Solving circuit using complex numbers

[masterjoda]

Homework Statement

Find the current in them in this circuit, if we know R=X_L, X_C and u=5sin(314t)

The Attempt at a Solution

First , 5=U_0, 314=\omega and voltage we can write as u=U_0cos(\omega t + \frac{\pi}{2}) and u=U_0 e^{i\frac{\pi}{2}}=iU_0. U is the voltage at the source U_1 in the branch and U_2 at the resistor. Now U=U_1+U_2 or U=IR+IZ where \frac{1}{Z}=\frac{1}{iL\omega}-\frac{C\omega}{i} or Z=\frac{iL\omega}{1-CL\omega^2}
Now the current is I=\frac{U}{R+Z} or when it is arranged I=\frac{iU_0(1-CL\omega^2)}{iL\omega (2-CL\omega^2)}. Now I don't know how to complete the calculation to reduce it to the form like this I_0sin(\omega t + \theta).

 

[gneill]

If it is true that R = X_L,X_C, then X_L = X_C. What does that tell you about the LC tank circuit?

 

[masterjoda]

No just R=X_L.

 

[SammyS]

Homework Statement

Find the current in them in this circuit, if we know R=X_L, X_C and u=5sin(314t)

The Attempt at a Solution

First , 5=U_0, 314=\omega and voltage we can write as u=U_0cos(\omega t + \frac{\pi}{2}) and u=U_0 e^{i\frac{\pi}{2}}=iU_0. U is the voltage at the source U_1 in the branch and U_2 at the resistor. Now U=U_1+U_2 or U=IR+IZ where \frac{1}{Z}=\frac{1}{iL\omega}-\frac{C\omega}{i} or Z=\frac{iL\omega}{1-CL\omega^2}
Now the current is I=\frac{U}{R+Z} or when it is arranged I=\frac{iU_0(1-CL\omega^2)}{iL\omega (2-CL\omega^2)}. Now I don't know how to complete the calculation to reduce it to the form like this I_0sin(\omega t + \theta).

Doesn't R=X_L, X_C, mean that R=X_L=X_C\ ?

 

[gneill]

No just R=X_L.

Ah, my bad. I read it as both being R, rather than as part of a list of known values. Where's a semicolon when you need one?

Anyways, you can write the complex impedances directly:

Z_R = R

Z_L = 0 + jR

Z_C = 0 - jX_C

These can then be used in formulas in the usual way to find voltages and currents (complex). Convert the values to phasor form at the end.

 

[masterjoda]

These can then be used in formulas in the usual way to find voltages and currents (complex). Convert the values to phasor form at the end.

I have used them and I got that last equation for I.

 

[gneill]

I'm not sure why you've gone to the "raw" L and C versions of the impedances when you've been given X_L and X_C. You should be able to find the total current as a complex function of u, R, and X_C.