Understanding Faraday's Law: Investigating Phase Shift in an AC Circuit

[2.718281828459]

I did an experiment with two solenoids, one inside the other. The inner solenoid was hooked up to an AC power source and ammeter. The outer coil had a voltmeter. This is a faraday's law experiment. Because the current is changing, we can find a formula for the emf. Here are my math steps:
View attachment untitled-1.pdf
So the phase shift for the emf minus the phase shift for the current should give pi/2, but it doesn't! The values I got were (1.103-1.977)=-0.874

What is going on?

 

[NascentOxygen]

How did you measure the phase relation of current to voltage?

 

[2.718281828459]

How did you measure the phase relation of current to voltage?

Loggerpro graphed both simultaneously (using a sin function for each). The general form for each equation was Asin(Bt+C)+D. I found C for each graph and then took the difference. Does that answer your question?

 

[NascentOxygen]

What size were the solenoids and how many turns? Was the excitation 60 Hz? What value (roughly) of voltage in the sensing coil did you experience?

 

[2.718281828459]

I can give you all of the information I have if it will help, but conceptually the exact numbers shouldn't matter, right?

Yes, 60 Hz

number of turns of outer coil- (2920 pm 1)
number of turns of inner coil-(235 pm 1)
diameter of outer coil-(3.4200E-2 pm 7.8E-5) meters
diameter of inner coil- (1.7300E-2 pm 7.8E-5) meters
Length of inner coil- (12.000E-2 pm 7.8E-4) meters
Length of outer coil- (11.000E-2 pm 7.8E-4) meters

All 95% tri pdf

Emf max=2.270V

I also have values for the resistance, capacitance and inductance (from the manual, not from measurement)

Outer coil #29 wire, approx .29mm

Inductance: 63 pm 3 mH
Resistance 76 pm 2 Ω
Capacitance: 124 pm 2 pF

Inner coil #18 wire approx 1mm

Inductance: 78 pm 22 μH
Resistance: 0.4 pm 0.1 Ω
Capacitance: 142 pm 2 pF

I have a suspicion that this has to do with the resistance of the coils, but I still don't understand why or how the model I used is flawed. The predicted result was almost a perfect match with the actual result except for this shift.

 

[NascentOxygen]

diameter of outer coil-(3.4200E-2 pm 7.8E-5) meters

How many mm is that?

Anyway, my suggestion for the phase not being as expected hinges on there being capacitive coupling between the coils, in addition to the inductive coupling. Capacitive coupling gives a phase shift different from the transformer coupling. Added to this, there is always inductive coupling from power cables in the wall, and this may not have the same phase as the sinewave that is driving the solenoid here. The latter interference is more pronounced in high impedance circuits, and I surmise that the load on your sensing coil is the high input impedance of a voltmeter?

 

[2.718281828459]

I'm still not getting this. Apparently there are some differential equations for this and the phase shift is frequency dependent. More help? Any idea what these equations are?