What Is the Correct Source Voltage Amplitude for Desired Power in an AC Circuit?

[superslow991]

Homework Statement

An ac series circuit consists of a voltage source of frequency f = 60 Hz and voltage amplitude V, a resistor of resistance R = 163 ohms and a capacitor of capacitance C = 6.2x10^-6 F. What must the source voltage amplitude V be for the average electrical power consumed in the resistor to be 529 watts? There is no inductance in the circuit.

Homework Equations

XC = 1 / (2 × π × f × C)
Z = √(R2 + X2)
P=V^2/R

The Attempt at a Solution

So first i calculated the reactance and got 1/(2*π*60*6.2*10^-6)= 428.05
next impedance-√(163^2 + 428.05^2)= 458.0
last i used the power equation - 529=v^2/458.0 and solved for v and got 492.2 V

My only question is I've checked different answers online and saw some as V = 1165 or 1166

just wondering if i did the math right on this question. Any help is appreciated.

 

[cnh1995]

for the average electrical power consumed in the resistor to be 529 watts

You are asked to find the active power in the circuit, which is dissipated in the resistor only.
You have calculated the apparent power, which is the sum of active and reactive powers.

What is the power factor of this circuit? How will you find the active power with the help of power factor?

 

[superslow991]

You are asked to find the active power in the circuit, which is dissipated in the resistor only.
You have calculated the apparent power, which is the sum of active and reactive powers.

What is the power factor of this circuit? How will you find the active power with the help of power factor?

Hmm isn't the power factor R/Z? Also after reading the question more wouldn't I used the average power equation?

 

[cnh1995]

Hmm isn't the power factor R/Z?

Yes.
So what is the expression for active power now that you know the apparent power and the power factor?

 

[superslow991]

Yes.
So what is the expression for active power now that you know the apparent power and the power factor?

Have no clue

P=Z*R? That wouldn't make sense I think

 

[cnh1995]

Have a read.
http://googleweblight.com/i?u=http:...r-in-ac-circuits.html&grqid=lrSto1Eh&hl=en-IN

 

[superslow991]

Have a read.
http://googleweblight.com/i?u=http:...r-in-ac-circuits.html&grqid=lrSto1Eh&hl=en-IN

I read it but I'm not seeing anything related to the active power. Only the reactive power Q=IV.

Also are you saying when I solved for the impedance that was solving for the apparent power?

 

[cnh1995]

Only the reactive **apparent** power Q **S**=IV.

Also are you saying when I solved for the impedance that was solving for the apparent power?

Yes.

I'm not seeing anything related to the active power

There is a formula in that article which you can directly use to solve this problem. Read 'instantaneous power in ac circuits' again.
That formula describes the relation among active power P, voltage V, impedance Z and power factor cosθ.

 

[superslow991]

Yes.

There is a formula in that article which you can directly use to solve this problem. Read 'instantaneous power in ac circuits' again.
That formula describes the relation among active power P, voltage V, impedance Z and power factor cosθ.

P=V^2/Z x cos(theta)?

 

[cnh1995]

P=V^2/Z x cos(theta)?

Right.

 

[superslow991]

Right.

So I tried 529=v^2/458 * 163/458 but I don't think that's right am I missing something?

 

[cnh1995]

but I don't think that's right

Why?

 

[superslow991]

Why?

Idk maybe cause some of the answers I saw online but I'll just work with the answer you provided em. Thanks a lot

 

[ehild]

So I tried 529=v^2/458 * 163/458 but I don't think that's right am I missing something?

The power is average power, calculated with the rms value of the AC voltage. P=V_rms²R/Z². V_rms=amplitude/√2. You got the rms voltage of the source, while the problem asked the amplitude of the generator voltage.

 

[cnh1995]

Idk maybe cause some of the answers I saw online but I'll just work with the answer you provided em. Thanks a lot

They are asking for the amplitude of the voltage. You are calculating the rms voltage. As ehild said, multiply it by 1.4142 and verify your answer with the online one.

 

[superslow991]

The power is average power, calculated with the rms value of the AC voltage. P=V_rms²R/Z². V_rms=amplitude/√2. You got the rms voltage of the source, while the problem asked the amplitude of the generator voltage.

They are asking for the amplitude of the voltage. You are calculating the rms voltage. As ehild said, multiply it by 1.4142 and verify your answer with the online one.

Yea got the answer that i saw online, 1166.
Thanks for the help
So to recap first i calculated for the reactance. Then i calculated for the impedence or the apparent power. Then i calculated for the rms of the voltage or the power dissipated through the impedance? Then i solved for the amplitude of the voltage using the equation v(rms)=amplitude/√2 and solved for amplitude

 

[cnh1995]

rms of the voltage or the power dissipated through the impedance?

Voltage does not dissipate. Active power dissipates in the resistance only and not in the reactance.

 

[superslow991]

Voltage does not dissipate. Active power dissipates in the resistance only and not in the reactance.

hmm ok so what exactly could you call the equation P=V^2/Z x cos(theta)? Apparent power dissipated through resistance? but what about the impedance?

 

[cnh1995]

Apparent power dissipated through resistance? but what about the impedance?

No.
Apparent power has two components: Active power and reactive power. Active power is dissipated in the resistive elements only.
Reactive power is associated with the reactive elements (inductance and capacitance). It oscillates between the source and the load.

 

[ehild]

hmm ok so what exactly could you call the equation P=V^2/Z x cos(theta)? Apparent power dissipated through resistance? but what about the impedance?

Power is dissipated (transforms to heat) on the resistors only. The equivalent complex impedance seen by the source has a real and an imaginary part:: Z=R+iX. The magnitude of the impedance is $Z=\sqrt{R^2+X^2}$. R=Zcos(theta), X=Zsin(theta), and the magnitude of the current is I=U/Z, where U is the voltage of the source. When calculating power, we use the rms value of both the voltage and current.
The power dissipated in the circuit is that dissipated on its equivalent resistance: the square of the rms current, multiplied by R:
$P=I^2 R=\left(\frac{U}{Z}\right)^2 R$. As R/Z=cos(θ), you can write $P=\frac{U^2}{Z}\cos(\theta)$.