Why Is There a Phase Shift in an LC Tank Circuit at Resonance?

[tomcorker]

In a lab session we were investigating the LC tank circuit of a superhet radio. The tank consisted of a ferrite coil antenna connected in parallel with a variable capacitor. A signal generator was connected to a second ferrite antenna (the drive antenna) and placed close to the first in order to induce a signal.

Now, my lab notes say that the LC tank circuit is at resonance when the voltage out is a maximum and the phase shift relative to the drive antenna is +-90 degrees. I'm fine with the maximum voltage, but why the phase shift? The L and C are parallel so the voltage over them must be common. So the phase shift must be something to do with the coupling between the two antennas? But I've no idea why...

 

[NascentOxygen]

What do you mean by +-90 degrees? What instrument were you using to measure the phase difference?

 

[tomcorker]

I mean the sine wave on the tank circuit is leading or lagging the sine wave supplied to the driver antenna by pi/2. The phase difference was measured with a dual channel oscilloscope.

 

[NascentOxygen]

I mean the sine wave on the tank circuit is leading or lagging the sine wave supplied to the driver antenna by pi/2.

You noted the phase difference, but didn't note which was the leading waveform? Was that an oversight?

The phase difference was measured with a dual channel oscilloscope.

Which was it set to: ALT or CHOP? Can you explain why this may be important to know?

 

[gneill]

Since the statement about the phase shift is an assertion made in the lab notes (presumably the lab instruction manual), the method of measurement of the phenomenon does not affect the underlying veracity or physics behind that statement.

I suppose one could write the differential equations for the circuit and solve for the transfer function. I'll bet that the transfer function changes sign when the driving frequency passes through a particular frequency (resonance). This would have the effect of flipping the phase shift 180°. The phase shift AT resonance would have to be precariously balanced between, or at +/- 90°.

 

[NascentOxygen]

I'll bet that the transfer function changes sign when the driving frequency passes through a particular frequency (resonance). This would have the effect of flipping the phase shift 180°. The phase shift AT resonance would have to be precariously balanced between, or at +/- 90°.

You'd lose your bet. There is no abrupt "flipping" of the phase at resonance. In fact, the phase shift varies smoothly and approximately linearly across the region of resonance, and this makes observing phase shift a most reliable method for determining a system's resonant frequency. Contrast this with monitoring the amplitude response to try to gauge the resonant point: this is far too imprecise, the amplitude of the gain being roughly level around the resonant frequency. I have no idea what "to be precariously balanced between, or at +/- 90 degrees" could mean. There is no risk of the system "flipping " between +/- 90 degrees; that sounds like behavior you'd see in a PLL.

 

[gneill]

You'd lose your bet. There is no abrupt "flipping" of the phase at resonance. In fact, the phase shift varies smoothly and approximately linearly across the region of resonance, and this makes observing phase shift a most reliable method for determining a system's resonant frequency. Contrast this with monitoring the amplitude response to try to gauge the resonant point: this is far too imprecise, the amplitude of the gain being roughly level around the resonant frequency. I have no idea what "to be precariously balanced between, or at +/- 90 degrees" could mean. There is no risk of the system "flipping " between +/- 90 degrees; that sounds like behavior you'd see in a PLL.

I may indeed be recalling the curve for something else. After giving it further thought realized that the transition would have to be smooth; For a very high Q tank the "region of resonance" can be very narrow (unless you expand your frequency axis around the resonance peak), but nevertheless the transition of the phase will be linear. I should have engaged the little grey cells before typing my reply!

Here's an AC frequency sweep of a tank circuit as performed by LTSpice. The plot shows the tank signal amplitude (dB) and phase.

I still think we need to derive the transfer function for Vout/Vin in order to pinpoint the reason for the +/- 90° statement --- how does the circuit choose which it will be?

 

[NascentOxygen]

I may indeed be recalling the curve for something else. After giving it further thought realized that the transition would have to be smooth; For a very high Q tank the "region of resonance" can be very narrow

Yes, and as the OP was investigating the region of resonance with a dual channel oscilloscope, I expect the region of close interest would be just a few hundred Hz either side of the peak.

Here's an AC frequency sweep of a tank circuit as performed by LTSpice. The plot shows the tank signal amplitude (dB) and phase.

Thank you for going to all that trouble of a SPICE simulation. OP seems to have lost interest, but I haven't.
While I was thinking of an explanation for OP's observed +/- 90 degrees, I'd be more comfortable after first being assured it was an accurate observation, not the result of misuse of oscilloscope triggering.

I still think we need to derive the transfer function for Vout/Vin in order to pinpoint the reason for the +/- 90° statement --- how does the circuit choose which it will be?

I'm still in the dark about what OP means. And as s/he hasn't replied, I assume focus of interest has shifted to a new topic.

 

[agsys]

I'd like to butt in here on account of (I believe) currently performing exactly the same lab and being irritated by the same question!

As the OP says, the circuit is an LC parallel tank driven by coupling from a nearby sine generator fed into an identical inductor.

At the frequencies investigated, the frequency generator impedance is negligible compared to the big ol' coil it drives so the Voltage at the first measurement (over the drive inductor) is lagging current by ~90 degrees.

The question is why, at resonance, is the voltage across the LC tank lagging (or leading, but I very much suspect lagging as it's a driven system) the voltage across the driving inductor by 90 degrees? It's unmistakeably the case, as the response amplitude did indeed peak at a +/-90 shift on the 'scope screen.

Since at resonance the L current is 90 degrees out of phase to the resonator Voltage, my suspicions would (wrongly!) have been that the phases would either match up or appear at 180 degrees... Hmm...

Does the weak coupling of the inductors have an effect? or is a 90 degree lead in magnetic field required for the first inductor to drive the resonator?

Alas I am stumped.

 

[agsys]

Back again for the benefit of future people having (I think) figured it:

To analyse this circuit, use the equivalent circuit for a transformer (http://en.wikipedia.org/wiki/Transformer#Equivalent_circuit") and neglect magnetising reactance and iron loss on account of the air core... Refer everything else to the primary and the result should be representable as a voltage source driving a series RLC circuit with some R, L and C (group them together) and the output taken only across the capacitor. Hence at resonance, the source sees an exclusively real impedance (reactances cancel out) so the source current and voltage share phase - but a quick complex potential divider equation will reveal the phase shift due to taking output voltage across the capacitor only!

This answer is very circuit-analysisey though: If anyone could come up with a more physical electromagnetics based explanation I'd be happy to give it a listen.