# Phasor circuit problem: 1 resistor, 1 capacitor, 1 inductor, 1 Voltage source

## Homework Statement

What value of $\omega$ will cause the forced response $v_0$ in the circuit below to be zero?

http://img248.imageshack.us/img248/1854/problem934ao5.jpg [Broken]

Phasor eqs.

## The Attempt at a Solution

$$i\,=\,\frac{50\,-\,V_1}{2\Omega}\,=\,\frac{V_1\,-\,V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}$$

If I substitute $V_1\,=\,0$, then I get:

$$25\,=\,-\frac{V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}$$

How do I use this to find $\omega$?

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## Answers and Replies

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SGT
If you want Vo to be zero, the impedance of the series connection of the capacitor with the inductor must be zero.
You got the impedance of the capacitor wrong.
$$Z_C = \frac{-j}{\omega C}$$
$$Z_L = j \omega L$$

http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg [Broken]

$$Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega$$

$$V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}$$

We want $V_{OUT}$ to be zero:

$$0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,20j\omega\,+\,2\Omega}\,V_{IN}$$

$$0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}$$

$$-\frac{1}{5j\omega}\,=\,20j\omega$$

$$\omega\,=\,\frac{1}{10}$$

Is that right?

Last edited by a moderator:
SGT
http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg [Broken]

$$Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega$$

$$V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}$$

We want $V_{OUT}$ to be zero:

$$0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,12j\omega\,+\,2\Omega}\,V_{IN}$$

$$0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}$$

$$-\frac{1}{5j\omega}\,=\,20j\omega$$

$$\omega\,=\,\frac{1}{10}$$

Is that right?
Yes, the impedance is zero at the resonant frequency $$\omega_0 = \frac{1}{(LC)^\frac{1}{2}} = \frac{1}{10}$$

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