05/19 - Lab 24: Inverting Voltage Amplifier

Lab 24: Inverting Voltage Amplifier

In this lab, we explored the use of op amps with capacitors in order to gain an understanding of the steady-state circuit response to sinusoidal inputs. By comparing input and output signals, the circuit can be seen to provide some amplitude gain between the input and output signals, as well as a phase shift between the phasors of input and output. We measured the gain and phase change response of an inverting op amp and compared them to our theoretical expectations.

The circuit used in this lab is shown in the diagram below.

Our pre-lab assignment was to determine the value of R such that the angular frequency obtained was 864 rad/s.

Next, we built the circuit from the diagram above. Because there were no resistors near 526 Ohms, we decided to place 2 1k resistors in parallel to achieve a resistance of 500 Ohms. The actual circuit we built is pictured below.

We were able to obtain a square wave graph from the output voltage with an amplitude of 4V. This means we were able to take a sinusoidal voltage input and get a voltage output that looks like a square wave, where voltage is essentially switching between 4V and -4V.

In Class Examples:

1. Determine vo (t) for the op amp circuit below if vs = 3 cos 1000t V.

2. Refer to the circuit depicted below. Find the average power absorbed by each element.

3. Find the value of ZL in the circuit of below for maximum power transfer.

05/17 - Lab 23: Passive RL Circuit Response

Lab 23: Passive RL Circuit Response

In this lab, we calculated the theoretical phase and gain response of an RL circuit and compared the results to actual measurements taken from a passive RL circuit. The circuit used was very simple and only contains a 47 Ohm resistor in series with a 1 mH inductor.

The cutoff frequency was given as R/L, and we were instructed to perform our calculations and measurements of the circuit for the cutoff frequency, 10 * cutoff frequency, and 1/10 * cutoff frequency. The actual resistance of the 47 Ohm resistor was 46.9 Ohms, and the true inductance of the inductor was 0.963 mH. Using these values, we calculated the cutoff frequency to be w = 48182.8 rad/s, and thus a frequency of f = 7669 Hz. The actual circuit used for this experiment is pictured below.

First, we used the cutoff frequency as our voltage input, with an amplitude of 2V. White board calculations and measurements from the actual circuit are pictured below.

The impedance of the inductor was calculated as jwL = 46.4j Ohms. Since the resistor is 46.4 Ohms, the total impedance was calculated to be sqrt(46.4^2 + 46.4^2) = 65.6 Ohms, with a phase shift of atan(1) = 45°, thus Z_tot = 65.6 Ohms, 45°. The current in the circuit can be calculated as V/Z, which is 2, 0°/65.6, 45° = 30.5, -45° mA. The voltage across the inductor was calculated by multiplying the current by the impedance of the inductor, which yielded V_L = 1.42, 45° V. The results obtained from measuring the actual circuit were nearly spot on to our calculations.

Oscilloscope at f = 7669 Hz (cutoff frequency)

Blue = Input Voltage

Red = Current

Oscilloscope at f = 7669 Hz (cutoff frequency)

Blue = Input Voltage

Yellow = Inductor Voltage

Red = Current

The wrong scale was used in the second set of graphs (it should have shown the yellow scale to show voltage across the inductor); however, looking closely it can be seen that the scale used for both voltages are 0.5V / division. The input voltage was 2 V, and the voltage across the inductor is slightly less than 1 division below the input voltage, thus the inductor voltage we calculated was correct at ~1.4 V. It can also be seen that the inductor voltage is out of phase with the input voltage by ~45°.

Next, we changed the input voltage frequency to be 1/10 the cutoff frequency; thus, w = 4818.3 rad/s, and f = 766.9 Hz. White board calculations as well as measurements taken from the circuit are pictured below.

Changing the input frequency affects the impedance of the inductor (by a factor of 1/10), and thus the total impedance was much smaller than the previous setup at Z_tot = 46.6, 5.71° Ohms. The current and voltage were calculated using the same method as before, yielding values of 43.0, -5.71° mA, and 0.2, 34.4°

Oscilloscope at f = 766.9 Hz (1/10 cutoff frequency) 

Blue = Input Voltage

Yellow = Inductor Voltage

Red = Current

Oscilloscope at f = 766.9 Hz (1/10 cutoff frequency)

Blue = Input Voltage

Yellow = Inductor Voltage

Red = Current

The first set of graphs shows the vertical scale in terms of mA, and it can be seen that the peak current measured across the circuit was just over ~42 mA with a very slight phase shift from the input voltage, which was nearly identical to our white board calculation. The second graph shows the same image but with a scale indicating the voltage. The voltage across the inductor was measured to be ~0.25 V with a slightly lower phase shift than the first setup.

Finally, we tested the same RL circuit setup but with the input frequency at 10 * the cutoff frequency, or w = 481,828 rad/s, and f = 76,695 Hz. The white board calculations as well as measurements taken from the actual circuit are pictured below.

Since w was increased by a factor of 10, the impedance of the inductor was also increased by the same factor, making Z_L = 464j Ohms, and thus Z_tot = 466.3, 84.3° Ohms. Using the same calculations as previous setups, the current and voltage were calculated to be 4.29, -84.3° mA

Oscilloscope at f = 76695 Hz (10 * cutoff frequency)

Blue = Input Voltage

Yellow = Inductor Voltage

Red = Current

Oscilloscope at f = 76695 Hz (10 * cutoff frequency)

Blue = Input Voltage

Yellow = Inductor Voltage

Red = Current

In the first series of graphs, the vertical scale indicates the voltage across the inductor. Just as we predicted, the voltage across the inductor is nearly the same as the input voltage, with a very slight phase shift of ~5°. The second set of graphs shows the same thing but with the scale depicting the current through the circuit in mA. We predicted a current with an amplitude of 4.3 mA, and the measurements taken show this result exactly. The phase shift of the current was calculated to be -84.3° from the input voltage, which is precisely what our measurements indicate.

In Class Examples

1. Find i x in the circuit shown using nodal analysis.

2. Determine current Io in the circuit below using mesh analysis.

3. We were asked to use mesh currents again for the previous example, but with only a current source that delivers I = -2.353 + j2.353 A

05/12 - Lab 22: Impedance

Lab 22: Impedance

The purpose of this lab was to gain understanding of the concept of impedance in resistors, capacitors and inductors. When AC current is applied to a circuit, these elements will exert a small amount of resistance due to the changing current/voltage of the AC input. Three different circuits were analyzed, one containing two resistors in series, one containing a resistor and inductor in series, and another containing a resistor and capacitor in series. The input voltage was a sinusoidal wave with amplitude 2 V and frequencies of 1000, 5000, and 10000 Hz were tested on each circuit.

In each circuit, R = 47 Ohms. The voltage across this resistor was recorded in each circuit and divided by 47 to determine the amplitude of the current in the circuit. All input and output voltages, and the function for the current, were recorded using the oscilloscope in the Waveforms program.

Actual values for each circuit element

Before starting the lab, a wall of math was created to determine the function for the current through each circuit for various input frequencies.

The first circuit tested was the one containing two resistors. This circuit contains one 47 Ohm resistor in series with a 100 Ohm resistor. The actual circuit, as well as the oscilloscope readings are pictured below.

1000 Hz

Current through the circuit in mA

5000 Hz

Current through the circuit in mA

10000 Hz

Current through the circuit in mA

According to our wall of math, the current across the 47 Ohm resistor should have been 13.4 mA; we obtained an actual current with an amplitude of 15 mA, which is very close to our prediction.

The second circuit tested contained the same 47 Ohm resistor in series with a 1 mH inductor. The actual circuit, as well as the recorded oscilloscope are pictured below.

1000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

5000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

10000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

According to the calculations in our wall of math, the amplitude of current should have been slightly less than 40 mA for each frequency, and the phase should shift slightly negative (to the right) as the frequency increased. The results shown above indicate that the phase slightly increases as the frequency increases, just as predicted. The actual currents turned out to be slightly off from what we predicted, but the trend that the current slightly decreases as frequency increases was still seen.

The final circuit analyzed contained the 47 Ohm resistor in series with a 47 micro Farad capacitor. The actual circuit, as well as the oscilloscope readings, are shown below.

1000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

5000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

10000 Hz

Blue = V_in  ,  Yellow = V_out  ,  Red = Current (mA)

According to our predictions, the amplitude of current should go from 1 mA at 1000 Hz to 4.6 mA at 5000 Hz and 9.1 mA at 10000 Hz. So, the amplitude scales by the same factor as the frequency of input voltage. Though the value of our amplitude of current does not match with the results shown in the oscilloscope, it can be seen that the current scales in the same manner as predicted. Also, the phase shift changes with the frequency just as we predicted. It goes from -89° at 1000 Hz to -84° at 5000 Hz, and to -77° at 10000 Hz. In other words, the phase starts out at ~ -90° and slowly shrinks down as the frequency increases, which brings the input and output voltages closer together on the oscilloscope readings.

In Class Examples

1. Find v(t) and i(t) for the circuit below:

2. Given that is(t) = 5 sin (1000t) A, find vc(t):

3. For the circuit below, find Zin for w = 1000 rad/s where the inductor is 20H

4. Determine vo(t) in the circuit shown.

5. What is the phase shift if R = C at a particular frequency?