What is the impedance of the circuit?

[Sean1218]

Homework Statement

A circuit contains two elements, but it is not known if they are L, R or C. The current in this circuit when connected to a 51.0 V, 60.0 Hz source is 3.60 A and leads the voltage by 75.0 degrees.

What is the impedance of this circuit?

Homework Equations

Z=sqrt(R^2 + (Xc)^2)
tan(phi)=Xc/R
Xc=1/(2pifC)
I=E/Z

The Attempt at a Solution

I figured it contains R and C because current leads voltage, so there's no Xl.

For a) Z = sqrt(R^2 + (Xc)^2), and I tried tan(phi) = Xc/R to get C for Xc = 1/(2pifC), but didn't get right answer, using phi = -75.0 degrees.

Any help?

 

[gneill]

Why not construct a phasor for the current given the known angle +75°? Put it in complex form and find the impedance Z = V/I as a complex number.

For power, you're given the voltage (no phase shift so it's a simple real number) and now you have the current. There's a "trick" to finding the power with voltage and current in complex form, do you know what it is?

 

[Sean1218]

I found power already with P=Irms*Vrms*cos(phi) if that's the trick you mean.

I'm not sure how to make phaser diagrams (haven't learned it yet), so I was hoping to just do it algebraically.

Why do you say the known angle is +75°, wouldn't it be negative because its leading?

 

[Sean1218]

Could I use cos(phi) = R/Z? I'm not sure if (51.0 V)/(3.6 A) gives me the right resistance though.

Z=R/cos(phi)
Z=(51/3.6)/cos(75 deg)
Z=54.7 Ohm, except this is the incorrect answer I got in the method I outlined in my original post.

edit: or maybe since the given voltage and current values are both rms values, I can just use V/I = Z? Yep, it was V/I = Z.

 

[gneill]

If the current waveform is leading the voltage waveform then the argument of the sine or cosine function that describes the current is "ahead" by the given angle. So add that angle to the ωt. So for example, cos(ωt + θ); Note that when t=0, the argument is θ ahead.

The usual equations like R = V/I work when the V and I magnitude values are rms AND the load is purely resistive; so no reactive components allowed! One can, however, use complex values to represent the voltage and current which incorporate the phase angles, and then you can use usual expressions with complex arithmetic (power, P = VI, is just a bit trickier).

Given the current's magnitude I_mag and angle θ you can create a complex value to represent the current (real and reactive parts):

$I = I_{mag}(cos(\theta) + j\;sin(\theta))$

The voltage waveform is assumed here to have zero phase angle, so it's just a real number (51 V I believe was the given value).

The complex impedance should then be Z = V/I. You can pick out the resistance (real) and reactive (imaginary) parts to determine appropriate component values if you wish.

When you calculate the power using the complex values, use the conjugate value of the current. So P = VI*, where I* is represents the complex conjugate (the conjugate is where the sign of the imaginary component is reversed. If the complex value is A + jB, then the conjugate is A - jB). The components of the result are the real and reactive power.