Maximize Average Power Transfer with Impedance ZL - Circuit Analysis Homework

[Color_of_Cyan]

Homework Statement

a) For the circuit shown, determine the impedance Z_L that results in maximum average power transfer to the Z_L

b) What is the maximum average power transferred to the load impedance?

Homework Equations

Thevenin, Norton procedures, voltage division, current division, etc etcMax. average power = 1/2(Vs/Rs + RL)² * RLP_L = (V/(R_s + R_L))²*R_L

R_s = R_L

J_s = -J_L

The Attempt at a Solution

I actually do not know how specifically for here how to get the equivalent impedance for the load, but I assume it's the same as finding Thevenin resistance so...

5Ω || 20Ω = 4Ω 4+3j || -6j =

=(1/4+3j + -1/6j)^-1=[ (6j-4-3j)/(24j-18) ]^-1

= 24j-18 / 3j-4multiplying by conjugate now:

=> * (-3j-4)/(-3j-4)

= (-72j² - 96j + 54j + 72)/(-9j² - 12j + 12j + 16)

= (72 + 72 -42j)/25= 144-42j/25

=5.76 - 1.68j

R_Th = 5.76Ω - 1.68jΩZ_L = 5.76 + 1.68jΩNow

average power = 1/2*(20/(5.76 + 5.76))²*5.76 = 8.68WNot sure if this is right

 

[gneill]

You didn't complete the Thevenin model; the voltage driving the load impedance through the Thevenin impedance will not be the original Vs. I don't get the result you did.

If it were me, rather than memorize another special case formula to determine the power delivered, I'd use the Thevenin equivalent and the determined load impedance to find the current I through the load, then find the complex power delivered as $P = I \cdot I^*\cdot Z_L$, where I* is the complex conjugate of I. The average power is the real component of the complex power.

 

[Color_of_Cyan]

Ok, my bad I flipped out there. When finding VTh you have to forget about RTh, and find parallel impedances all over again.

The load impedance is still the same as R_Th though right? With the imaginary component negated? So for VTh now, the load cut out will put -j6Ω + j3Ω = -j3Ω

Now the circuit's impedance (just for finding V_Th would look like

( -j3Ω || 20Ω) + 5ΩSo ( -j3Ω || 20Ω)

= (1/20Ω - 1/3Ωj)^-1

= [ (3Ωj - 20Ω)/(60Ω²j) ]^-1

= (60Ω²j)/(3Ωj - 20Ω)Multiplying by complex conjugate now

*(-3Ωj - 20Ω)/(-3Ωj - 20Ω)= (-180Ω³j² - 1200jΩ³)/(-9Ω²j² + 400Ω²)

= (180Ω³ - 1200jΩ³)/(9Ω² + 400Ω²)

= (180Ω³ - 1200jΩ³)/(409Ω²)

= 0.44Ω - 2.93Ωj Now (0.44Ω - 2.93Ωj) is in series with 5ΩSo with voltage division:

V_Th = 20∠0 *(0.44 - 2.93j)/(5.44 - 2.93j)

In polar form:

V_Th = 20∠0 *(2.96∠-81.4)/(6.18∠-28.3)

V_Th = 9.57∠-53.1 degSo with this and RTh connected to the load, would the current then be I=V/R, and assuming the load is almost the same as Rth with the imaginary component negated and them both combined now:= (9.57∠-53.1)/(5.76Ω + 1.68Ωj + 5.76Ω - 1.68Ωj),

(Which would then be)

= (9.57∠-53.1)/(11.76∠0)

= (0.81∠-53.1)A ?

How would the complex conjugate be applied to this now?

 

[gneill]

Your approach to finding the Thevenin voltage doesn't make sense to me. You want to find the open circuit potential at terminals a-b, yet your first step seems to "disappear" terminals a and b by incorporating the capacitor's impedance with the inductor's.

One approach to finding the Thevenin equivalent for this sort of layout, which I find useful, is to work from the source forward to the open terminals incorporating the components successively into a Thevenin model. So begin with just the voltage divider presented by the source and two resistors.

Here's a diagram where the first step has been completed:

 

[Color_of_Cyan]

Now I'm kind of confused. I thought you were supposed to disappear / cut out the load to find V_Th? I guess I still have trouble understanding it (or for whatever this case is). Or maybe I just took the voltage divider for the wrong thing? The voltage division above was for the j3, 20, and -j6 simplified together (with the load cut out).

I take it though that a-b only just meant across the -j6 then? If so (and assuming the work itself still isn't wrong) I think I could still find it since I think I still found the Vin for all of that to be

9.57∠-53.1 deg

with the voltage divider

so then I think

= 9.57∠-53.1°*(-6j / -6j + 3j)

= 9.57∠-53.1°*(-6j/-3j)

= 9.57∠-53.1°*(6∠-90° /3∠-90°)

=9.57∠-53.1°*(2∠0°)=(19.14∠-53.1°)V ? Or still all wrong?

 

[gneill]

Now I'm kind of confused. I thought you were supposed to disappear / cut out the load to find V_Th? I guess I still have trouble understanding it (or for whatever this case is).

Yes, remove the load but don't "lose" the connection points for the load -- you want to find the voltage at those open connection points.

Or maybe I just took the voltage divider for the wrong thing?

Could be.

The voltage division above was for the j3, 20, and -j6 simplified together (with the load cut out).

I take it though that a-b only just meant across the -j6 then?

Yes.

If so (and assuming the work itself still isn't wrong) I think I could still find it since I think I still found the Vin for all of that to be

9.57∠-53.1 deg

with the voltage divider

Still doesn't look right.

Let's try going step by step in the manner that I proposed.
What do you get for E1 and Z1 considering only the original source and the resistor divider? (looking towards the source from the point of view (1) ).

 

[Color_of_Cyan]

Not sure what you mean, but I guess you simplify out the 20Ω somehow.

Can you switch the left there to a Norton equivalent and then back again to a Thevenin equivalent with a different voltage than before?

 

[gneill]

Not sure what you mean, but I guess you simplify out the 20Ω somehow.

Can you switch the left there to a Norton equivalent and then back again to a Thevenin equivalent with a different voltage than before?

Norton and Thevenin models are always interchangeable. But really, why would you go to the trouble of finding the Norton equivalent first when you've got such a simple voltage divider?

What's the Thevenin equivalent of the above?

 

[The Electrician]

Now I'm kind of confused. I thought you were supposed to disappear / cut out the load to find V_Th? I guess I still have trouble understanding it (or for whatever this case is).


Or maybe I just took the voltage divider for the wrong thing?


The voltage division above was for the j3, 20, and -j6 simplified together (with the load cut out).

I take it though that a-b only just meant across the -j6 then?


If so (and assuming the work itself still isn't wrong) I think I could still find it since I think I still found the Vin for all of that to be

9.57∠-53.1 deg This is the correct voltage at the junction of the 5Ω, 20Ω and j3Ω, although I get 9.6 rather than 9.57

with the voltage divider

so then I think

= 9.57∠-53.1°*(-6j / -6j + 3j)

= 9.57∠-53.1°*(-6j/-3j)

= 9.57∠-53.1°*(6∠-90° /3∠-90°)

=9.57∠-53.1°*(2∠0°)


=(19.14∠-53.1°)V ? Or still all wrong? This is the correct Vth, although, again, I get 19.20 rather than 19.14

So far, so good.

 

[Color_of_Cyan]

Norton and Thevenin models are always interchangeable. But really, why would you go to the trouble of finding the Norton equivalent first when you've got such a simple voltage divider?

What's the Thevenin equivalent of the above?

Okay I see what you mean now; when finding VTh you can break them all up into other Thevenin models.RTh of only the 20 and 5 with the source is

(1/5 + 1/20)^-1 = 4Ω

and then the VTh of that is

20*(20/25) = (16∠0°)V ?

Would the voltage division for the actual VTh of the problem then be

= (16∠0°)V*(-6jΩ/(4Ω + 3jΩ - 6jΩ) ?

= (16∠0°)V*(-6jΩ/(4Ω - 3jΩ)

= (16∠0°)V*(6∠-90°/5∠-36.8°)

= (19.2 ∠-53.2°)V ? Almost the same as I had originally.

 

[gneill]

Okay I see what you mean now; when finding VTh you can break them all up into other Thevenin models.

Yup. That's the idea.

RTh of only the 20 and 5 with the source is

(1/5 + 1/20)^-1 = 4Ω

and then the VTh of that is

20*(20/25) = (16∠0°)V ?

Yes.

Would the voltage division for the actual VTh of the problem then be

= (16∠0°)V*(-6jΩ/(4Ω + 3jΩ - 6jΩ) ?

= (16∠0°)V*(-6jΩ/(4Ω - 3jΩ)

= (16∠0°)V*(6∠-90°/5∠-36.8°)

= (19.2 ∠-53.2°)V ? Almost the same as I had originally.

Yup. That's right.

 

[Color_of_Cyan]

So

R_Th = 5.76Ω - 1.68jΩ

so Z_L = 5.76Ω + 1.68jΩ

V_Th = (19.2 ∠-53.2°)VAre all of the above connected in series now?If so the current I will be:

=(19.2 ∠-53.2°)V/(5.76Ω - 1.68jΩ + 5.76Ω + 1.68jΩ)

=(19.2 ∠-53.2°)V/(11.52Ω)

=(19.2 ∠-53.2°)V/(11.52Ω ∠0°)

I = (1.69∠-53.2°)A

find the current I through the load, then find the complex power delivered as $P = I \cdot I^*\cdot Z_L$, where I* is the complex conjugate of I. The average power is the real component of the complex power.

Would I have to convert the current back to rectangular form? Or is there a way to bypass that and multiply I with I* in polar form? Would the conjugate of the current just be (1.69∠53.2)A?

 

[gneill]

So

R_Th = 5.76Ω - 1.68jΩ

so Z_L = 5.76Ω + 1.68jΩ

V_Th = (19.2 ∠-53.2°)V


Are all of the above connected in series now?

Yes they are. They form a classic voltage divider with an ideal source, a source impedance, and a load impedance.

If so the current I will be:

=(19.2 ∠-53.2°)V/(5.76Ω - 1.68jΩ + 5.76Ω + 1.68jΩ)

=(19.2 ∠-53.2°)V/(11.52Ω)

=(19.2 ∠-53.2°)V/(11.52Ω ∠0°)

I = (1.69∠-53.2°)A

Okay, that looks pretty good. You might want to remember to hang onto more digits of precision through intermediate steps (round only results for presentation). This is especially true when there are a lot of steps involved, like switching back and forth from rectangular to polar form. The third digit for the magnitude and angle are both off a tad.

Would I have to convert the current back to rectangular form? Or is there a way to bypass that and multiply I with I* in polar form? Would the conjugate of the current just be (1.69∠53.2)A?

Sure. Think about what happens to a phasor on the complex plane when you form the conjugate; since only the imaginary coordinate is negated the phasor is reflected about the real axis. This is
exactly the same as negating the angle in polar form.

 

[Color_of_Cyan]

Okay so

P = (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω + 1.68jΩ)If I'm just finding average power though can you just say

P_avg = (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω), because the 5.76Ω is the real resistance?

So P_avg = 19.4W (or approximately) ?

 

[gneill]

Okay so

P = (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω + 1.68jΩ)

Okay.

If I'm just finding average power though can you just say

P_avg = (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω), because the 5.76Ω is the real resistance?

So P_avg = 19.4W (or approximately) ?

The imaginary parts of all the expression "interact" with the real ones, so you should carry out the full calculation (convert the load impedance to polar form for convenience). The average power will then be the magnitude of the result multiplied by the cosine of the result's angle (that is, the real component of the complex power). Or, if you do the calculation in rectangular form, just pick out the real part of the result for the real (average) power delivered.

 

[Color_of_Cyan]

Ah, thank you so much for that.

= (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω + 1.68jΩ)

= (1.69∠-53.2°)A*(1.69∠53.2°)A*(6∠16.2°)Ω

P = 17.1366∠16.2°

r = (mag)cosθ

P_avg = 17.1366cos16.2

P_avg = 16.45WThanks again

 

[The Electrician]

You need to follow gneill's advice to carry more digits in your computations.

The correct answer is exactly 16 watts.

 

[Color_of_Cyan]

Is there any chance I can know where the error in significant figures is?I just did the whole problem over again and got the exact same answer, the only thing I got different was rounding off the angle for the current, so it was 53.13 instead of 53.2 but I would just cancel that out.

 

[The Electrician]

Is there any chance I can know where the error in significant figures is?


I just did the whole problem over again and got the exact same answer, the only thing I got different was rounding off the angle for the current, so it was 53.13 instead of 53.2 but I would just cancel that out.

In post #12, you have:

"If so the current I will be:

=(19.2 ∠-53.2°)V/(5.76Ω - 1.68jΩ + 5.76Ω + 1.68jΩ)

=(19.2 ∠-53.2°)V/(11.52Ω)

=(19.2 ∠-53.2°)V/(11.52Ω ∠0°)

I = (1.69∠-53.2°)A"

That final result should be I = (1.66667∠-53.13010°)A because 19.2/11.52 is 1.66667, not 1.69, and the angle is slightly different.



Then in post #16, you have:

"Ah, thank you so much for that.

= (1.69∠-53.2°)A*(1.69∠53.2)A*(5.76Ω + 1.68jΩ)

= (1.69∠-53.2°)A*(1.69∠53.2°)A*(6∠16.2°)Ω

P = 17.1366∠16.2°

r = (mag)cosθ

Pavg = 17.1366cos16.2

Pavg = 16.45W"


This should be:

Ah, thank you so much for that.

=(1.66667∠-53.13010°)A*(1.66667∠53.13010°)A*(5.76Ω + 1.68jΩ)

= (1.66667∠-53.13010°)A*(1.66667∠53.13010°)A*(6∠16.26020°)Ω

P = 16.66667∠16.26020°

r = (mag)cosθ

Pavg = 16.66667cos16.26020°

Pavg = 16.00W

 

[Color_of_Cyan]

Thanks again. That's not rounding error but dumb mistakes I keep on making ;/