Phasors: Passive RL Circuit Response

Before proceeding to the lab, we did a number of exercises with phasors and phasor arithmetic.

Circuit diagram with calculation of cutoff frequency and voltage gain for a number of frequencies.

Implemented circuit, analog discovery connected to measure input voltage and inductor voltage.

Measured input and output voltage and calculated current at 470krad/s

Summary

Calculating gains as Vo/Vi and reading phase angles from oscilloscope output, then comparing to theoretical gain and phase angle arctan(w L / R), we find errors of less than 5% for gain in all frequencies tested. Phase angles, however, had errors of 10-20%. Overall, our theoretical expectations are largely consistent with the practical implementation.

Thevenin's Theorem

A number of exercises on Thevenin equivalents accompanied the lab,

in addition to a brief treatment of Norton equivalents.

Circuit diagram and calculation of Thevenin Equivalent.

Implemented circuit, multi-meter configured to measure the Thevenin equivalent voltage and, with the sources short circuited, resistance

Circuit reconfigured with potentiometer as load resistor, multi-meter measuring pot resistance.

Table of measured load resistances and voltages.

Plot of power vs load resistance, calculated as V^2/R

Summary

We find Thevenin's Theorem to be accurate, the theoretical equivalent voltage and resistance matching the measured values within 1%. Additionally, we find some support for maximum power being obtained when load resistance is equal to the Thevenin resistance. While more data points are necessary to properly confirm, the form of the data appears to be quadratic with a peak in the neighborhood of the Thevenin resistance.

Superposition

Before the lab, we did a number of exercises including

mesh analysis with dependent sources and linearity

Circuit diagram including both sources. Below are calculated voltages

on the left and measured voltages on the right.

The implemented circuit, multi-meter configured to measure total voltage with both sources in place.

Summary

The results of this experiment are not satisfactory to confirm the validity of the superposition principle. While the measured total voltage differs from the theoretical by a mere 10%, the single source voltages are of concern. Firstly, they are each in error of 50%. Worse, they do not add to the total measured voltage, the error there being 36%. In conclusion, we see that the total voltage calculated via superposition is correct within reasonable error, but that the singe source voltages are wildly inaccurate. This is likely the result of mistakes in the implementation of the circuit or the supplies.

Nodal Analysis

Circuit diagram and calculation of expected voltages via nodal analysis.

Implemented circuit, multi-meter configured to measure V2.

Multi-meter configured to measure V1

Beginning exercises in mesh analysis followed the lab.

Summary

While the theoretical values are uncertain (there was quite some variation across the class), the measured values appear to support nodal analysis as an effective means of evaluating the function of a circuit. There is some error, which may be attributed to variation in resistance values or mistakes in the implementation of the circuit, such as reversed supplies, none of which were verified at the time of the experiment.

Dusk-to-Dawn Light

Circuit diagram, where variable photocell resistance controls base voltage of BJT and, in turn, emitter voltage for LED.

Experimental setup, multi-meter configured to measure base voltage while photocell

is under full brightness and subsequently while occluded.

Multi-meter configured to measure emitter voltage under both photocell conditions

Video of circuit in operation.

Summary

The circuit behaves as expected. Subject to light conditions, the photocell has low resistance, dividing the supply voltage and leaving the BJT inactive. When occluded, the photocell has high resistance, resulting in a higher base voltage where the BJT is in the active region and the LED is sufficiently supplied to turn on. Confirmation of the base voltage under specific photocell resistances, while preferable, was not possible due to the high sensitivity under experimental conditions. This could be rectified by varying ambient light or constructing a stable occluding apparatus instead of occluding the photocell by hand.

Dependent Sources and MOSFETs

 Circuit diagram, where MOSFET receives variable gate supply voltage

and the resultant drain current may be found

The implemented circuit, with current measurement while MOSFET is in saturation.

 Collected data, consisting of supply voltages and corresponding current measurements.

Plot of data.

Summary

While a relationship between gate voltage and drain current can be seen, the collected data does little to reveal its nature. From our knowledge of semiconductors, it is likely that the MOSFET turned on and promptly entered the saturation region as gate voltage was increased. As a result, we do not have enough information to confirm the behavior of the MOSFET as a voltage controlled current source nor can we calibrate operation were we to implement it into a larger circuit. This could be rectified by revisiting the experiment and selecting finer voltage steps within the active region.

Ohm's Law

Experimental circuit, consisting of a variable voltage source and 100 Ohm load resistor.

Experimental setup, multi-meter connected in parallel to measure voltage.

Multimeter connected in series to measure current 

Collected data, organized in a table of corresponding voltages and currents

(also an exercise on the relationship between circuit branches, loops, and nodes)

 Plot of data, with linear fit.

Summary

With the definition of Ohm's Law, V=IR, rearranged into the form R=V/I, resistance can be found as the slope of a V vs I plot, as our data has been fit. The slope found from our experimental data was 0.1 V/mA, corresponding to a resistance of 100 Ohms, exactly as expected. Also of note is that all data points fall on the fit line, showing little deviation from the expected linear behavior. We can conclude that Ohm's Law has been experimentally verified as accurate.