Given RLC circuit to find V over load & other complications

[hdp12]

Homework Statement

In my Intro to EE class we have a homework assignment with the following problem:

I think I finished part a but want to make sure that I am doing the problem correctly before I move on to the next part.

Homework Equations

The Attempt at a Solution

 

[gneill]

You'll want to work with phasors and impedances; You can't mix Ohms and Henries and Farads in your node equations. So first thing to do is determine the impedances of your reactive components (L and C). These will be imaginary values, in Ohms. Then use these in your node equations.

 

[hdp12]

but the given current is in terms of t and impedances are in terms of s or jω
what do I do about that?

 

[gneill]

but the given current is in terms of t and impedances are in terms of s or jω
what do I do about that?

The current is specified as a cosine function of time. It has a frequency, so there's your ω for determining impedances. Its phasor will be just the magnitude of the cosine function since there's no phase shift involved.

 

[hdp12]

Yeah?

 

[gneill]

No, for a couple of reasons. First, you're mixing Laplace domain and time domain quantities in the same equations (the cosine as a function of time along with 's' implied differentiation and integration in the Laplace domain). That's not going to work, and it's really too much mathematical machinery for the problem at hand: you aren't looking for transient and steady state response of the circuit, you're looking for just the steady state response, which is much simpler!

o Start by determining the operating frequency of the circuit: pull ω out of the time domain definition of the source.
o Write the current as a phasor value: For a cosine it's just the magnitude, so it's really simple.
o Use the ω from above and determine the impedances of the reactive components (jω stuff). Write them onto the circuit diagram.
o Write the node equations (or whatever other method you choose to solve for the required values).

 

[hdp12]

What do I do with that i(t)? I think I'm supposed to know based on the phasor but I'm unsure

 

[gneill]

Okay, you're getting there. Replace i(t) on your figure with the phasor current 20 A. Keep in mind that this represents the peak value, not the rms value. Later you'll be asked to find rms values and powers dissipated, so you'll need to remember this. You'll see

Calculate values for the impedances of L and C. You've got the frequency and the component values, so do the calculations. Write those values onto your diagram.

 

[hdp12]

Now what?

 

[gneill]

Hmm. Something's gone awry in your workings. Your value for V2 doesn't look right. I think you've done fine up to where you've found an expression for VA in terms of V2.

When you've sorted that, the value for V2 will be the complex form of the phasor voltage for V2. You can find its magnitude and phase from the complex value.

 

[hdp12]

I found where I made the error, it was when I plugged in Va in terms of V2 into (1), so after solving it (correctly this time I think) I got

61.54j-61.54

does that look right?

 

[gneill]

I found where I made the error, it was when I plugged in Va in terms of V2 into (1), so after solving it (correctly this time I think) I got

61.54j-61.54

does that look right?

Yes, that looks much better!

So now you have the phasor for V2 in complex form. You can convert it to polar form: magnitude and phase, then write the time domain version from that.