LCR circuits - phase difference between C and L

[Darkmisc]

I'm writing up a prac that involved LCR circuits.

One part of the prac involved measuring the phase difference between the voltage across the capacitor (V_C) and the inductor (V_L), while the frequency of the input signal was varied.

This was done by measuring the distance between peaks for (V_C) and (V_L) on an oscilloscope.

I got varying values for the phase difference, with a phase difference of zero when the input frequency matched the resonant frequency.

V_C may be expressed as (-1/Cw)I cos wt.

V_L may be expressed as Lw I cos wt.


When looking at these equations, I don't understand why the phase difference depended on the signal's frequency. The way I interpret the equations is that V_C and V_L are permanently 180 degrees out of phase (both being cos of wt, but one negative, the other positive). I can understand that their amplitudes won't always be the same, but it seems to me they should always be 180 degrees out of phase.

Where have I gone wrong in my understanding?

Thanks

 

[Bob S]

Your equations for the voltages across the capacitor and inductor are over-simplified. The proper equations, in the engineering notation, are (using I(ω) = I₀cos ωt)

V_C = (+1/jωC) I(ω) = (-j/ωC) I(ω) = (+1/ωC)I₀sin ωt

and V_L = jωL I(ω) = -ωL I₀sin ωt

[Added - Note the 180-degree phase difference between V_C and V_L]

J is a shorthand operator for the derivative operator j that changes the phase of the current by 90 degrees. so

jsinωt --> +cosωt, and jcos(ωt) --> -sin(ωt).

These are both derived from the basic equations V_L = L dI_L/dt, and I_C = C dV_C/dt

Bob S