Impedance

Today we derived the impedance of resistor, capacitor and inductor. The impedance of a resistor is R; the impedance of a capacitor is -j/ωC or 1/jωC; the impedance of an inductor is jωL. It is obvious that the impedance of a capacitor or an inductor changes when different frequency is applied.

Impedance can be expressed as Z = R + Xj. Z is called impedance. R is called resistance. X is called reactance.

We also have the definition of their reciprocals. Y = G + Bj. Y is called admittance. G is called conductance. B is called susceptance.

Then we did a lab on impedance. We applied different frequencies to different types of circuit elements.

Resistor

R (Ω)	47
R0 (Ω)	100
Z (Ω)	147
f (kHz)	1
ω (rad/s)	6283.185307
Vin (V)	2
i (A)	0.01360544218
Vout (V)	1.360544218
Gain	0.6802721088
Phase Difference (º)	0

Capacitor 1kHz

R (Ω)	47
C (μF)	0.1
XC (Ω)	1591.549431	-90
Z (Ω)	1592.243258	-88.30849159
f (kHz)	1
ω (rad/s)	6283.185307
Vin (V)	2	0
i (A)	0.001256089477	88.30849159
Vout (V)	1.999128492	-1.691508405
Gain	0.9995642459
Phase Difference (º)	-1.691508405

Capacitor 5kHz

R (Ω)	47
C (μF)	0.1
XC (Ω)	318.3098862	-90
Z (Ω)	321.7610661	-81.60068958
f (kHz)	5
ω (rad/s)	31415.92654
Vin (V)	2	0
i (A)	0.006215792434	81.60068958
Vout (V)	1.978548182	-8.399310418
Gain	0.9892740911
Phase Difference (º)	-8.399310418

Capacitor 10kHz

R (Ω)	47
C (μF)	0.1
XC (Ω)	159.1549431	-90
Z (Ω)	165.9496789	-73.54761774
f (kHz)	10
ω (rad/s)	62831.85307
Vin (V)	2	0
i (A)	0.01205184616	73.54761774
Vout (V)	1.91811089	-16.45238226
Gain	0.959055445
Phase Difference (º)	-16.45238226

Inductor 1kHz

R (Ω)	47
L (mH)	1
XL (Ω)	6.283185307	90
Z (Ω)	47.4181233	7.614427915
f (kHz)	1
ω (rad/s)	6283.185307
Vin (V)	2	0
i (A)	0.04217796616	-7.614427915
Vout (V)	0.2650119773	82.38557208
Gain	0.1325059886
Phase Difference (º)	82.38557208

Inductor 5kHz

R (Ω)	47
L (mH)	1
XL (Ω)	31.41592654	90
Z (Ω)	56.53282622	33.75971669
f (kHz)	5
ω (rad/s)	31415.92654
Vin (V)	2	0
i (A)	0.03537767584	-33.75971669
Vout (V)	1.111422465	56.24028331
Gain	0.5557112325
Phase Difference (º)	56.24028331

Inductor 10kHz

R (Ω)	47
L (mH)	1
XL (Ω)	62.83185307	90
Z (Ω)	78.46554505	53.20247414
f (kHz)	10
ω (rad/s)	62831.85307
Vin (V)	2	0
i (A)	0.0254888945	-53.20247414
Vout (V)	1.601514474	36.79752586
Gain	0.8007572372
Phase Difference (º)	36.79752586

Summary: The impedance of resistors is independent from frequency. The magnitude of impedance of capacitors and inductors are related to the frequency. Larger frequency will cause the capacitor impedance to be smaller and the inductor impedance to be bigger. Smaller frequency will cause the capacitor impedance to be bigger and the inductor impedance to be smaller.