# Impedance matching derivation (next best after complex conjugate method)

(This is not a question I was given to solve, it is a question about the course notes.)

## Homework Statement

In impedance matching, what is the next best method after the complex conjugate method?
If the source has V_s and Z_s, what should Z_L be?
V_s, Z_s, and Z_L are in series.

## Homework Equations

The best impedance match is when Z_s = (Z_L)*
And this was derived from maximum power transfer, or
P = (I_rms)^2 * R_L = 1/2 * (|V_s|/( |Z_s + Z_L| )) = ...

## The Attempt at a Solution

The notes say
Z_L = R = |Z_s|

This says the next best choice is having Z_L be a resistance equal to the magnitude of Z_s.

However, from the math I did, this assumes that there cannot be a reactance for Z_L.
Why this assumption?

Edit: Why is this the "second-best method"? Why is there the assumption that there is no load reactance?
(I do not have any criteria either. It is from a handout. I will ask the professor after the 1st day of class, I guess.)

Last edited:

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rude man
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Tetraoxygen;4051178[h2 said:
The Attempt at a Solution[/h2]
The notes say
Z_L = R = |Z_s|

This says the next best choice is having Z_L be a resistance equal to the magnitude of Z_s.

However, from the math I did, this assumes that there cannot be a reactance for Z_L.
Why this assumption?
By defintion, |Z_s| is a real quantity, i.e. a resistance, not a complex impedance.

As to what constitutes "second-best" matching impedance I have no idea. No criterion(a) was/were offered.

By defintion, |Z_s| is a real quantity, i.e. a resistance, not a complex impedance.

As to what constitutes "second-best" matching impedance I have no idea. No criterion(a) was/were offered.
I am aware that this method yields a real number.

My question is, why is this the second best matching impedance.

No criteria were offered to me either...

Anyone with real-world experience?

rude man
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OK I think I know.

What is meant is: if a complex load is not permitted, i.e. if it has to be resistive, then the highest power transfer is when Z_L = R = |Z_s|. And that is correct.

(Has to be that: otherwise you could just say, OK, let Z_L = R_s - jX_s + ε where Z_s = R_s + jX_s and ε is an arbitrarily small resistance!)

So: can you proceed with that info? Hint: what is |i|, the current? Then what is power P_L, then how do you determine optimum R_L for max P_L?

OK I think I know.

What is meant is: if a complex load is not permitted, i.e. if it has to be resistive, then the highest power transfer is when Z_L = R = |Z_s|. And that is correct.

(Has to be that: otherwise you could just say, OK, let Z_L = R_s - jX_s + ε where Z_s = R_s + jX_s and ε is an arbitrarily small resistance!)

So: can you proceed with that info? Hint: what is |i|, the current? Then what is power P_L, then how do you determine optimum R_L for max P_L?
I'm sorry, but that is not what I meant.

I want to know Why is a complex load not permitted in the second best method? Why is this the second best method?

I've already done the math showing that this R_L corresponds to maximizing P_L when Z_L is only a resistance, not a reactance.

rude man
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Gold Member
I'm sorry, but that is not what I meant.

I want to know Why is a complex load not permitted in the second best method? Why is this the second best method?

I've already done the math showing that this R_L corresponds to maximizing P_L when Z_L is only a resistance, not a reactance.
There used to be a commercial on TV about a hair dye product that went: "Only your hairdresser knows for sure".

Only your instructor knows for sure.

So the bad news: The professor does not remember any more why this is so. :( He said it was just best practices.

And the better news: He says it may have something to to do with minimizing the reflection coefficient.

Thoughts?

rude man
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