Network Maximum Power Transfer Calculation

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Discussion Overview

The discussion revolves around calculating the load impedance for maximum power transfer in a given electrical network, utilizing Thevenin's theorem. Participants explore the implications of frequency on impedance and the application of the Maximum Power Transfer Theorem (MPTT) in the context of AC circuits.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether a given value labeled as E in the problem statement is a printing error, suggesting it should be in volts rather than hertz.
  • Another participant clarifies that impedance depends on frequency, not voltage, and provides the formula for angular frequency.
  • Calculations for Thevenin impedance are presented, with one participant providing specific resistance values and calculations.
  • Concerns are raised about the imaginary component of inductance, with a participant emphasizing the need to consider complex impedance.
  • Participants discuss the application of the MPTT, with some asserting that the load impedance should equal the Thevenin impedance, while others argue that it involves the complex conjugate of the source impedance.
  • Frustration is expressed by some participants regarding the complexity of the problem and the lack of examples that match their specific circuit configuration.
  • There is a suggestion to assume a unit voltage source for simplification, which some participants find confusing.

Areas of Agreement / Disagreement

Participants express various viewpoints on the application of the MPTT, particularly regarding the relationship between load impedance and Thevenin impedance. There is no consensus on the correct approach or interpretation of the problem, and multiple competing views remain throughout the discussion.

Contextual Notes

Participants note the complexity of deriving the MPTT for complex impedances compared to real-valued resistances. There is also mention of the challenge posed by the lack of a specified voltage source in the problem, which complicates the calculations.

Who May Find This Useful

This discussion may be useful for students and individuals studying electrical engineering concepts, particularly those focusing on circuit analysis, Thevenin's theorem, and the Maximum Power Transfer Theorem in AC circuits.

shaltera
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Homework Statement


Determine, for the network shown in fig 4 the value of the load
Impedance that will dissipate the maximum power and the value of
this power.

Homework Equations


Using Thevenin theorem to simplify the circuit then,I believe maximum power transfer theorems apply.

P=I2R
R=√(R2+x2)
I=E/Zt
Zt=z+R

The Attempt at a Solution


Homework Statement



But E is in Hz not in Volts?Do you think it could be a printing error?

Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution

 

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hi shaltera! welcome to pf! :smile:
shaltera said:
But E is in Hz not in Volts?Do you think it could be a printing error?

no

the impedance of a circuit (or of a component) does not depend on the voltage (the emf)

but it does depend on the frequency, ω, doesn't it? :wink:
 
Thevenin Impedance

Thank you for your reply.I calculated Thevenin impedance.Calculation below:

R1=220Ω
R2=1KΩ
R3=150Ω

(R1xR2)/(R1+R2)=(220ΩX1000Ω)/(220Ω+1000Ω)=180.32Ω

Resistance of L1
Zl=2πfL=2xπ50x700x10-3=220Ω (round)

R3+RL1=150Ω+220Ω=370Ω

ZTH=180Ω+370Ω=550Ω

And then?
 
Careful with the inductor. The inductor has impedance, an imaginary value, not a real values resistance. ZL = jωL.
 
How ω can be calculated?There is no ω?
 
shaltera said:
How ω can be calculated?There is no ω?

ω is the angular frequency. ##\omega = 2\pi f##.
 
Thank you
 
ZL = jωL=j2πfL=j220
R3+RL1=150+j220
ZTH=180+150+j220=330+j220
 
OK.I calculated Zth what should I do next?Thank you
 
  • #10
shaltera said:
ZL = jωL=j2πfL=j220
R3+RL1=150+j220
ZTH=180+150+j220=330+j220

Looks good. Don't forget that the units of the impedance is still Ohms, just like resistance.
 
  • #11
Sorry I forgot to add Ω at the end

330+j220Ω
 
  • #12
Okay! So now you either apply the Maximum Power Transfer Theorem, or you try to derive it from scratch. It's simple for real-valued resistances, but it gets a bit "mathy" for complex impedance values. The result is very simple, but getting there is tedious! You might want to do a little investigation on the web to see how its done.
 
  • #13
You mean:

ZL=ZTH
Pmax=|VTH|2/8RTH

Thanks
 
  • #14
Nope ZL is not Zth for maximum power transfer! That's the tricky bit. It's very closely (amazingly, really) related to Zth though...

Do a web search on "Maximum Power Transfer Theorem".
 
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  • #15
Thanks again.I keep looking and looking.All examples have a voltage source.
 
  • #16
shaltera said:
Thanks again.I keep looking and looking.All examples have a voltage source.

A voltage source and a series resistance or impedance. In other words, a Thevenin model...

Check out the Wikipedia entry on the MPTT. It has a section devoted to impedance.
 
  • #17
Searched in Wiki on the MPTT,weird results.I'm fed up now,spent too much time on this silly problem.Thanks anyway
 
  • #19
Yes I did, but examples are with a voltage source.I'm sure you know what you are talking about,its probably me.I really wanted to do it myself but its getting ridiculous and I have to move to Plan B.
 
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  • #20
I started college 4 weeks ago, and they want me to solve this problem,ridiculous. That's why people noways do not remember simple things.Colleges,Uni give so much information for short time.Electrical and Electronic Theory in some countries is divided in 3 part (12 months), in UK in 7 weeks.
 
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  • #21
shaltera said:
Yes I did, but examples are with a voltage source.I'm sure you know what you are talking about,its probably me.I really wanted to do it myself but its getting ridiculous and I have to move to Plan B.

Your circuit also has a voltage source. It's an AC voltage source, only they haven't specified a particular voltage; they've just supplied the frequency of the AC source. The frequency is the important piece of information for this problem since it determines the impedance of any reactive components (inductors or capacitors). You can assume a 1V AC source if you wish.

While the total power depends on the voltage, the fraction of the total that ends up in the load depends upon the source impedance (your Thevenin equivalent) and the load impedance. The source voltage is just a scaling factor. The frequency determines the impedances, and so determines how the power will be split between the source impedance and load impedance.
 
  • #22
what a headache?

I can't find a solution
 
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  • #23
If the problem is just looking for the required load impedance then you are free to use the result of the Maximum Power Transfer Theorem and be done with it. Just state that that is what you used; it's a well known theorem. It's like using KVL or KCL or Ohm's law, which you don't derive every time you use them.

If you are required to derive the MPTT from scratch, that is another matter.
 
  • #24
The maximum power transfer theorem states: A load will receive maximum power from a network when its resistance is exactly equal to the Thevenin resistance of the network applied to the load.

image.jpg
 

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  • #25
Yes, that's true for resistances. There's a bit of a twist when impedances are involved since there are separate real and imaginary quantities that interact through the mathematical operations. The result is still very simple though. Look at the last line of the Wikipedia article...
 
  • #26
I've looked at it,Zl=Zs.Too complicated
 
  • #27
shaltera said:
I've looked at it,Zl=Zs.Too complicated

Not Zs. Zs*. The "*" designates the complex conjugate (which you've used before, I'm sure, when you want to clear complex values from denominators).

Zs is the source impedance: your Thevenin impedance.
 
  • #28
I think is something very easy, but I don't get it.
 
  • #29
shaltera said:
I think is something very easy, but I don't get it.

Is there something in particular that stumping you?

The setup in the Wiki article is identical to that of your simplified circuit: a voltage source with a series impedance driving a load impedance. The only difference is the variable names assigned.
 
  • #30
Just don't get.I give up.
 

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