How Do You Calculate Complex Impedance in AC Circuits?

[scorpius1782]

The circuit and questions are attached.

Homework Statement

First: Calculate the complex impedance of the circuit.
Second:Limiting cases
Third:I_max
Lastly: Phase

Homework Equations

The Attempt at a Solution

I've never had any ac circuits and the DC circuits I did before transferring were extremely simple and more than a couple years ago. So, I'm lost.

I asked a friend for help and he indicated that the initial equation should look something like:

$Z=(\frac{1}{R} -\frac{1}{wCi})^{-1}+iwL$

If this is correct then I believe I have a,b,c covered. Can someone comment to let me know?

Once I have that squared away then for part D am I understanding it correctly that they want that I do this with the real equation instead of the complex one?

 

[scorpius1782]

I've tried figuring this out myself and I get a different impedance than my friend.

$Z=(\frac{1}{R} +\frac{1}{X_C})^{-1}+X_L$
So for the complex impedance I got:
$Z=(\frac{1}{R} +wCi)^{-1}+iwL$

This is where I'm stuck. I'm just not sure which (if either) is correct! If mine is correct I have a lot of algebra to do in order to get it to the form they want.

 

[sandy.bridge]

You have a resistor in parallel with a capacitor. The node connecting these two is in series with the inductor.

$Z_{total} = Z_{C}||Z_R + Z_L$

You have to be careful when determining the impedance of the parallel network due to the imaginary number associated with the capacitor.

 

[scorpius1782]

I've calculated impedance using the first equation I was told was correct and now the one I've come up with. I don't understand what you're trying to tell me though. Like I said, I've done only RC circuits and years ago so I'm entirely lost with this.

 

[sandy.bridge]

I've tried figuring this out myself and I get a different impedance than my friend.

$Z=(\frac{1}{R} +\frac{1}{X_C})^{-1}+X_L$
So for the complex impedance I got:
$Z=(\frac{1}{R} +wCi)^{-1}+iwL$

This is where I'm stuck. I'm just not sure which (if either) is correct! If mine is correct I have a lot of algebra to do in order to get it to the form they want.

The bottom one is correct.

 

[scorpius1782]

Thank goodness! Glad I didn't waste my time redoing it. I thought the first one looked fishy. I separated the equation so that its in the format they asked for. It's messy but wolfram got the same:

$Z=\frac{1}{C^2Rw^2+\frac{1}{R}}+i(Lw-\frac{Cw}{C^2w^2+\frac{1}{R^2}})$

You asked in the other thread setting up the complex impedance for figuring out the phase. I do not know what the relationship is between the two.

Thank you for your help. Wish I could have started school here instead of transferring. I'm behind in every class it seems.