Obj: to determine the phase angles between the input and the output of a simple RC series circuit.

Vout will lag Vi because of capacitor.

in the lab with measure value of Vout, and measure the phase difference.

Rs+ jXs = RL -jXL

nominal 10k Ω resistor: 9.988 Ω
Vi= 10.00Vpp
f=10kHz

derive: V0/Vi = 1/(1+jωRC)

its a voltage divider:
VO = ViZC/ZC+ZRVO/Vi=-jωC/-jωC+RVO/Vi=1/RjωC/1/RjωC+1V0/Vi = 1/(1+jωRC)

C, F	measure Δt, sec	calc θ=Δt*f*360, deg	expected θ=tan-1-2πfR(C+47pF), deg	% diff
1μ	254μ	-91.4	-89.1	2
300n	244μ	-87.8	-87.0	1.5
100n	236μ	-84.9	-81.0	5
30n	172μ	-61.9	-62.1	0.8
10n	92μ	-33.1	-32.2	3
1n	20μ	-7.2	-3.76	50
300p	12μ	-4.3	-1.25	70
100p	8μ	-2.9	-.53	82
0	4μ	-1.4	-.17	85

calculation: θ=360fΔt, deg 254*10^-6*10^3*360= 91.44 244*10^-6*10^3*360= 87.84 236*10^-6*10^3*360= 84.96 172*10^-6*10^3*360= 61.92 92*10^-6*10^3*360= 33.12 20*10^-6*10^3*360= 7.2 12*10^-6*10^3*360= 4.3 8*10^-6*10^3*360= 2.9 4*10^-6*10^3*360= 1.4 θ= tan^-1 -2*πf*R(C+ 47pF), deg 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+1000))= -89.13295339989629447636 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+300))= -87.00625743142780314286 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+100))= -80.99767047169381985426 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+30))= -62.10989689274871720362 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+10))= -32.26629979874937002503 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+1))= -3.76378228634969788050 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+.3))= -1.24900231885805943401 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+.1))= -.52918496753049275503 180/3.14*a(-2*3.14*10^3*10^4*10^-9*(.047+0))= -.16919950864900798605 discussion: In this lab we verified the general tendency for voltage across a capacitor to lag behind input voltage. More specifically we showed the phas angle will vanish as the capacitance does also. Our observations are closely in line with theory, except for the small capacitance values were the phase angle become approximately zero and essentially unmeasurable with the oscilloscope. We neglected any inductance and capacitance in both the function generator and in the oscilloscope, which would have introduced phase distortions of their own. This appears to have been a reasonable simplification.