Solving AC Homework: e(t),i(t),Ul(t),Ur(t),Uc(t),Urc(t),Url(t)

[builder_user]

Hello.This is my new task.

Homework Statement

Find e(t),i(t),Ul(t),Ur(t),Uc(t),Urc(t),Url(t).
Find reactive power and complex power.

Homework Equations

What to do with Ulc(t)?

The Attempt at a Solution

C=200microfarad
L=3millihenry
r=4 Ohms
Ulc(t)=17.89sin(1000t-64)

I've done this
Zc=-j/wC
Zl=jwL
I've deleted L and C and added Zc and Zl to scheme.What is next?
I can't find w because I do not have e(t).

Here my original scheme.

 

[gneill]

There's no Ulc(t) shown in the first figure. Are the diagrams supposed to represent the same circuit redrawn, or are there two circuits to solve?

What are the units for the given Ulc(t)? Is the amplitude in volts? Is t in seconds? Is '64' an angle in degrees or in radians? How about the '1000'? Is it in radians per second? Degrees per second? Something else?

 

[builder_user]

There's no Ulc(t) shown in the first figure. Are the diagrams supposed to represent the same circuit redrawn, or are there two circuits to solve??

It's one circuit.

What are the units for the given Ulc(t)?

V

Is the amplitude in volts?Is t in seconds?Is '64' an angle in degrees or in radians? How about the '1000'? Is it in radians per second? Degrees per second? Something else?

17.89 - amplitude
t - seconds
64- degrees
1000 -w(sec-1)

 

[gneill]

Well, you can pick the value of ω out of the given expression for Ulc(t). Presuming that the 1000 is 1000 degrees/second, you can convert that to radians per second for ω.

That will allow you to calculate the impedance for the series LC. If you then express Ulc as a complex phasor, you should have no difficulty computing the complex current phasor for the current through LC. Since the whole circuit is series connected, the same current flows through the R as well, and you can determine e as a complex phasor.
Convert back to sin(ω*t + θ) form if desired.

Given the current and voltages, you should have no trouble calculating the power.

 

[builder_user]

64*=0.35pi?
1000 = 5.5pi?

w will be the same for all u(t) and i(t) and e(t)?

 

[gneill]

Yes, ω will be the same across the board.

You might find it to be a good idea to carry a few more decimal places for intermediate results. Round the final results at the end. So

64° --> 1.117 radians
1000°/sec --> 17.453 radians/sec

 

[builder_user]

Is I(t)=Ulc/zl+zc=
17.89sin(5.5pi*t-0.35pi)/(j*5.5pi*L+(-j/5.5pi*L)=17.89sin(17.5t-1.12)/(17.5jL-j/(17.5L)?

 

[gneill]

I think a couple of your L's should be C's in the above.

Convert the Ulc(t) to phasor form so you're working entirely in the frequency domain. But you're on the right approach.

 

[builder_user]

Convert the Ulc(t) to phasor form .

exp?
17.89*(exp^j(17.5t-1.12)-exp^-j(17.5-1.12))/(2*j(17.5j-j/17.5C))

To get u for every element of the circuit i(t) must be multiplyed by every resistance?

 

[gneill]

In the frequency domain the time dependent angle disappears. In complex form you can write the voltage as:

Ulc = 17.89V*(cos(φ) + j*sin(φ))

where φ is your phase angle, -64° = -1.117 radians

Solve for the current by dividing this voltage by the impedance ZL+ZC as you indicated earlier. Then current x individual impedances for the individual voltages.

You didn't say whether the given voltage Ulc was peak or rms. Did the original problem statement mention it?

 

[builder_user]

Then current x individual impedances for the individual voltages?

So,current is the same for all elems.
For Urc = I*(Zr+Zc)?

And How to find e(t)?

You didn't say whether the given voltage Ulc was peak or rms. Did the original problem statement mention it?

I don't know what is rms.All I know is Ulc - instanteous voltage

 

[gneill]

The current is the same for all series connected components. So the overall voltage is given by I*Ztotal. That will equal your e (in complex phasor form).

It seems that your two circuit diagrams have the order of the components changed, so that makes it hard to understand what is meant by Urc or Ulc.

 

[builder_user]

It seems that your two circuit diagrams have the order of the components changed, so that makes it hard to understand what is meant by Urc or Ulc.

It's made to find LC.I(t) will not change because all elems are the same.

How to find complex power and reactive power?Amplitude e(t)/Ze,
where Ze=Zl+Zr+Zc?

Amplitude for every U&e -> x *(cos...)?

 

[builder_user]

How to find active and reactive power?

 

[gneill]

How to find active and reactive power?

If you have the voltage source, e, in complex phasor form, and the current, I, also in complex phasor form, then the apparent power is p = eI*, where I* is the complex conjugate of I.

The components of p are the real and reactive components of the power. Do a search on "power triangle" if you need more information about apparent, real, and reactive power.

 

[builder_user]

where I* is the complex conjugate of I.

It's means that if I have
e(t)=B*sin(wt-q)
and I(t)=A*sin(wt-q) the result will be like this

p=B*sin(wt-q)/A?

 

[gneill]

It's means that if I have
e(t)=B*sin(wt-q)
and I(t)=A*sin(wt-q) the result will be like this

p=B*sin(wt-q)/A?

No. First of all, the voltage and current waveforms will not have the same phase angle, because of the impedance having both real and imaginary parts. Secondly, you want to be dealing with complex values in the frequency domain, not sines and cosines in the time domain. There should be no t's involved.

Perhaps I'm making an assumption that I shouldn't be making. Do you know what phasors are? How about their representation as complex values?

 

[builder_user]

I=B*(cosq+jsinq)
e=I*Z=B*Z*(cosq+jsinq)
and
p=e*Bjsinq?

 

[gneill]

Okay. Why don't you put some numbers to those expressions?

What values do you have for e, I, and Z?

 

[builder_user]

I'm not sure but from previous posts and the task it's seems to be
Z=-289j+0.05j+4
I=17.89*(0.45+j*0.89)/(0.05j-289j)
e=I*Z

 

[gneill]

Your value for Z does not look right. Can you show your work for finding the impedances of the inductor and capacitor?

 

[builder_user]

С=-j/WC=-j/5.5pi*200*10e-6

L=jwL=j*5.5pi*3*10e-3

 

[gneill]

Okay. You should keep a few more decimal places for intermediate values. I see the capacitive reactance as -286.4 Ohms, and for the inductive reactance, 0.0524 Ohms.

This is assuming that the "1000t" in the time domain expression for Ulc implies 1000 degrees per second.

So, with Z_L and Z_C in hand, what numerical value are you getting for the current (normalized complex form)? How about for e?

 

[builder_user]

I=0.03j-0.05Is 17.89 the amplitude of I?

 

[gneill]

I'm getting a different value for I. In particular, I'm seeing (0.0561 + 0.0274j) Amps, or if you prefer your current in milliamps, (56.1 + 27.4) mA. Perhaps you should review your calculation. If you don't get this value, show your calculation in detail.

The amplitude of the current would be |I| = |56.1 + 27.4| = 62.5 mA

 

[builder_user]

С=-286.4j
L=0.0524j

I=17.89*(0.45+0.89j)/(0.0524j-286.4j)=-0.0625*(0.45+j0.89)/j=0.0625j*0.45+j*j*0.89*0.0625=0.028j-0.056

 

[gneill]

The expression that you're using for Ulc is suspect: 17.89*(0.45 + 0.89j). The phase angle is -64 degrees, making cos(φ) and sin(φ) 0.438 and -0.899 respectively. Where did you get 0.45 and +0.89?

 

[builder_user]

I forgot that angle is -64 not 64.

 

[builder_user]

|I| = |56.1 + 27.4| = 62.5 mA

Why?

 

[gneill]

$$|I| = \sqrt{56.1^2 + 27.4^2} = 62.4 mA$$

(I think when I originally calculated it I was keeping more decimal places, so the ".4" crept up to ".5" on rounding)