5/19/16 Day 26: Op Amp AC Circuits, Oscillators, Instantaneous and Average Power and Max Power Transfer

We first quickly reviewed how to solve for op amp ac circuits, which involve nodal analysis and complex algebra. We the did a problem involving ac in op amp circuits. We then learned about oscillators, which convert dc inputs to ac outputs. This is done by using an op amp circuit with a gain of one or greater and by resulting in no phase shift between input and output. Another interesting part of oscillators is that their op amp has two feedbacks and not just one. We then reviewed a Wein-bridge oscillator, the simplest oscillator. After, we did a lab on the op amp relaxation oscillator. Lastly, we went over average and instantaneous power. Average power was found to be 1/2VIcos(v-i), where v and i are phase shifts for the voltage and current, respectively. We then did a problem involving power. Lastly, we quickly reviewed max power transfer for ac circuits, which occurs when the load impedance is the complex conjugate of the Thevenin impedance. The max power is also given by (V_th)^2/(8R_th).  

LECTURE:

In this example, the output voltage of this op amp ac circuit was determined. As in all op amp circuits, nodal analysis was used. The exception in this case is that impedances and time-varying voltages were taken into account. The answer was determined to be 1.029cos(1000t+59.04) V.

In this derivation above, the simplest Wein-bridge oscillator was solved for. It was found that for the resistance to be equal to the capacitance, the gain of such an oscillator is three and its angular velocity is 1/3. For such an oscillator, there is no imaginary part, only real values.

In this problem, the objective was to determine the average power each circuit element absorbs. Right off the bat we can tell that the average power absorbed by the capacitor and inductor are zero, since only resistors absorb an average power; capacitors and resistors absorb and release and equal amount of power, making the average zero. Then, the voltages and currents across each resistor were determined, and by this, the power was found. It was found that the source supplies 7.5 W of power, the 4 ohm resistor absorbs 5 W, and the two ohm resistor absorbs 2.5 W. This is expected, since the sum of the power in the circuit must be zero.

LAB:

Op Amp Relaxation Oscillator:

Purpose:

The purpose of this experiment was to generate a relaxation oscillator and to analyze its output behavior. A relaxation oscillator that would generate oscillations close to the expected (given from EveryCircuit) was also another goal of this lab.

Prelab:

The purpose of the prelab was to determine the resistance value of inverting feedback resistor needed to generate a frequency of 159 Hz in the relaxation oscillator. The other two parallel resistors were assumed to be equal. Beta was found to be a half and from this the resistance was determined from the period function T = 2 R C ln ((1+beta)/(1-beta)). 

Then, the determined resistance (2.86 kOhm) was tested using EveryCircuit by determining the resulting frequency of adding that value of resistance. Based on EveryCircuit, the frequency was found to be 158 Hz using such a resistance, which is very close to the needed value of 159 Hz.

Procedure:

The circuit was put to the test by first building it, as seen above. The schematic of this circuit is seen in the prelab above. For the two parallel resistors, a value of 1 kOhm was used. In addition, a 1 uF capacitor was also used. Then, using Waveforms, the output voltage and the voltage across the capacitor were measured on the oscilloscope window. The windows are shown below in the data section. Also, a 3 kOhm resistor was used instead of 2.86 kOhm, since that was the closest resistance value available.

Data:

Above is the oscilloscope window for the voltages as functions of time across the capacitor and the output voltage. The yellow graph is the voltage across the capacitor, and the blue is the output. As can be seen, the op amp goes repeatedly through positive and negative saturation, creating a square wave. In addition, comparing this window to the obtained graph in EveryCircuit, they are very similar, which shows that the procedure was performed correctly.

Data Analysis / Conclusion:

Based on this window, the frequency was experimentally calculated. The calculation is shown below: 

 As can be seen, the experimental frequency (163.7 Hz) was close to the expected (159 Hz), which shows that the method used to analyze oscillators is correct. There is a percent difference of 2.96%, and most, if not all, of this small difference is due to using a 3 kOhm resistor instead of a 2.85 kOhm.