Verifying Complex Power Calculations

[eehelp150]

Homework Statement

Calculate the complex power delivered by the source
V = 12cos(wt) V

Homework Equations

V = IR

The Attempt at a Solution

1. I combined 8ohm resistor and 8j ohm inductor in parallel to get 4+4j ohms
2. I combined that with 4ohm resistor in series to get a Zth of 8+4j ohms
3. V = IR -> I = V/R = 12/(8+4j) amps
4. I = 6/5 - 3i/5

Complex power = V * I conjugate =
12 * (6/5 + 3i/5)
= 14.4 + 7.2i

Did I do this right?

 

[cnh1995]

Complex power = V * I conjugate =
12 * (6/5 + 3i/5)
= 14.4 + 7.2i

Hmm..Your calculations are all correct but I am not sure about this one. When we say complex power "absorbed" by the load, we take the conjugate of the current. So for lagging loads, the complex power comes out to be P+jQ, with positive imaginary part, indicating absorption of reactive power. When the load is leading, the complex power "absorbed" by the load becomes P-jQ, where imaginary part is negative, indicating it is supplying reactive power. In your problem, the source is supplying reactive power and the load is absorbing that reactive power. Hence, IMO, complex power supplied by the source should be VI and not VI*. So it should be S=14.4-j7.2.

 

[eehelp150]

Hmm..Your calculations are all correct but I am not sure about this one. When we say complex power "absorbed" by the load, we take the conjugate of the current. So for lagging loads, the complex power comes out to be P+jQ, with positive imaginary part, indicating absorption of reactive power. When the load is leading, the complex power "absorbed" by the load becomes P-jQ, where imaginary part is negative, indicating it is supplying reactive power. In your problem, the source is supplying reactive power and load is absorbing that reactive power. Hence, IMO, complex power supplied by the source should be VI and not VI*. So it should be S=14.4-j7.2.

The current is I = 6/5 - 3i/5
Wouldn't the complex conjugate be 6/5 + 3i/5?

 

[cnh1995]

The current is I = 6/5 - 3i/5
Wouldn't the complex conjugate be 6/5 + 3i/5?

Yes but I think we should not take the conjugate here. The expression S=VI* gives the complex power "absorbed" by the load. The question is asking the complex power supplied by the source. Here, reactive power is being supplied by the source, hence, it should be negative.

 

[gneill]

I'm pretty sure that the question author intended that the power delivered by the source is to be interpreted as being identical to the power "handled" by the load. I'd use the complex conjugate of the source current to find that power, and leave any interpretations of the imaginary component for later.

 

[eehelp150]

Yes but I think we should not take the conjugate here. The expression S=VI* gives the complex power "absorbed" by the load. The question is asking the complex power supplied by the source. Here, reactive power is being supplied by the source, hence, it should be negative.

So source delivers 14.4 - 7.2i and load absorbs 14.4 - 7.2i?

 

[gneill]

You had it right in your first post.

 

[eehelp150]

You had it right in your first post.

How would I know which (conjugate or not) to use on an exam?

 

[cnh1995]

I think you should use the formula VI* as gneill said earlier. It holds for both sending and receiving ends. I think the interpretation of reactive power I wrote is incomplete (or wrong, maybe). I'll search and post if I find any explanation for this.

 

[cnh1995]

I think you should use the formula VI* as gneill said earlier. It holds for both sending and receiving ends. I think the interpretation of reactive power I wrote is incomplete (or wrong, maybe). I'll search and post if I find any explanation for this.

Well, after doing some math, it turns out that the conjugate part has nothing to do with reactive power but it affects the active power P. VI* gives the true active power P dissipated in the circuit. If VI is used, the value of P (real part of complex power) is different from the actual active power dissipated in the circuit. So, always use S=VI*. S=VI is not even a valid expression. My apologies for misguiding you with this equation.

 

[eehelp150]

Well, after doing some math, it turns out that the conjugate part has nothing to do with reactive power but it affects the active power P. VI* gives the true active power P dissipated in the circuit. If VI is used, the value of P (real part of complex power) is different from the actual active power dissipated in the circuit. So, always use S=VI*. S=VI is not even a valid expression. My apologies for misguiding you with this equation.

It's alright, I really appreciate your help!

 

[eehelp150]

You had it right in your first post.

Well, after doing some math, it turns out that the conjugate part has nothing to do with reactive power but it affects the active power P. VI* gives the true active power P dissipated in the circuit. If VI is used, the value of P (real part of complex power) is different from the actual active power dissipated in the circuit. So, always use S=VI*. S=VI is not even a valid expression. My apologies for misguiding you with this equation.

Apparently the correct formula was:
1/2 * S * I*

Any idea why?

 

[cnh1995]

Apparently the correct formula was:
1/2 * S * I*

Any idea why?

Where did you get this formula?

 

[eehelp150]

Where did you get this formula?

Solutions were posted. Solution for this problem is:
S = 1/2 * Vm * Im* = 7.2 + j3.6 VA

I looked through my notes and this was in it:
S = V * I* = 1/2 * Vm * Im<θv-θi = Vrms * Irms <θv-θi

Also in my notes:
Vrms = Vm/sqrt(2)
Irms = Im/sqrt(2)

I'm assuming the voltage source provided (12cos(wt)) was not in RMS, so in order to use S = V * I*, we have to convert it to RMS first? It's weird because my textbook uses the formula S = V * I* and not S = 1/2 * V * I*

I will ask the professor in person but do you have any idea when I'm supposed to used S = V * I* and when I'm supposed to use S = 1/2 V * I*?

 

[cnh1995]

S = 1/2 * Vm * Im*

Ah.. that's right. I missed that the voltage given is peak voltage and not rms. I read only the 12∠0° part and didn't notice V=12coswt written above the diagram, which indicates 12V is actually the peak voltage. So when the voltage equation is given in terms of peak voltage, use

S = 1/2 * Vm * Im*

Or, convert it into rms and then use
S=VI*.

 

[eehelp150]

Ah.. that's right. I missed that the voltage given is peak voltage and not rms. I read only the 12∠0° part and didn't notice V=12coswt written above the diagram, which indicates 12V is actually the peak voltage. So when the voltage equation is given in terms of peak voltage, use

Or, convert it into rms and then use
S=VI*.

So if I were given only the diagram, (12<0 V), should I assume peak or RMS? When the function is given (12cos), should I always assume peak unless it specifically says RMS?

 

[cnh1995]

When the function is given (12cos)

It means the peak voltage is 12V. RMS value is never written in terms of a sinusoidal equation since it's the dc equivalent of the ac waveform. So whenever you see a sinusoidal equation, it gives the peak of the sinusoid. If the voltage is given directly as 12∠0°, it could be rms or peak. It is usually mentioned in the problem whether to read it as rms or peak. But in some of my books, it is assumed to be rms by default.

 

[eehelp150]

It means the peak voltage is 12V. RMS value is never written in terms of a sinusoidal equation since it's the dc equivalent of the ac waveform. So whenever you see a sinusoidal equation, it gives the peak of the sinusoid. If the voltage is given directly as 12∠0°, it could be rms or peak. It is usually mentioned in the problem whether to read it as rms or peak. But in some of my books, it is assumed to be rms by default.

So in other words, if I see the sinusoidal equation, I should always convert to RMS first.

 

[gneill]

So in other words, if I see the sinusoidal equation, I should always convert to RMS first.

If you intend to work with power, yes. Otherwise it's not really an issue, particularly if you're expected to supply an answer for currents or voltages in the same form as given (you'd just end up removing the same scaling factor at the end).