Network Maximum Power Transfer Calculation

[shaltera]

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

 

[tiny-tim]

hi shaltera! welcome to pf!

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?

 

[shaltera]

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
Z_l=2πfL=2xπ50x700x10^-3=220Ω (round)

R₃+RL₁=150Ω+220Ω=370Ω

Z_TH=180Ω+370Ω=550Ω

And then?

 

[gneill]

Careful with the inductor. The inductor has impedance, an imaginary value, not a real values resistance. Z_L = jωL.

 

[shaltera]

How ω can be calculated?There is no ω?

 

[gneill]

How ω can be calculated?There is no ω?

ω is the angular frequency. $\omega = 2\pi f$.

 

[shaltera]

Thank you

 

[shaltera]

ZL = jωL=j2πfL=j220
R3+RL1=150+j220
ZTH=180+150+j220=330+j220

 

[shaltera]

OK.I calculated Zth what should I do next?Thank you

 

[gneill]

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.

 

[shaltera]

Sorry I forgot to add Ω at the end

330+j220Ω

 

[gneill]

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.

 

[shaltera]

You mean:

Z_L=Z_TH
P_max=|V_TH|²/8R_TH

Thanks

 

[gneill]

Nope Z_L is not Z_th for maximum power transfer! That's the tricky bit. It's very closely (amazingly, really) related to Z_th though...

Do a web search on "Maximum Power Transfer Theorem".

 

[shaltera]

Thanks again.I keep looking and looking.All examples have a voltage source.

 

[gneill]

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.

 

[shaltera]

Searched in Wiki on the MPTT,weird results.I'm fed up now,spent too much time on this silly problem.Thanks anyway

 

[gneill]

Searched in Wiki on the MPTT,weird results.I'm fed up now,spent too much time on this silly problem.Thanks anyway

Did you come across this URL?

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

 

[shaltera]

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.

 

[shaltera]

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.

 

[gneill]

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.

 

[shaltera]

what a headache?

I can't find a solution

 

[gneill]

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.

 

[shaltera]

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.

 

[gneill]

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...

 

[shaltera]

I've looked at it,Zl=Zs.Too complicated

 

[gneill]

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.

 

[shaltera]

I think is something very easy, but I don't get it.

 

[gneill]

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.

 

[shaltera]

Just don't get.I give up.

 

[shaltera]

Thanks.And sorry for wasting your time

 

[gneill]

Thanks.And sorry for wasting your time

You're welcome. No worries, glad to help (or at least try ) If you want to continue or try again, we'll be here.

 

[shaltera]

Thanks.This problem is already solved..

 

[shaltera]

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.

What do you me by to "assume a 1V AC source"?

E_th=1V?

 

[gneill]

What do you me by to "assume a 1V AC source"?

E_th=1V?

Sure. 1 V @ 50 Hz would do nicely.

If a particular value is not critical to the details of the analysis, it's generally simplest to assume a unit quantity.