How to find the resonant frequency of this circuit?

In summary, the resonant frequency occurs when the imaginary part of the impedance is equal to zero. However, in order to isolate the imaginary part of the impedance from the real part, you need to do some complex math.
  • #1
Nat3
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Homework Statement



Given this circuit:

[PLAIN]http://img546.imageshack.us/img546/2921/resonance.png [Broken]

Homework Equations



How does one find the resonant frequency? (Since there are no values for the components, I assume this means find the formula).

The Attempt at a Solution



I read that the resonant frequency occurs when the imaginary part of the impedance is equal to zero, so I tried solving it that way but was not able to isolate the imaginary part of the impedance from the real part.

Any help would be very much appreciated!
 
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  • #2
Can you show us your calculations?
 
  • #3
Hello skeptic,

Thank you for your reply.

I tried this:

[PLAIN]http://img546.imageshack.us/img546/2921/resonance.png [Broken]

Z = L || (C+R) = [tex]\frac{(L)(C+R)}{L+(C+R)} = \frac{(j\omega L)(\frac{1}{j\omega C}+R)}{j\omega L+\frac{1}{j\omega C}+R}=\frac{\frac{L}{C} + j\omega LR}{j\omega L+\frac{1}{j\omega C}+R}[/tex]

But I don't know how to separate the real part from the imaginary part.
 
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  • #4
If it was something simple, like:

[tex]Z = R + j\omega L + \frac{1}{j\omega C}[/tex]

then I could just factor out the j:

[tex]Z = R + j(\omega L - \frac{1}{\omega C})[/tex]

and set the imaginary/complex part to zero:

[tex]\omega L - \frac{1}{\omega C} = 0[/tex]

and solve for [tex]\omega[/tex] to get the formula for the resonant frequency, right?

But I can't figure out what to do for the circuit with L in parallel with R and C.
 
  • #5
I would really appreciate any help anyone might be able to give on this :smile:
 
  • #6
You're doing alright with your impedance math. In order to separate into real and imaginary parts you first want to clear the imaginary part of the denominator. To do that, multiply the numerator and denominator both by the complex conjugate of the denominator. In other words:

[tex] \frac{a + jb}{c + jd} \cdot \frac{c - jd}{c - jd} = \frac{ac + bd + j(bc - ad)}{c^2 + d^2} [/tex]

You can then pick out the imaginary part quite easily.
 
  • #7
Thanks a lot! I didn't think of multiplying by the conjugate :uhh: After doing that it was kind of messy, but worked out.

Thanks again.
 

1. What is resonant frequency?

Resonant frequency is the frequency at which a circuit or system naturally vibrates or oscillates with the least amount of external energy. It is the frequency at which the circuit or system will resonate or amplify the input signal.

2. How do I calculate the resonant frequency of a circuit?

The resonant frequency of a circuit can be calculated using the formula: fr = 1/(2π√(LC)), where fr is the resonant frequency, L is the inductance of the circuit, and C is the capacitance of the circuit.

3. What is the importance of finding the resonant frequency of a circuit?

Finding the resonant frequency of a circuit is important for designing and optimizing circuits for specific frequencies. It also helps in preventing unwanted oscillations or interference in the circuit. Additionally, it is essential for applications such as tuning radio receivers and filters.

4. Can the resonant frequency of a circuit be changed?

Yes, the resonant frequency of a circuit can be changed by adjusting the values of inductance and capacitance in the circuit. This can be done by adding or removing components, or by using variable components such as variable capacitors or inductors.

5. What factors can affect the resonant frequency of a circuit?

The resonant frequency of a circuit can be affected by the values of inductance and capacitance, as well as by the quality or type of components used. Other factors such as temperature and external interference can also impact the resonant frequency of a circuit.

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