Maximum Power Transfer with Negative Thevenin Resistance?

gta100

Hello everyone! I have a general question about how to find out the maximum power transfer to a load resistor with a negative Thevenin Resistance (Rth < 0). I know for positive cases, P = R*(Vth/(R + Rth))^2, in which R is the load resistor connected to the Thevenin equivalence circuit, and Pmax = (Vth)^2/(4*Rth). But what if Rth < 0? I don't think that the Pmax equation still holds true in negative cases for Rth. Can someone help me explain this phenomenon please? I very much appreciate your help! Thank you.

Sincerely,
gta100

berkeman

Welcome to the PF.

What is a negative Thevenin resistance? Is it physical? Or is it related to some complex impedance (in which case there is a good answer)?

gta100

Hello berkeman! Thank you very much for your reply!

By negative Thevenin resistance, I mean Rth < 0 in the Thevenin equivalent circuit. According to my textbook, negative Rth may appear when there are ONLY DEPENDENT sources inside "the circuit". By "the circuit", I mean the circuitry with CONSTANT PHYSICAL QUANTITIES FOR ALL ELEMENTS and with two terminals connected to a load resistor with a VARYING RESISTANCE (rheostat). I hope that is clear? Thank you!

gta100

berkeman

So to figure this out, we should take a typical circuit that exhibits negative output resistance, and figure out how to get maximum power to a load impedance (by varying the load impedance in this case).

Can you post a typical circuit that exhibits negative resistance at its two output terminals? And then see if you might be able to answer your own question by trying some different resistances connected to those terminals?

gta100

Hi Berkeman! Sure! Let me go ahead and post a sample circuit for you (Please see attachment).

So in this circuit, I find Rth = -15000/11 ohms = - 1363.636364 ohms , and Vth = - 2680/11 V (Also negative). I need to find the maximum power transfer to the rheostat R as indicated.

Thank you for your help!

gta100

Attached Thumbnails

 

rude man

Forgetting Thevenin equivalent reistance for a moment, have you determined i(R) for arbitrary R?

gta100

Hello rude man! Thank you for your reply.

I have determined i(R) for sure. It is Vth/(R + Rth), right? I actually included it in the formula for the power transferred to the load resistor, namely, P = R*(Vth/(R + Rth))^2.

I even plotted the equation in MATLAB, and it turned out that P will reach infinitely high when R = - Rth (Recall that Rth < 0 now, so R = -Rth > 0). So it seems like there is no max power. Could you help me investigate if I calculated the wrong Rth? I attached the sample problem in my last post. Thank you very much!

gta100

Ratch

gta100,

I solved this problem using node analysis, as shown in the attachment. I don't see Thevenin helping very much. Check my node calculations. I show that when R=1363.636364, then the power dissipated in R is 2.962740001E21, a very large number. Included is a plot of the power dissipated.

Ratch

Attached Thumbnails

 

uart

Yes, the correct value of [itex]R_{thv}=[/itex] is [itex]-\frac{15}{11}k \Omega[/itex]. (Hand calculated the good old fashioned way. )

Yes the maximum power is unlimited (infinity if you like), as R_L approaches -R_thv.

The direction that Berkeman was trying to point you toward was to consider what type of real circuit components can produce a negative resistance. When you do this you'll find that in all cases the negative resistance is a result of some active component that is only valid over a limited operating range. There is no such thing as a true (or intrinsic) negative resistance.

Common examples of negative resistances are.

1. Small signal linearized examples like the tunnel diode, where both V and I are actually in the same direction, but [itex]dV/dI[/itex] is negative over some limited range of its V/I characteristics.

2. Active circuits using opamps and transistors. (remember that these always have a limited linear operating range).

3. Electro-mechanical systems like motors and generators. Again these will always be subject to a finite operating range.

gta100

Hello uart! Thank you very much for your reply! That is very inspirational and reassuring! Thank you for quoting some real-world examples which would produce negative Thevenin resistances!

gta100

Ratch

uart,

Did you check your negative resistance value by finding the power dissipated by R at that negative value? If you did, you would find that the power is a small negative value, which means that mathematically, R is generating power at that negative value. However, at R=1363.636364, the power dissipated is 2.962740001E21, not infinity. See the power plot I posted above.

Ratch

uart

Hi Ratch. Your results are simply an artifact of the simulation software only evaluating the response at a finite number of discrete values of R.

For any circuit that has a negative Thevenins resistance (and a non zero Thevenins voltage) it is easy to see that if you make the external resistance equal to the absolute value of the Thevenins resistance that the total resistance becomes zero. The current, and therefore the load power, must be infinite.

As I explained to gta100 however, in reality this can never happen, as the active components forming the negative resistance will limit or saturate at some finite level.

Ratch

uart,

I did not use simulation software. As shown in post #8 of this thread, I found the voltage across R by using node analysis. Then using the well known formula V^2/R, I plotted and calculated the R value at the highest power dissipation. That occurs when R is positive, not negative. Anytime R is negative, the resistor is not dissipating power, because V^2/R will always be negative. I stand by my answer of R=1363.636364, and again invite you to calculate the power the resistor dissipates at your value and mine.

Ratch

uart

Ok your plotting software then.

What are you talking about, they're basically the same value, you just had a decimal approximation. I said to use R = -R_thv, which is positive 15/11 k-ohm. As a decimal this is [itex]1363.\dot{6}\dot{3}\, \Omega[/itex] (that's the .63 repeated).

Ratch

uart,

My plotting software does "evalute the response at a finite number of discrete values of R". However, that is not what I used to determine the value of R.

I don't know about the Thevenin value, but if you agree that R is positive, then the matter is resolved.

Ratch

uart

I do know about the Thevenin value, and I also know that -(-15/11) is a positive number. So yes this is resolved.

uart

Ratch. Are you honestly saying that you cannot see that the maximum value of,
[tex]\frac{2.173 R}{(0.00605R - 8.25)^2}[/tex]
increases without bound as R approaches [itex]\frac{8.25}{0.00605}[/itex]?

* Equation taken from your own working (with factor of [itex]10^{24}[/itex] removed top and bottom).