5/11 Phasors: Passive RL Circuit Response

Today, we talked about sinusoids and phasors.
A sinusoid is a signal that has the form of the sine or cosine function.

where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
ωt = the argument of the sinusoid

A sinusoid can be expressed in either sine of cosine form. With these identities, it is easy to express both as either sine or cosine with positive amplitudes.

By using these relationships, we can transform a sinusoid from sine form to cosine form or vice versa.
The picture below is the graphical approach.

To add A cos ωt and B sin ωt, we note that A is the magnitude of cos ωt while B is the magnitude of sin ωt.
The magnitude and argument of the resultant sinusoid in cosine form is readily obtained from the triangle.


Next, we talked about phasors, which are more convenient than sinusoids to work with sine and cosine functions.
A phasor is a complex number that represents the amplitude and phase of a sinusoid.


To deal with the complex numbers calculations, the following operations are important.

We did a little example of how to transform rectangular form to polar form.

Also, we did a example of the calculation of polar form.

Next, we introduced how to determine which one the leading direction and lagging direction are.

Then, we did a example of two sinusoids function addition.


The derivative v(t) is transformed to the phasor domain as jωV
The integral of v(t) is transformed to the phasor domain as V/jω
These are useful in finding the steady-state solution, which does not require knowing the initial values of the variable involved.
The picture below is the example of the transformation of the derivative of v(t) and the integral of v(t) to the phasor domain.

The table below summarizes the time-domain and phasor-domain representations of the circuit elements.

Then, we did lab.

Phasors: Passive RL Circuit Response
Pre-lab

The picture above is the amplitude gain and phase difference between the input voltage and the input current with different frequency.

The picture below is the basic set up for this lab.


Result:

With the cutoff frequency, the phase difference is (16.87)/(135)*360 = -44.9degree, and the gain is 13.28/668.3 = 0.01987


With ten times the cutoff frequency, the phase difference is (3.102)/(13.5)*360 = -82.72degree, and the gain is 2.155(mA)/223.2(mV) = 0.009655


With one tenth of cutoff frequency, the phase difference is (0.03)/1.45 = -7.448degree, and the gain is 0.001675/1.013 = 0.0016535


The picture above is the comparison of experimental and theoretical data.
From the data, we can see that we successfully finished the lab, especially on the phase difference.

Summary

We learned about sinusoids and phasors, which are different ways to deal with AC circuit. Also, we learned how to determine the phase difference and gain according to the graph. By comparing the data, we can see the experiment matched what we expected.

5/9 RLC Circuit Response

Today, we talked about Schmitt Trigger, whihc is a logic input type that provides hysteresis or two different threshold voltage levels for rising and falling edge. This is useful because it can avoid the errors when we have noisy input signals from which we want to get square wave signals. (From Day 19 2nd Order Circuits Contc.pdf)
The picture below is what the operational amplifier based schmitt trigger looks like:

Next, we introduced the non-symmetrical schmitt trigger.

The picture below is what the the non-symmetrical schmitt trigger looks like.

Then, we talked about the step response of a series and parallel RLC circuit.

Review questions:

Next, we did lab.

RLC Circuit Response

Pre-lab

The picture is the differential equation relating Vin and Vout for the system.

By using the equation, we can determine the damping ratio and natural frequency of the circuit.

The damping ratio is 1563.8 and natural frequency is 10105.8Hz.

The picture below is the basic set up for this lab.

Result:

The damping ratio from the graph is 0.25, rise time is 2.15*10^-4, DC gain is 8.

After lab, we talked more about the second order circuits.

Summary

We learned about what the Schmitt Trigger is, and reviewed about RLC circuit which is second order circuit.  In order to recover the transmitted analog

signal, the output is smoothed by letting it pass through a “smoothing” circuit, as illustrated at right. An RLC circuit may be used as the smoothing circuit.

5/2 Series RLC Circuit Step Response

Before, we considered the circuits with only one element (a capacitor or a inductor), and the circuits are first-order.
Today, we talked about the circuits with two storage elements which is known as second-order circuits because their responses are described by differential equations that contain second derivatives.
The main idea of this kind of problems is to get the value of  v(0), i(0), dv(0)/dt, di(0)/dt, i( ∞ ), and v( ∞ ). By determining these value, we can solve the problems easily.
Next, we talked about the source free series RLC circuit.
 
By determining  α  and ωo, we can see what type of this circuit is.
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.
We did a example of the source free series RLC circuit.


Series RLC Circuit Step Response
Pre-lab

This is the prediction of differential equation, damping ratio, natural frequency, and damped natural frequency of this lab.

The picture below is the basic set up for this lab.

The graph below is the result for this lab.

The  rise time of the graph is 1.241ms.
The overshoot is 3.5ms.
The oscillation frequency is 23485Hz.
Out estimated damping ratio, natural frequency and damped natural frequency is in the pre-lab.
Our estimated DC gain is 2.383 (in the beginning of the transient).

Summary

From today's lecture, we learned the source free series RLC circuit and the source free parallel RLC circuit. By determining α and ω 0 , we can determine which case the circuit is. From the lab, we learned how to determine the rise time, overshoot time, and the oscillation frequency. And from the graph, we can see that there is a sudden change when we apply the voltage to the circuit. An automobile ignition system takes advantage of this feature. By creating a large voltage (thousands of volts) between the electrodes, a spark is formed across the air gap, thereby igniting the fuel.

4/25 Inverting Differentiator

Today, we introduced an integrator which is an op amp circuit whose output is proportional to the integral of the input signal, and a differentiator which is an op amp circuit whose output is proportional to the rate of change of the input signal.
We derived a equation for the integrator.

Different input voltage will have different output voltage.

Then, we talked about the switching function including step function, impulse function, and ramp function.
The unit step function u(t) is 0 for negative values of t and 1 for positive values of t.


Next, we did lab.

Inverting Differentiator
Pre-lab

The frequency should be 234Hz to get a gain of -1.
The actual resistor we used is 665ohms.

Apply a sinusoidal input voltage with frequency = 100Hz, amplitude = 1V, and offset = 0V to the circuit of Figure 1.

Apply a sinusoidal input voltage with frequency = 250Hz, amplitude = 1V, and offset = 0V to the circuit of Figure 1.

Apply a sinusoidal input voltage with frequency = 500Hz, amplitude = 1V, and offset = 0V to the circuit of Figure 1.

The table below is the comparison of the expected output voltage and the experimental output voltage.

Summary

We learned about integrator and differentiator, and step function, impulse function, and ramp function. By doing the lab, we can see that how differentiator work and how it affects the circuit by looking at the result of the graph and the table. 

4/18 Passive RC Circuit Natural Response & Passive RL Circuit Natural Response

We started with inductors with series and parallel.
It's the same as resistors.
The equivalent inductance of series-connected inductors is the sum of the individual inductances, and the equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

Here is the example we did in the class.

The equivalent inductance for this question is 15H.

Next, we talked about first order RC and RL circuits.

A source-free RC circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor is released to the resistors. A circuit response is the manner in which the circuit reacts to an excitation.

(From Day 16 1st Order Circuits.pdf)

There are two important relationships:

Both relations take 5 time constant to reach the final state or the steady state.

We did a example.

Passive RC Circuit Natural Response

Pre-lab

Estimation the time constant for the circuits of Figures 2(a) and 2(b).

For figure 2(a) the 5 time constant = 0.352s, and for figure 2(b) the 5 time constant = 0.0756s

The actual resistance for R1 = 0.977 ohms and R2 = 2.16k ohms.

This is the basic set up for this lab.

Figure 2(a)

We got almost the same value for the 5 time constant. (Forgot to take a picture for the initial time that started to drop)

Measured 5 time constant = 0.343s

Calculated 5 time constant = 0.352s

% error = 2.56%

Figure 2(b)

We got almost the same value for the 5 time constant. (Forgot to take a picture for the initial time that started to drop)

Measured 5 time constant = 0.08158s

Calculated 5 time constant = 0.0756s

% error = 7.91%

Passive RL Circuit Natural Response

Don't have time to do pre-lab

The picture below is the result for figure 2(a)

Summary

We learned how to deal with inductors with series or parallel, RC and RL circuits, and how to determine the time constant in a real world circuit.

Recently, if the class had the second lab at that day, the time would not be enough to finish the second lab. We spent to much time on the first lab. Limiting the time for the first lab would help to solve this problem.

4/13 Capacitor Voltage-current Relations & Inductor Voltage-current Relations

We talked about capacitors and Inductors today.

Capacitor

A capacitor consists of two conducting plates separated by an insulator (or dielectric). (From Day 15 Capacitors and Inductorsb.pdf)

The unit is farads (F).

We started with figuring out the unit of the capacitance. 

C is proportional to area/distance which is the unit m.

In order to get the correct unit, we need to have a constant which is the permittivity of the dielectric material between the plates.

Next, we did a example.

The video is when we apply too much voltage to the capacitor. We exploded a capacitor.

Capacitor Voltage-current Relations

Pre-lab

We predicted that the capacitor voltage and the capacitor current if the capacitor voltage is a sinusoidal wave and a triangular wave.

The picture below is the prediction.

The picture below is the basic set up for this lab.

The actual resistance is 99.2 ohms.

The picture below is when applying a sinusoidal input voltage with frequency = 1kHz, amplitude = 2V, and offset = 0V to the circuit.

The picture below is when applying a sinusoidal input voltage with frequency = 2 kHz, amplitude = 2V, and offset = 0V to the circuit .

The picture below is when applying a triangular input voltage with frequency = 100 Hz, amplitude = 4V, and offset = 0V to the circuit.

Next, we leaned how to deal with the problem that the capacitors are series or parallel. It's the opposite of resistors.

For the example above, the equivalent capacitance would be 20 micro farads.

Inductor

An inductor consists of a coil of conducting wire. (From Day 15 Capacitors and Inductorsb.pdf)

The unit is henrys (H).

We did a example. 

By applying the equation, we got the energy stored in the inductor at t = 1s would be 35.7J.

Then, we did the second lab.

Inductor Voltage-current Relations

The actual resistance is 99.2 ohms.

The picture below is the basic set up for this lab.

The picture below is when applying a sinusoidal input voltage with frequency = 1kHz, amplitude = 2V, and offset = 0V to the circuit.

The picture below is when applying a sinusoidal input voltage with frequency = 2 kHz, amplitude = 2V, and offset = 0V to the circuit.

Summary

We leaned how capacitors and inductors work in the circuits, and how to calculate the equivalent capacitance when the capacitors are in series or parallel, and why we use capacitors and inductors in a circuit. Also, we learned the important equations for calculating the energy stored in the capacitors and the energy stored in the inductors.