4-21: Inverting Differentiator

Inverting Differentiator

In this lab, we use an op amp (OP 27) and a capacitor to create an inverting differentiator. This means that the output voltage V_o is (a multiple of) the negative derivative of the input voltage V_in, or V_o = -RC*(dv_i/dt), where R is the feedback resistance, C is the capacitor capacitance, and v_i is the input voltage. If the input voltage v_i = Acos(ωt), where A is the input amplitude and ω is the angular frequency, then its derivative is (dv_i/dt) = -Aωcos(ωt). Thus the output voltage is V_o = RCAωsin(ωt).

 Here are the formulas related to the change of the voltage and current with time for capacitors, and inductors

Pre-Lab: We determine the Vo, V1, V3 ,use the data under.

When f=1kHz, A=1V, offset=0V, The display the waveforms for Vo A= 1.144V and Vin A=1V.

When f=500Hz, A=1V, offset=0V, The display the waveforms for Vo A= 0.586V and Vin A=1V.

When f=2kHz, A=1V, offset=0V, The display the waveforms for Vo A= 2.204V and Vin A=1V.

Theoretical value for the Vout  is much clossed to the experiment value. The min % different is 2.3% the max %different is 8.3%. 

 it has a  amplitude = 2.204V and the same pi/4 phase shit.

Summary: In this lab, we learned  the output voltage has a derivate with respect to the time of the input to the circuit relationship. With a higher frequency the the op amp will have a higher gain for the output, and with a lower frequency the op amp will have a lower gain for the input. 

4-28: Series RLC Circuit Step Response

Series RLC Circuit Step Response

Pre-Lab: Based on the characteristic equation, we can figure out the natural response of the RLC circuit. Alpha is defined as R/2L, and omega is defined as square root of 1/LC. The circuit is overdamped when alpha > omega, critically damped when alpha = omega, and underdamped when a < omega.

Theoretical value for the w, wd  is much clossed to the experiment value, but a has big error. 

This is the resulting output graph from the capacitor: We experimental alpha is 1.5*10^6, which is about172% of our theoretical alpha value of 550000. Possible source of error is from our input variables, the way we measured the circuit.


Summary: In this lab, we went over the basic principles of series RLC circuit, and did a lab to see how this type of circuit works when critically damped.

4-30: RLC Circuit Response

RLC Circuit Response

The purpose of this lab is to see the measured response of a second order circuit based on damping ratio and natural frequency. In this lab we are testing the step response and comparing the measured values to theoretical values that we derived from equations in class. These values include the damping ratio and natural frequency of the circuit.

We built the circuit with a 1.1Ω resistor, a 47 Ω resistor, a 10uF capacitor and a 1 uH inductor. 47 ohm resistor and the inductor are in series while they are in parallel with the other components in the circuit.

                                   

In the lab, the 47 ohm resistor has a real value of 49 ohm. The 1.1 ohm resistor has a real value of 4.3 ohm. The 10 uF capacitor has a real value of 9.92 uF, and we cannot measure the 1 uH inductor.  

We calculated our experimental value of α to be 3588.7, while our theoretical value is 1063.8. It has a precent difference of -237.4%. 

This is the input of our power. It is a square wave at frequency at 100 Hz, and amplitude of 2V at offset 0

Summary:

In this lab, we learn how to analyze circuits containing a resistor, capacitor and an inductor in parallel. There are three possible cases that may occur in a RLC circuit which include over damped, underdamped, or critically damped. Overall, we can see the relationship the circuits have in each other depending on the arrangement of the circuit and how this will affect the damping and natural frequency of the circuit.

5-07: Impedance

Impedance

In this lab we measured impedances of resistors, capacitors, and inductors. We then compared them to their expected values. The impedance of a resistor is just Z = R, so the impedance is really just the resistance of the resistor. For a capacitor, the impedance can be calculated as Z = 1/(jωC) , where j is the imaginary component, ω is the angular velocity of the input, and C is the capacitance of the capacitor. The impedance of an inductor is Z = jωL, where L is the inductance of the inductor. The phase angles for a resistor, capacitor, and inductor are 0°, -90°, and 90°, respectively.

The impedance and admittance of resistors, capacitors, and inductors.

Pre-Lab: We determined the resistor impedance = 47 + R , the real resistor values came out 48.7 ohm and 100.1 ohm, and therefore the expected impedance for resistors = 148.8 ohm  Inductor impedance = 48.7 + 0.00628j  Capacitor impedance = 48.7 - 1711.34j 

Output for circuit with input frequency of 1 KHz

part A: When frequencies are: 1kHz, 5kHz, 10kHz the V and i. And the time different is 0.

Output for circuit with input frequency of 5 KHz

Output for circuit with input frequency of 10 KHz

Part B: When frequencies are: 1kHz, 5kHz, 10kHz the V , i, time different  and z.

Output for circuit with input frequency of 1KHz

Output for circuit with input frequency of 5 KHz

Output for circuit with input frequency of 10 KHz

In the three graphs we see that the current lags the voltage by 90° as expected. Also, the voltage across the inductor increases as the frequency increases.

Part C: When frequencies are: 1kHz, 5kHz, 10kHz the V , i, time different  and z.

Summary: In this lab,  we went over impedance and admittance, and how to solve for them within circuits involving Resistors, Inductors, and Capacitors. We also did a lab to see how impedance works in real life.

5-12: Phasors: Passive RL Circuit Response

Phasors: Passive RL Circuit Response

In this lab, we measured the gain and phase responses of a passive RL circuit and with these experimental values to compare with theoretical values mathematically.

Pre-Lab: We did a pre-lab analysis to predict the amplitude gain and phase difference with 3 different frequency: Wc, Wc/10, and 10Wc (Wc = 2.2*10^5 Hz). When W= Wc, expected amplitude gain = 0.032

and expected phase difference = -45degree. When frequency = Wc/10 expected amplitude gain = 0.045

and expected phase difference = -5.7 degree. When frequency = 10Wc expected amplitude gain =24.5*10^-3 and expected phase difference = -84 degree

When W= Wc these were our results:

When W= Wc/10 these were our results:

When W= 10Wc these were our results:

The circuit as illustrated previously in the schematic

The data for exp value. The amplitude gain is different, becaese we used  I/V to calculate the  amplitude gain in pre-lab, and we use the Vout/Vin in the final calculation. 

Summary:In this lab,  we went over how to solve sinusoidal circuits, did a lab to see the gain and phase change in a RL circuit, and learned to solve for Thevenian and Norton equivalent of AC circuit.

3/31 :Summing Amplifier, Difference Amplifier

 Summing Amplifier & Difference Amplifier

 In this lab, we used an op amp to sum two input voltages.
Practice problem on solving op amp circuits before Lab:  We leaned how to used node to calculate the V0, V2, and Vs.

Buffer Amplifiers, which doesn’t have much change in voltage when there is a big change in output voltage, therefore Vin = Vout.Inverting Amplifier, which produces an output with opposite sign, Vout = -Rf/Ri * Vi. 

Pre-lab for Summing Amplifier : we used the Vout = - R3/R1 * (Va+Vb) to get the R3 should less than 600ohm.  

This is the result table of our data, with Va varying from -4V to +5V, and Va keeps constant at +1V. We do fine on this lab. 

To build the circuit, we used two 1k resistors, one 220  resistor, one OP 27, one Analog Discovery device to provide the two input voltages, and a multimeter to measure the output voltage:

Difference Amplifier: How to used node to calculate the Va, Vb, and Vo.

Pre-lab for Difference Amplifier: We figure out that if we use four identical resistors, the out put voltage is simply Vout = Va - Vb.

This is the set up of this experiment.  We build the circuit with four 10k resistors, one Op Amp 27, an Analog Discovery device to provide +/-5V and two input voltage. 

This is the result table of our data, Va keeps constant at +1V. We do fine on this lab. We get a minimum % difference of 1% and a maximum of 3%,the results were very close to what we expected 

Summary: In this Lab we learn more about about different types of OP Amp and their distinct utilities, and did two labs demonstrating how OP Amps can be used to combine and subtract two input signals.

Lab 10 (03/26): Inverting Voltage Amplifier

 Inverting Voltage Amplifier

Class practice

Per-lab: We use R1=2.2K R2=4.4K.

We built following non ideal op amp circuit.  Measured vout values and compared it with v expected.


Mearsurment for the Vin and Vout

Graph of Vout vs Vin.