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

In summary, by setting the impedance of the capacitor and inductor equal to each other and solving for \omega, we can find the resonant frequency at which the forced response in the circuit will be zero.
  • #1
VinnyCee
489
0

Homework Statement



What value of [itex]\omega[/itex] will cause the forced response [itex]v_0[/itex] in the circuit below to be zero?

http://img248.imageshack.us/img248/1854/problem934ao5.jpg

Homework Equations



Phasor eqs.

The Attempt at a Solution



[tex]i\,=\,\frac{50\,-\,V_1}{2\Omega}\,=\,\frac{V_1\,-\,V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}[/tex]

If I substitute [itex]V_1\,=\,0[/itex], then I get:

[tex]25\,=\,-\frac{V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}[/tex]

How do I use this to find [itex]\omega[/itex]?
 
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  • #2
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.
[tex]Z_C = \frac{-j}{\omega C}[/tex]
[tex]Z_L = j \omega L[/tex]
 
  • #3
http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg

[tex]Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega[/tex]

[tex]V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}[/tex]

We want [itex]V_{OUT}[/itex] to be zero:

[tex]0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,20j\omega\,+\,2\Omega}\,V_{IN}[/tex]

[tex]0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}[/tex]

[tex]-\frac{1}{5j\omega}\,=\,20j\omega[/tex]

[tex]\omega\,=\,\frac{1}{10}[/tex]

Is that right?
 
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  • #4
VinnyCee said:
http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg

[tex]Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega[/tex]

[tex]V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}[/tex]

We want [itex]V_{OUT}[/itex] to be zero:

[tex]0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,12j\omega\,+\,2\Omega}\,V_{IN}[/tex]

[tex]0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}[/tex]

[tex]-\frac{1}{5j\omega}\,=\,20j\omega[/tex]

[tex]\omega\,=\,\frac{1}{10}[/tex]

Is that right?

Yes, the impedance is zero at the resonant frequency [tex]\omega_0 = \frac{1}{(LC)^\frac{1}{2}} = \frac{1}{10}[/tex]
 
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What is a phasor circuit?

A phasor circuit is a type of electrical circuit that deals with alternating current (AC) signals. It is represented by complex numbers called phasors, which are used to simplify calculations and analysis of the circuit.

What components are typically found in a phasor circuit?

A phasor circuit typically includes one resistor, one capacitor, one inductor, and one voltage source. These components are used to create a circuit that can handle and manipulate AC signals.

How do you solve a phasor circuit problem?

To solve a phasor circuit problem, you must first convert all components into their respective phasor values. Then, you can use Kirchhoff's laws and Ohm's law to analyze the circuit and find the voltage and current at different points. Finally, you can convert the phasor values back into their time-domain equivalents to get the final solution.

Why is a phasor circuit useful?

A phasor circuit is useful because it allows for simplified analysis and calculations of AC circuits. It also helps in understanding the behavior of the circuit at different frequencies and allows for easy comparison of different circuits.

What are some real-world applications of phasor circuits?

Phasor circuits are commonly used in power systems, telecommunication systems, and electronic devices that use AC signals. They are also used in industries such as aerospace, automotive, and renewable energy for various applications such as power distribution, signal processing, and control systems.

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