Prove this Total Voltage at Resonance

[fromthepast]

Prove that at the resonance of a series RLC circuit, instantaneous velocity across the capacitor is equal to negative instantaneous velocity across the inductor, and thus, the total voltage across C and L is zero.

Thanks!

 

[gneill]

Prove that at the resonance of a series RLC circuit, instantaneous velocity across the capacitor is equal to negative instantaneous velocity across the inductor, and thus, the total voltage across C and L is zero.

Thanks!

Please follow the posting template. You must show some work before help can be given (we can't just do your homework -- we can only assist you with your work).

 

[fromthepast]

I know that impedance across the inductor is equal to the angular frequency times the inductance, and that the impedance across the conductor is equal to the reciprocal of the angular frequency times the capacitance.

I have the equation: voltage across inductor (instantaneous) = amplitude of voltage across inductor times cos(angular freq x time + 90). There is an equivalent for voltage across conductor (instantaneous).

I realize that instantaneous voltage across capacitor is equal to the negative instantaneous voltage across the inductor. Where I get lost is, how do I know what angular freq x time is to determine that the total voltage across C and L is zero?

Thanks

 

[gneill]

The condition given in the problem statement is "...at the resonance of a series RLC circuit,...". So you're interested in the case where the circuit is in resonance. That means that the frequency is the resonant frequency. What's the resonant frequency for a series RLC circuit?

 

[fromthepast]

angular frequency/2(pi)

 

[fromthepast]

or 1/(square root (LC) x 2pi)

 

[gneill]

or 1/(square root (LC) x 2pi)

So, at the resonant frequency what are the impedances of the L and C components?

 

[fromthepast]

Impedance of L = angular frequency times inductance = 2 x pi x f x L

Impedance of C = 1/(angular frequency x capacitance) = 1/( 2 x pi x f x C)

 

[gneill]

Impedance of L = angular frequency times inductance

Impedance of C = 1/(angular frequency x capacitance)

Plug in your angular frequency.

 

[fromthepast]

Impedance of L = angular frequency times inductance = 2 x pi x f x L

Impedance of C = 1/(angular frequency x capacitance) = 1/( 2 x pi x f x C)

 

[gneill]

Impedance of L = angular frequency times inductance = 2 x pi x f x L

Impedance of C = 1/(angular frequency x capacitance) = 1/( 2 x pi x f x C)

Yes, but you've just calculated the resonant frequency... so plug it in!

 

[fromthepast]

:)

Impedance of L = 2 x pi x (angular frequency/2pi) x L

Impedance of C = 2 x pi / (2 x pi x angular frequency x C)

 

[gneill]

:)

Impedance of L = 2 x pi x (angular frequency/2pi) x L

Impedance of C = 2 x pi / (2 x pi x angular frequency x C)

Seriously, fromthepast? Going in circles isn't helping.

You need to GET RID OF the variables f and/or "angular frequency" from your impedance expressions by plugging in the expression you found for the resonant frequency in terms of L and C. Simplify and compare the impedances.

 

[fromthepast]

Doing that doesn't relate the instantaneous velocity of the inductor and the instantaneous velocity of the capacitor.

vC = -vL

 

[gneill]

Doing that doesn't relate the instantaneous velocity of the inductor and the instantaneous velocity of the capacitor.

vC = -vL

What is the magnitude of the voltage across equal magnitude impedances in a series circuit? Remember, in a series circuit all the components have identical current at all times.

It only remains to show that the voltages are phase shifted +90° and -90°, for a total of 180°.

(The problem would be much simpler if you used the complex impedances).

 

[fromthepast]

So how does saying that the phase angle is 180 connect vC = - vL to having total voltage across C and L equal zero?

 

[gneill]

Because they are 180 degrees out of phase and equal. Draw a sine curve. Draw a sine curve that 180° shifted. Note that their values at any given time always sum to zero.
sin(x + pi) = -sin(x).

 

[fromthepast]

That still doesn't reason how when instantaneous voltage across a capacitor =
negative instantaneous voltage across an inductor, total voltage across C and L is zero.

...

 

[gneill]

That still doesn't reason how when instantaneous voltage across a capacitor =
negative instantaneous voltage across an inductor, total voltage across C and L is zero.

...

They are in series, so the potential differences add.

 

[fromthepast]

Yes, but what makes them add to zero?

 

[gneill]

Yes, but what makes them add to zero?

At every instant in time they have equal and opposite values. What other value could they sum to?

 

[fromthepast]

"At every instant in time they have equal and opposite values. What other value could they sum to?"

That was perspicaciously helpful.

Thanks