1:1 transformer. Do you understand the meaning of the two dots shown on the transformer?
 
  • #3
What do the dots mean? Is it CW vs CCW windings?
Positive becomes negative and negative becomes positive?
 
  • #4
tarakan said:
What do the dots mean? Is it CW vs CCW windings?
Positive becomes negative and negative becomes positive?

Yes, the dots indicate the polarity of the input and output AC voltages. If the dots are both at the top, then the output voltage waveform is in phase with the input voltage waveform. If the dots are on opposite ends like in this figure, then the AC output voltage waveform is 180 degrees out of phase with the input voltage waveform.

Can you take a cut at the mesh KVL or node KCL equations now...?
 
  • #5
I would use a SPICE simulator for that, personally.
Even if I know how to solve for resistor grids, what resistance do transformer windings offer?
What is the shift in current /voltage in this 'ideal' transformer?
Thanks.
 
  • #6
Hello, it is seen on the picture that its 1:2. :) And I didnt understand what you meant with transfer function :( You mean those stuff with complex variables (s)? Or is there a formula?
 
  • #7
Cetullah said:
Hello, it is seen on the picture that its 1:2. :) And I didnt understand what you meant with transfer function :( You mean those stuff with complex variables (s)? Or is there a formula?

Ah, I missed the 1:2. By transfer function, I just mean how the AC voltage makes it through an ideal transformer. Does the amplitude change and phase inversion make sense to you? Can you start writing the equations now?
 
  • #8
Cetullah said:
I want to write the mesh equations but I simply don't know what should I do with the transformator in the middle when writing the equation for upper mesh.

Don't use the upper "mesh". Use the loop comprised of the periphery of the circuit, along with the left and right meshes.

Also, you probably will need to assume that the two transformer windings are perfectly coupled so that the mutual inductance, m, is just SQRT(L*4L) = 2L, where L is the inductance of the left winding and 4L is the inductance of the right winding. It may be reasonable to assume that the DC resistance of both windings is zero.
 
  • #9
Cetullah said:
Hello people, I ve recently started to solve questions about magnetic coupled circuits, but I am stuck at a question that I ve seen for first time. I want to write the mesh equations but I simply don't know what should I do with the transformator in the middle when writing the equation for upper mesh. I seek your guidance, thank you.

The question is in the attachment.

Assume the transformer is ideal, then L and 4L don't enter the picture. Just assume the secondary voltage is 2x the primary voltage and the secondary current is 1/2 the primary current. The rest should be straightforward. As berkeman pointed out in post #2, observe the polarity due to the dots.
 
  • #10
rude man said:
Just assume the secondary voltage is 2x the primary voltage and the secondary current is 1/2 the primary current.


The OP said "I want to write the mesh equations..."

Your recommended assumptions can certainly lead to an ad hoc solution with the additional assumption that the transformer is ideal, but how can they be included in mesh equations such that all the equations remain mesh (or loop) equations?
 
  • #11
Can this circuit be represented like this at avrious stages of the AC cycle?

We are solving for RMS? "Rectified" voltage on that resistor. Am I correct?

AC on the primary and on the secondary of a transformer is in phase?

http://en.wikipedia.org/wiki/Transformer: [Broken]
"With sinusoidal supply, core flux lags the induced emf by 90°."
Is this part of the ideal model? Exactly 90° or is it ignored?

Does this cause transformers to shift phase ≈90° between primary and secondary?

Transformer_Analog.jpg


I was taught to draw a DC analogs for AC circuits.
I am not an expert, I had a similar problem 5 years ago in college and failed an exam because I never solved it.
 
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  • #12
The Electrician said:
The OP said "I want to write the mesh equations..."

Your recommended assumptions can certainly lead to an ad hoc solution with the additional assumption that the transformer is ideal, but how can they be included in mesh equations such that all the equations remain mesh (or loop) equations?

Why 'ad hoc'?

I would go with summing currents to zero at 3 independent nodes. 3 equations, 3 unknowns plus the transformer voltage and current relations I gave. In fact, I already have. The result for the output voltage is quite elaborate, but I don't think any other method would be better.

I see no impediment to writing 3 mesh equations if that is what is insisted upon. Since the OP did not label his resistors I can't describe them readily but they should be obvious: 1 includes the primary, one around 3 outside resistors, and the third includes the secondary. Wouldn't do it that way myself & don't see why the OP should either.
 
  • #13
tarakan said:
Can this circuit be represented like this at avrious stages of the AC cycle?

We are solving for RMS? "Rectified" voltage on that resistor. Am I correct?

AC on the primary and on the secondary of a transformer is in phase?

http://en.wikipedia.org/wiki/Transformer: [Broken]
"With sinusoidal supply, core flux lags the induced emf by 90°."
Is this part of the ideal model? Exactly 90° or is it ignored?

Does this cause transformers to shift phase ≈90° between primary and secondary?

Transformer_Analog.jpg


I was taught to draw a DC analogs for AC circuits.
I am not an expert, I had a similar problem 5 years ago in college and failed an exam because I never solved it.

Never heard of drawing dc analogs. Sounds dangerous! Certainly would not work if a non-ideal transformer is assumed.

Anway there is no phase shift betw. primary and secondary voltage or current except possibly 180 deg. Primary and secondary inductances are assumed infinite, leakage fluxes and lead resistances zero. Coupling is unity.
 
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  • #14
rude man said:
Never heard of drawing dc analogs. Sounds dangerous! Certainly would not work if a non-ideal transformer is assumed.

Anway there is no phase shift betw. primary and secondary voltage or current except possibly 180 deg. Primary and secondary inductances are assumed infinite, leakage fluxes and lead resistances zero. Coupling is unity.

So can I use this model to solve a problem that Cetullah posted?
 
  • #15
tarakan said:
So can I use this model to solve a problem that Cetullah posted?

You mean assume an ideal transformer I hope? As I said, I'd forget any dc analog model asap.
Assume an ideal transformer unless someone tells you differently.

You can use KVL or KCL or any other valid method of analysis.
 
  • #16
But I don't know how to treat the 8 ohm resistor that this problem is about. It is neither part of primary nor the secondary circuit.
 
  • #17
Treat it like any other component. It's part of the circuit, what's the problem?

The transformer is just a device where output current is 1/N times the input current and output voltage is N times the input voltage. In your case N=2. That is all you need to solve the equations, whether you use KCL, KVL or whatever (I personally always sum currents at each independent node to zero).
 
  • #18
How can I sum currents in every node if there are two different current sources and they are tied with a resistor that has effect on both nodes? To sum currents to zero they all need a sign (+ or -). I don't understand this either.

I took a class on this to get a piece of paper that determines one's social status, but I had no time to learn subjects in depth while in college.

Thank you for trying to help me, Rude Man!
 
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  • #19
OK, I'm going to give you one nodal equation, the one at the transformer primary V1.

Let

V1 = xfmr. primary high (dotted end)
V2 = xfmr. secondary high (not the dotted end)
V3 = right-hand node (3 resistors meet)
i1 = current flowing into xfmr primary dotted end
i2 = current flowing into xfmr secondary dotted end
Then you can write 5 equations in 5 unknowns: V1, V2, V3, i1 and i2.
Here's the first one for you:

(60 - V1)/4Ω = I1 + (V1 - V3)/8Ω

This is a KVL equation or close to it (I don't pay attention to nomenclature; I I just sum currents to zero at each independent node). Now come up with the other four. Remember what I said about the relationship betw. i1/i2 and V1/V2 for an ideal transformer; that's 2 equations.
 
  • #20
V1 = {I1 + [( (V1-V3)/8Ω)*4Ω)-60} * (-1)

V2 = (-2) * {I1 + [( (V1-V3)/8Ω)*4Ω)-60}

I1 = ( (V1-V3)/8Ω) *4Ω -60+V1

I2 = { ( (V1-V3)/8Ω) *4Ω -60+V1 }/2

I will finish V3 tomorrow. Than I solve a system of equations by substitution and it is solved?

Thank you.
 
  • #21
tarakan said:
V1 = {I1 + [( (V1-V3)/8Ω)*4Ω)-60} * (-1)

V2 = (-2) * {I1 + [( (V1-V3)/8Ω)*4Ω)-60}

I1 = ( (V1-V3)/8Ω) *4Ω -60+V1

I2 = { ( (V1-V3)/8Ω) *4Ω -60+V1 }/2

I will finish V3 tomorrow. Than I solve a system of equations by substitution and it is solved?

Thank you.

That's the idea. You can substitute I2 = 0.5 I1 and V2 = -2 V1 to reduce your equations to a more manageable three (in V1, V3 and I1 only). Use Wolfram Alpha or other software to avoid math mistakes. That way you don't need to rewrite all your equations for V2, I1, I2 etc. the way you did, thereby avoiding math errors. The idea is to concentrate on making sure you have the original equations correct, then leave the math to software.

(Check I2. I2 goes into the dotted end so the current going into the top end is -I2.)
 
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  • #22
rude man said:
Let

V1 = xfmr. primary high (dotted end)
V2 = xfmr. secondary high (not the dotted end)
V3 = right-hand node (3 resistors meet)
i1 = current flowing into xfmr primary dotted end
i2 = current flowing into xfmr secondary dotted end
Then you can write 5 equations in 5 unknowns: V1, V2, V3, i1 and i2.

Why solve for 5 unknowns? The problem is to solve for the voltage across the 8Ω resistor so all that is needed is V1 and V3. Two equations in two unknowns would seem to be sufficient.
 
  • #23
The Electrician said:
Why solve for 5 unknowns? The problem is to solve for the voltage across the 8Ω resistor so all that is needed is V1 and V3. Two equations in two unknowns would seem to be sufficient.

I didn't say to solve for 5 unknowns. I said to solve 5 equations and obviously if you only need V3 then that's the only one you solve for.

The equations for 5 unknowns are there whether you like them or not. You cannot solve a 5-variable problem with 2 equations.
 
  • #24
rude man said:
I didn't say to solve for 5 unknowns. I said to solve 5 equations and obviously if you only need V3 then that's the only one you solve for.

The equations for 5 unknowns are there whether you like them or not. You cannot solve a 5-variable problem with 2 equations.

I know that.

Thank you for helping me.

Transformer winding 1 has no resistance or infinite resistance?
What is the current flowing through?
Thanks.
 
  • #25
If I short the secondary, the current through the short will be infinite even though it is 1/2 of the primary current?
Thanks.
This is my last question because I still don't understand what I am doing.
 
  • #26
rude man said:
I see no impediment to writing 3 mesh equations if that is what is insisted upon. Since the OP did not label his resistors I can't describe them readily but they should be obvious: 1 includes the primary, one around 3 outside resistors, and the third includes the secondary. Wouldn't do it that way myself & don't see why the OP should either.

Just because we can't see why the OP would ask for mesh equations doesn't mean that there isn't a good reason for doing so.

I've seen circuit analysis problems on forums where the student is required to use a particular method for solution, where that method is plainly the worst possible method to use. The instructor is presumably requiring the method to give practice.

When someone posts with a particular request, it seems appropriate to be responsive to that request if possible, even if we don't know their reasons for the request. That doesn't mean that a responder can't offer other methods, but why not be responsive to the particular request?

The OP's problem appears to have an ideal transformer. The method of summing currents at nodes works for this case, but will not work well for a non-ideal transformer. Perhaps the OP's problem was given as a prelude to more general problems with non-ideal transformers, where mesh solutions are the best approach. Showing the OP how to solve this problem (with its ideal transformer) using mesh equations would be helping him prepare for the more general problems with non-ideal transformers.
 
  • #27
tarakan said:
Transformer winding 1 has no resistance or infinite resistance?
What is the current flowing through?
Thanks.

Ideal transformers have:

zero winding resistance
zero leakage flux
infinite winding inductances
unity coupling ratio (all the primary flux cuts the secondary also)
V2/V1 = N
I2/I1 = 1/N
N = ratio of secondary to primary winding

I1 is the current flowing into the primary winding, dotted end
I2 is the current flowing itno the secondary winding, dotted end

You don't need to worry about anything except the statements regarding V1, V2, I1 and I2.
 
  • #28
I am a grad student and I am asking out of interest because I forgot all this.
For sum of voltages to work, transformer winding #1 has 0 resistance or infinite resistance?
 
  • #29
Thank you. Zero resistance - that's interesting. Far from what I would consider "ideal".
Thank you, Rude Man, Thank you The Electrician.
 
  • #30
tarakan said:
If I short the secondary, the current through the short will be infinite even though it is 1/2 of the primary current?
Thanks.
This is my last question because I still don't understand what I am doing.

For an ideal transformer with a shorted secondary, the primary input impedance would also be zero, so you would be trying to feed a finite voltage into zero impedance which you can't do.

If you fed a constant sinusoidal current into the primary, the secondary current would be 1/N times the primary current. The primary voltage would be zero.
 
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  • #31
So is it (Lesser voltage - Greater voltage) / resistance that separates two voltages of two nodes?

Or what is the logic in terms of what voltage is subtracted from what voltage before the division?
 
  • #32
tarakan said:
So is it (Lesser voltage - Greater voltage) / resistance that separates two voltages of two nodes?

Or what is the logic in terms of what voltage is subtracted from what voltage before the division?

See here:

http://en.wikipedia.org/wiki/Nodal_analysis

for an explanation of why that form is used.
 
  • #33
I wonder if the OP has had his question answered, or if he's even still around?

Cetullah, are you there?
 
  • #34
It would be nice if you give me the right solution so I can study it. I read the wikipedia article and I don't understand anything.
 
  • #35
tarakan said:
It would be nice if you give me the right solution so I can study it. I read the wikipedia article and I don't understand anything.

I don't want to post the full solution unless we're sure that Cetullah no longer wants help with his problem.

Have a look at this reference:

http://www.electronics-tutorials.ws/dccircuits/dcp_6.html

Are you having a problem with the method of writing nodal equations generally, or is it the presence of the transformer that puzzles you?

It might be better to start a new thread for you.
 
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<h2>1. What is a transformer and how does it work?</h2><p>A transformer is an electrical device that is used to transfer electrical energy from one circuit to another through electromagnetic induction. It consists of two or more coils of wire (known as windings) that are linked by a magnetic field. When an alternating current (AC) flows through one of the windings, it creates a changing magnetic field which induces an alternating current in the other winding.</p><h2>2. What is the purpose of a transformer in a circuit?</h2><p>The main purpose of a transformer is to step up or step down the voltage in a circuit. This is important because different electrical devices require different voltages to operate. Transformers are also used to isolate circuits and provide galvanic isolation, which protects against electric shock and interference.</p><h2>3. What is a mesh current and how is it calculated?</h2><p>A mesh current is a current that flows in a closed loop in a circuit. It is used in circuit analysis to determine the current and voltage in each branch of the circuit. Mesh currents are calculated using Kirchhoff's Voltage Law (KVL), which states that the sum of the voltages around a closed loop in a circuit must equal zero.</p><h2>4. How do transformers and mesh currents relate to each other in circuit analysis?</h2><p>In circuit analysis, transformers and mesh currents are used together to determine the voltage and current in a circuit. Transformers are used to step up or step down the voltage, and mesh currents are used to calculate the current in each branch of the circuit. The voltage and current values obtained from the transformer and mesh current analysis can then be used to solve for other circuit parameters.</p><h2>5. What are some common problems encountered when analyzing circuits with transformers and mesh currents?</h2><p>Some common problems encountered when analyzing circuits with transformers and mesh currents include incorrect polarity of windings, incorrect assumptions about the direction of current flow, and incorrect application of KVL. It is important to carefully label the windings and follow the proper steps for solving mesh current equations to avoid these issues.</p>

1. What is a transformer and how does it work?

A transformer is an electrical device that is used to transfer electrical energy from one circuit to another through electromagnetic induction. It consists of two or more coils of wire (known as windings) that are linked by a magnetic field. When an alternating current (AC) flows through one of the windings, it creates a changing magnetic field which induces an alternating current in the other winding.

2. What is the purpose of a transformer in a circuit?

The main purpose of a transformer is to step up or step down the voltage in a circuit. This is important because different electrical devices require different voltages to operate. Transformers are also used to isolate circuits and provide galvanic isolation, which protects against electric shock and interference.

3. What is a mesh current and how is it calculated?

A mesh current is a current that flows in a closed loop in a circuit. It is used in circuit analysis to determine the current and voltage in each branch of the circuit. Mesh currents are calculated using Kirchhoff's Voltage Law (KVL), which states that the sum of the voltages around a closed loop in a circuit must equal zero.

4. How do transformers and mesh currents relate to each other in circuit analysis?

In circuit analysis, transformers and mesh currents are used together to determine the voltage and current in a circuit. Transformers are used to step up or step down the voltage, and mesh currents are used to calculate the current in each branch of the circuit. The voltage and current values obtained from the transformer and mesh current analysis can then be used to solve for other circuit parameters.

5. What are some common problems encountered when analyzing circuits with transformers and mesh currents?

Some common problems encountered when analyzing circuits with transformers and mesh currents include incorrect polarity of windings, incorrect assumptions about the direction of current flow, and incorrect application of KVL. It is important to carefully label the windings and follow the proper steps for solving mesh current equations to avoid these issues.

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