Find load impedance / (Thevenin / Norton too) with RLC & AC circuit?

Click For Summary

Discussion Overview

The discussion revolves around determining the load impedance (ZL) that results in maximum average power transfer in an RLC AC circuit. Participants explore the application of the Maximum Power Transfer Theorem, Thevenin and Norton equivalents, and the calculation of power in the context of the given circuit.

Discussion Character

  • Homework-related
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions whether ZL refers to the impedance through the inductor or the load, suggesting it may involve total impedance including RL and the capacitor.
  • Another participant confirms that the circuit is already in Thevenin form, indicating a voltage source and source impedance.
  • Discussion includes a formula for power transfer, with some participants attempting to derive or confirm the formula used for maximum power.
  • There is uncertainty about the correctness of the derived power expressions, with one participant questioning the inclusion of "1/2" in the formula.
  • Another participant suggests that if the reactances cancel, the power can be expressed in terms of the resistances and source voltage.
  • Participants express varying levels of confidence in their derived formulas and calculations, with some acknowledging reliance on external sources for their expressions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach or formulas for calculating maximum power transfer, with multiple competing views and expressions presented throughout the discussion.

Contextual Notes

There are unresolved questions regarding the derivation of power formulas and the assumptions made about reactance cancellation. Some participants express uncertainty about the formulas they are using, indicating a need for clarity on the underlying principles.

Who May Find This Useful

This discussion may be useful for students studying circuit analysis, particularly those interested in the Maximum Power Transfer Theorem and its application in RLC circuits.

Color_of_Cyan
Messages
386
Reaction score
0

Homework Statement

For the circuit shown, what value of ZL results in maximum average power transfer to ZL? What is the maximum power in milliwatts?

Homework Equations



P max = (VTh)2/4RL

Thevenin, Norton procedures, voltage division, current division, etc etc

The Attempt at a Solution



This is the exact question; kind of confusing if it means find ZL through the inductor or ZL through the load. But I assume it means find the total impedance through the load which includes RL and the capacitor.Really not sure how I would start for this though, so assuming it's the formula above I need to use, then I guess find the Thevenin Resistance here, and even more so for VTh.

Assuming finding RTh is the same way too (ie, if I'm supposed to find the resistance from A to B with the V source shorted and the load cut out),

then would RTh be (3000 + 4000jΩ)? (Or ZTh because it's impedance?). Not sure about what VTh might be either though. Any tips on where to start here would be helpful, thanks.
 
Last edited by a moderator:
Physics news on Phys.org
Looks like a situation where you want to apply the Maximum Power Transfer Theorem. Look it up :smile:
 
This circuit at nodes A and B is already in Thevenin form: a voltage source of 10 V + source impedance of 3 + j4 kOhm...

How would you choose the capacitance to make max power to be delivered to the load ZL = R + jXc? Hint: you want the max current to flow in this circuit, and use the fact that Xc is always negative.
 
So is the formula for it supposed to be

PL = (1/2)|Vs|2(RL)/(Rs + RL)2

XL = -Xs

RL = Rs

?So

Vs = 10V + 0j

Rs = 3000

I take it XL = -j4000ΩSo for the load is it

3000 - j4000Ω ?
 
Do you want us to confirm a guess?
 
My apologies, but I'm trying to find out how this works. It seems way easier than it looks if that's the way then.
 
Color_of_Cyan said:
My apologies, but I'm trying to find out how this works.


It seems way easier than it looks if that's the way then.

It's a matter of understanding the Maximum Power Transfer theorem (MPTT), and in particular, how it applies to impedances (as opposed to simple resistances for the source and load). Once that's done, your "guess" should become a confident statement :smile:

Yes, you want the load reactance to be the negative of the source reactance, because then the current and voltage will be in phase, so I*V will be maximized in the load. And the usual proofs associated with the MPTT tell you what the load resistance should be (that's where real power is delivered: into resistances where power is dissipated rather than stored and returned).
 
Plugging the first formula into find maximum power I got:

PL = (1/2)102*(3000/60002)

PL = 0.00416 W

= 4.16 mW
 
Color_of_Cyan said:
Plugging the first formula into find maximum power I got:

PL = (1/2)102*(3000/60002)

PL = 0.00416 W

= 4.16 mW

The result is not correct. Can you explain the "1/2" in that formula? Perhaps you should try to derive it symbolically.
 
  • #10
I would need help with that. Not sure what you mean, unless it's really the wrong formula.
 
  • #11
Suppose that the reactances of the source and load cancel each other. They effectively disappear from the circuit. You're left with the source and load resistances and the source voltage. Can you write an expression for the power dissipated by the load resistor?
 
  • #12
I'm thinking it's

[V/(RS + RL)]2*RL then?

So

PL =- (10/6000)2*3000

PL = 0.00833W

= 8.33mW
 
  • #13
Color_of_Cyan said:
I'm thinking it's

[V/(RS + RL)]2*RL then?
I get the impression that you're not confident about that expression. Did you find it somewhere or did you derive it for the circuit? Being able to derive required expressions from basic principles is an essential skill for circuit analysis; the circuits are not always so straightforward as the one here, and "canned" formulas just won't be available to memorize.

Anyways, it so happens that your formula above is correct for the present situation :smile:

So

PL =- (10/6000)2*3000

PL = 0.00833W

= 8.33mW

Yup. That's fine.
 
  • #14
Honestly, I got it from somewhere. The first one formula I posted is for average maximum power, I think, and kept to a formula like that because you still said to disregard the reactive impedance. I am looking at it again though and it seems to make sense since P = I2R and I = V/R and the R which is actually the real impedance only is Rs + RL

so P = (V/Rsource+Rload)2*RL. Thank you again anyway though, you have helped a lot.
 

Similar threads

Engineering Calculating Norton Equivalent Circuit for Complex Load Impedance
  • · Replies 2 ·
Replies
2
Views
2K
Engineering How do I find Vth with the node voltage method?
  • · Replies 10 ·
Replies
10
Views
4K
Engineering Thevenin Equivalent of AC RLC Circuit w/ Dependent Voltage
  • · Replies 2 ·
Replies
2
Views
4K
Thevenin's Theorem, Superposition & Norton's Theorem
  • · Replies 28 ·
Replies
28
Views
7K
Engineering How do I find the Thevenin Equivalent of this circuit?
  • · Replies 10 ·
Replies
10
Views
3K
Engineering Find the Thevenin equivalent circuit
  • · Replies 42 ·
2
Replies
42
Views
7K
Engineering Finding the Current Source of an RLC AC Circuit
  • · Replies 2 ·
Replies
2
Views
2K
Engineering Understanding Maximum Power in Circuit: Comparing Load and Circuit Approaches
  • · Replies 7 ·
Replies
7
Views
3K
Engineering Finding Current in a Thevenin Circuit
  • · Replies 5 ·
Replies
5
Views
2K
Engineering Finding the Thevenin Equivalent Circuit for a Network with a Load Resistor
  • · Replies 1 ·
Replies
1
Views
2K