L and C to Achieve Single Resonant Frequency in Series-Parallel Circuit?

[kl055]

Finding L and C for which there is only one resonant frequency in cct

Homework Statement

an inductor in series with a paralle of (a capacitor and a resistor)
The inductor and resistor are fixed while C is variable

Z = sL + R/(1+RsC)

Find a value of L and C such that there is only one value of C for which there is a resonant freq

Homework Equations

R = 10k
resonant freq = 40k * pi

The Attempt at a Solution

Set Z = R at resonance and find that w^2RLC + jw(R^2C-L) = 0
at resonance, the bracket term becomes zero. But I am confused by the fact that the w is outside of the brackets. Do I just set R^2 C - L = 0 and find solve for L and C? But I only have one equation and two unknowns.

 

[Simon Bridge]

R = 10k
resonant freq = 40k * pi

... you mean R=10kΩ ?
... what is the resonant frequency: 40π kHz ?

How do you know that is the resonant frequency?

Z = sL + r/(1+RsC)

... do you have a value for the little r in that equation?
Should that be an R?

The question implies that there must be more than one resonant frequency - bourne out by the quadratic - I suspect there is a term missing there otherwise the resonant frequency is zero (DC case). However:

I only have one equation and two unknowns.

Is there only one combination of L and C that gives rise to a single resonant frequency or are there a range of them? Does the question ask you to find a unique value or a relationship? Is there more information? i.e. are you provided a target value for the resonant frequency as well?

 

[kl055]

Sorry, I made a mistake in part 1 of my post. I corrected it now. r = R and the problem is to "Find a value of L and C such that there is only one value of C for which there is a resonant freq"

the resonant frequency is given in the problem statement.

Using only Locus diagrams and (trigonometry on those diagrams), I determined that L = 1/(2Rw) and C = R/(jw) or L = -1/(2Rw)) and C = -R/(jw). I am not sure if this is correct, but looking at the Locus diagrams it looked correct conceptually to me.

The second part of the problem is to verify my result by using equations only.


So I am given a target resonant frequency of 40k*pi rad/s and and that R = 10k ohms. I need to find L and C such that there is only one value of the variable capacitor for which the circuit is resonant. No other information is given.

 

[Simon Bridge]

You have a resonant frequency, you have a relation between L and C ... the question kinda suggests that you should expect there to be two or more possible values of C that satisfy the resonance condition except for one particular value of L. Is that what you have?

(i.e. the resonance condition should be a quadratic in C.)

 

[kl055]

Sort of. I had a circle in the impedance plane of the locus diagram. There were two intersections (when only the real part of the impedance exists at resonance) meaning that for a given value of L there were two values of C for which the circuit was resonant.

Then I shifted the circle up by L = 1/(2Rw) making the circle intersect at only one point (at the very bottom), meaning that for that particular value of L there was only one value of C for which the circuit is resonant. Similarly when I shifted downwards.

Thus when L = + or -1/(2Rw) there is only one value of C for which the circuit is resonant. I then found the corresponding C values using trigonometry.

Now I am required to do the same thing using only equations and no locus diagrams.

 

[Simon Bridge]

Yes - and that is what you have shown us in post #1 and that is what I was referring to.
You have derived a relationship, from the equations, that must be satisfied for resonance to occur.
Is the relationship you have derived a quadratic in C?

 

[kl055]

I suppose not since there is no C^2. So I need to derive another equation?

 

[Simon Bridge]

I'd revisit the the way you derived the relation, yes.
Check your assumptions and go carefully.

[edit]

 

[kl055]

Okay, thanks

 

[Simon Bridge]

Basically you have worked out the impedance of the circuit, then used your knowledge of how the circuit behaves at resonance to work out the resonance condition on the impedance ... this gives you a relationship between the L R C and ω₀ ... you are given R and ω₀ so the relation is between L and C.

I think your equation for impedance is correct (I could be mistaken)
$$ z = j\omega L + \frac{R}{1+j\omega RC}$$
... so how does the circuit behave under resonance - in terms of relative phase (say)?