Advice And Tutorials: Impedance

This, rather long, drawn out, post was written in response to a person questioning the nature -- and validity -- of the impedance matching theorem. This being one of those weekly outpourings of postage I gave it some significant girth as you can see ... thus earning a spot on my website. In case it's too long to be clear, it boils down to the facts that 1. output impedance and power matching are only strictly related for passive systems, and 2. loading a power amplifier at its output impedance generally puts the output devices (distinctly non-passive components!) outside of their design limits, so although it might produce more power, it also generally dissipates more.
A reference is made to Hyperphysics. The original poster started the argument with a misleading drawing seen on the website.

What you are misunderstanding is a basic concept of maximum power transfer vs. output impedance. Power transfer is maximized when the load impedance is equal to the Thevenin source impedance of the signal source. Let me put it this way:

Connect an ideal voltage source V in series with a resistor RI. Now connect a load resistor RL to complete the loop. The current can be expressed as voltage V through the sum resistance, I = V / (RI+RL). The voltage developed across RL is proportional to that current: VL = RL * [V / (RI + RL)] = V*RL / (RI+RL). Power is voltage times amperage, or PL = VL * I = [V / (RI+RL)]*[V*RL / RI+RL)], or

P = V2*RL / (RI+RL)2 .

This says power goes up with the square of voltage, but the rest of the equation isn't easy to figure out by inspection. You could graph the equation with RL as the variable and see how it works, but there's a better way to find what we want. I'll take the derivative ("rate of change") of P with respect to RL. When dP/dRL = 0, we know we are at either a peak or valley in the above equation, because for a small change in RL, there's basically no change in P. Nice, eh?

This is a quotient rule problem so I won't write the steps; suffice it to say,
dP/dRL = V2 * [(RI-RL) / (RI+RL)3]
That has even less meaning, but if we take dP/dRL = 0 and solve for RL, we can find all the peaks and valleys in the original equation. Now this part gets real fun -- I mean it, the answer just plops out in all its elegance:
First take distributive, seperate the +RI and -RL terms (they still have a common denominator):
0 = V2*RI / (RI+RL)3 - V2*RL / (RI+RL)3
Add thee negative term to both sides (move it to the left):
V2*RL / (RI+RL)3 = V2*RI / (RI+RL)3
Now cancel some things. What can we cancel? Heck, almost the entire equation! V2 drops, and the denominator too. We are left with:
RL = RI.

This was found by taking the derivative, which means we used the same RL, RI, etc. as in the original equation. Nothing's changed, so we can go back to it:

P = V2*RL / (RI+RL)2

Since RL = RI for dP/dRL = 0, we know when RL = RI is either a maxima or minima. We don't know which! There are two ways to tell: 1. take the second derivative and determine if it is positive or negative; 2. look at values of P at, less than, and greater than, RL = RI. (Remember RL is the variable and P is a function of RL, i.e., P(RL).)
I don't feel like taking the derivative again so let's try plugging values into P. Not real numbers, we don't need a specific case.. let's just set RL to some logical values. First of all, we need to know P at RL = RI. This comes to P = V2 / 4RI. First of all, power and resistance cannot be negative because such would be meaningless. Second, even if voltage were negative, V2 is always positive. This tells us P(RI) (i.e., RL = RI) is positive, so to prove it is a minima, we need to find values greater than V2 / 4R.
A logical starting point is taking extremes: RL = 0 and ∞ (i.e., short and open circuit). We know from electronics that both of these situations have zero power dissipated, but let's check just to make sure.
P(0) = V2*(0) / (RI + 0)2
Anything times zero is zero, and zero divided by anything is zero. Zero is clearly less than V2 / 4RI, so this end checks out.
For large values of RL, we are tempted to take a specific case and say RL is some large value, say 1Gohm, but how do we know RI isn't a teraohm? There are two ways to handle this: we can let RL equal to some arbitrary multiple of RI, say two times, or let RL approach some insane value, like infinity. Both work: the first, taking RL = 2RI, results in a coefficient of 2/9ths, which is less than the 1/4th coefficient we got for RL = RI. If we take the limit as RL approaches infinity (you can't actually let RL equal infinity because infinity isn't a number), P goes to zero. (Think of it this way: as RL becomes much bigger than RI, the denominator looks like RL squared. This cancels with the RL in the numerator giving V2/RL [for large RL]. For even larger RL, P becomes a very small fraction, and eventually, zero. Since zero at both ends is less than 1/4 V2/R, we can conclude that power is at a maximum at RL = RI.

There. I did all the math, algebra, and even some calculus for 'ya to prove that Pmax is at RL = RI.

What's that you say, so why do power amps have Zo ≠ RL?

Yep, I just spent half an hour talking about ONLY Thevenin equivalent circuits. A power amplifier does not necessarily have an equivalent representative circuit inside it. In fact they rarely do (which does at least make Hyperphysics' diagram erroneous).

So what then? You have to understand the difference between maximum power and output impedance. So what is output impedance? It's the exact electrical equivalent of me putting on a policeman's clothes. Just because I look the part doesn't mean I have any legal power over you... In the same way, you can adjust any *active* electrical circuit to appear as as miliohm output impedance, but that doesn't mean you'll be pulling kiloamperes from it when you short the output!

Amplifiers also have to worry about efficiency. Take this example: say you generate a square wave with a pair of MOSFETs. MOSFETs are essentially electrical switches with a well-defined ON resistance. Say the squarewave is 10V tall and the FETs are one ohm each. If you designed the circuit for 90% efficiency, then you intended that the load be about 9 ohms -- 10V aross 9+1 ohms is 1 ampere, which is 9V across the output and 1V across whichever FET is turned on, and likewise, 9W output, 1W dissipated in the FET, hence 90% efficiency. This current situation has damping factor = 9, 9W power output and 90% efficiency. If you reduce load to 1 ohm, it matches the Thevenin resistance of the generator and the 10V supply is shared evenly across both: 5V in the FETs, 5V on the load. This makes 5 amps, which is 25W -- far more power than before -- but also puts 25W into the FETs, astronomically more than the *1W* that was chosen at design time!

In a linear amplifier, you can account for output impedance effects by detecting the loss and applying more signal in proportion. This is negative feedback. You still have the same old amplifier, so Zo can go stone dead zero perfection, but you're still left with the same limits on voltage and current.

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