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.

Phasors: Passive RL Circuit Response

We have completely finished the first part of the class, DC circuits. Now we move on to a new part of this class, AC circuits. Today we are introduced to a new powerful tool to analyze AC circuits, phasor. Phasor is a representation of a sinusoidal wave without the time dependence. Also, capacitors and inductors can also be considered as resistors with imaginary impedance. Ohm's Law still applies to AC circuits.

Then we did a lab on passive RL Circuit Response. Here is a schematic of the lab

We applied three different sinusoidal waves to the voltage input and calculated the gain and phase difference.

ω = 47krad/s

	Experimental	Theoretical	Percent Error
Angular Frequency (krad/s)	47.00	47.00	0.00%
Frequency (kHz)	7.48	7.48	0.00%
Gain	0.0149	0.015	1.28%
Phase Difference (º)	-43.1459	-45.000	4.12%

ω = 470krad/s

	Experimental	Theoretical	Percent Error
Angular Frequency (krad/s)	470.0	470.0	0.00%
Frequency (kHz)	74.8	74.8	0.00%
Gain	0.0021	0.0021	0.14%
Phase Difference (º)	-86.1859	-84.289	2.25%

ω = 4.7krad/s

	Experimental	Theoretical	Percent Error
Angular Frequency (krad/s)	4.700	4.700	0.00%
Frequency (kHz)	0.748	0.748	0.00%
Gain	0.021	0.021	2.89%
Phase Difference (º)	-4.933	-5.711	13.61%

Summary: 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. This causes the different gain and phase difference in each case. Using phasors can help us analyze AC circuits in the same way as DC circuits.

RLC Circuit Response

We are introduced to Schmitt Trigger. The primal function of the Schmitt Trigger, to convert noisy square waves, sine waves or slow edges inputs into clean square waves. One way to implement Schmitt Trigger is to use operational amplifier.

Then we did a lab on the step response of an RLC circuit. Here is a schematic of the circuit

We calculated the theoretical natural frequencies, undamped oscillation frequency, damped oscillation frequency and damping factor.

Here is the setup of the circuit

We calculated the damping ratio, DC gain, natural frequency, rise time and steady state response.

Damping Ratio	0.155
DC Gain	0.023
Natural Frequency (rad/s)	10105.823
Rise Time (ms)	0.622
Steady State Response (V)	0.046

Summary: Even though this RLC circuit is not a regular series RLC circuit, we can still use the same method to get the response. The difference is that the damping factor and undamped natural frequency would not be R/2L and 1/sqrt(LC). The correct α and ω0 can be found by using the characteristic equation. To get the characteristic equation, a typical way is to use mesh analysis or nodal analysis.

Series RLC Circuit Response

Today we started second order circuits. Second order circuits are RLC circuits or a simplified version of them, LC circuits. The first step is finding boundary values. i(0+), v(0+), di(0+)/dt, dv(0+)/dt, i(∞), v(∞) are examples of boundary values. We can then use a second order differential equation to solve for a second order circuit. The solutions to the characteristic equation, s1 and s2, are called natural frequencies, in nepers per second (Np/s). s1 = -α + sqrt(α^2 - ω0^2), s2 = -α - sqrt(α^2 - ω0^2). α is called the never frequency or damping factor. ω0 is the resonant frequency or the undamped natural frequency.

There are three types of solutions

1. If α > ω 0 , we have the overdamped case.
2. If α = ω 0 , we have the critically damped case.
3. If α < ω 0 , we have the underdamped case.

For overdamped case, 

i(t)=A1 es1t +A2 es2t 

For critically damped case, 

i(t)=(A2 +A1 t)e−αt 

For underdamped case,
i(t)=e−αt(B1 cosωd t+B2 sinωd t)
ωd is called the damped natural frequency, ωd = sqrt(ω0^2 - α^2)

Then we did a lab on an RLC circuit in series. Here is a schematic of the circuit

Here is the square wave we applied to the circuit

 We were able to observe a natural response as following

We compared the oscillation frequency from the graph to the theoretical value.

	Experimental	Theoretical	Percent Error
Oscillation Frequency (Hz)	101341.6985	99998.74999	1.34%

 

Summary: RLC circuits can be represented by using second order differential equation. Similar to damping oscillation in mechanics, RLC also have three different cases of oscillation. Our lab demonstrates the pattern of an underdamped case where the damping factor is smaller than undamped natural frequency.

Inverting Differentiator

Today we went over differentiators and integrators. They are based on the inverting amplifier. A differentiator circuit can be obtained by replacing the feedback resistor with an inductor or replacing the input resistor with a capacitor. An integrator circuit can be obtained by replacing the feedback resistor with a capacitor or replacing the input resistor with an inductor. Since inductors are expensive and rarely used, we only use capacitor to implement differentiator or integrator.

Then the concept of singularity is introduced. There are three common singularity functions. They are unit step function, unit impulse function and unit ramp function. They can be used to represent step response.

We did a lab on inverting differentiator. Here is a schematic of the circuit

Here is the setup of the circuit

We measured the experimental resistance of the resistor

We applied three different voltage inputs to the circuit. They are 100 Hz sinusoid, 250 Hz sinusoid, and 500 Hz sinusoid. We observed the output voltage on the oscilloscope

100 Hz

250 Hz

500 Hz

The phase shift on the graph is exactly π/2, which corresponds to the phase difference between a sinusoidal function and its derivative. We also compared the amplitude of the output voltage curve and the theoretical output.

100 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	100.00	100.00	0.00%
ω (rad/s)	628.32	628.32	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	0.43	0.43	0.64%

250 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	250.00	250.00	0.00%
ω (rad/s)	1570.80	1570.80	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	1.07	1.07	0.17%

500 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	500.00	500.00	0.00%
ω (rad/s)	3141.59	3141.59	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	2.12	2.14	0.76%

Summary: Differentiators and integrators can be implemented by using capacitors and resistors. It can be used to shift the phase of a sinusoidal input. Additionally, differentiators and integrators are also operational amplifiers circuits. It shows that operational amplifiers can not only be used to implement basic operations such as addition, subtraction, multiplication and division, but also calculus operations such as differentiation and integration.