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.