AC Circuit and Thevenin's theorem

[archaic]

Homework Statement

I'm asked to find the thevenin equivalent circuit, the load being $R_1$

$e(t)=E_0\cos{(\omega t)}$
$R_1=1k\Omega \ R_2=R_3=1k\Omega$
$C=220nF \ f = 250Hz \ E_0=10V$

Homework Equations

The Attempt at a Solution

$Z_{th} = Z_{R_2}//(Z_c+Z_{R_3}) = \frac{R_2R_3C\omega-jR_2}{(R_2+R_3)C\omega -j}$
As for $E_{th}$ :
$\overline{\rm E_{th}} = \frac{Z_c+Z_{R_3}}{Z_c+Z_{R_2}+Z_{R_3}}\overline{\rm e} = \frac{R_3-j\frac{1}{C\omega}}{R_2+R_3-j\frac{1}{C\omega}} \overline{\rm e}$
First of all, are the starter results correct?
Second, how should I proceed after this, replace the variables with their actual values then substitute $\overline{\rm e}$ with $E_0e^{\omega t}$, develop and get the real part?

 

[gneill]

First of all, are the starter results correct?

Looks good so far.

Second, how should I proceed after this, replace the variables with their actual values then substitute $\overline{\rm e}$ with $E_0e^{\omega t}$, develop and get the real part?

Nah. The real part alone does not fully describe the network, so it would no longer be a Thevenin equivalent.

Find the magnitude and angle for your $E_{th}$ above and write it in the form $E \cos(\omega t + \phi)$, and for $Z_{th}$ you might just leave it as an impedance in complex form, or, for extra points, show the impedance can be represented by the series connection of a resistor and one reactive component.

 

[archaic]

Looks good so far.

Nah. The real part alone does not fully describe the network, so it would no longer be a Thevenin equivalent.

Find the magnitude and angle for your $E_{th}$ above and write it in the form $E \cos(\omega t + \phi)$, and for $Z_{th}$ you might just leave it as an impedance in complex form, or, for extra points, show the impedance can be represented by the series connection of a resistor and one reactive component.

Won't the magnitude of $Z_{th}$ be equal to $R_{th}$?

 

[gneill]

Won't the magnitude of $Z_{th}$ be equal to $R_{th}$?

No. Impedance has real and imaginary parts. The real part corresponds to resistance, the imaginary part to reactance.

You can assign the real part to a resistor. The imaginary part you need to decide whether an inductor or a capacitor will fill the bill.