Day 9: Maximum Power Transfer Theorem and Resistance Measurement

In class we learned about maximum power transfer theorem, which allows us to find the maximum power that a circuit could deliver to a resistor based on thevenin voltage and resistance. In order for a resistor to draw the most power it could out of a circuit, its resistance must equal the thevenin resistance, which was derived in class by taking the partial derivative and setting equal to zero the equation for power as a function of the load resistance. The equation is found below:

From the derivative with respect to load resistance, it was found that thevenin resistance must equal load resistance. Plugging in thevenin resistance for load resistance, we obtain:

Then, we did a problem involving finding max power in a variable resistor. A problem was also done involving applying these concepts to a robotics application to find the internal resistance in a battery. In addition another problem finding the load resistance for max  power in that resistor was done.

We then did a lab on Maximum Power Transfer, which is explained in more detail in the lab section. Lastly, we learned about source modeling and how engineers take into account internal resistance for sources so that the source provides the expected current or voltage. We also did an example on this application. In addition, we learned about a Wheatstone bridge circuit and how it can be used to measure internal resistance via a galvanometer; an example involving this application was performed along with a review problem that puts all of today's concepts together.

LECTURE:

This is the derivation to find the load resistance needed to obtain the maximum power in that resistor. It was done by taking the derivative with respect to the load resistance and setting it to zero since at max power the slope is zero. Again, it was found that the load resistance must equal the Thevenin resistance.

In this problem the Thevenin equivalents were found along with the current through the circuit if the load resistance was 8 ohms. Then, the resistance needed for max power was found (12 ohms) and the power at that resistance was calculated (33.33 W). In addition, power values at different load resistances was found to give us an idea of the curve for power as a function of load resistance.

This problem is the robotics application of max power. The motors in parallel were treated as resistors, and the Thevenin resistance was found. The max power was then found, and using the value that energy needed and amphours used to run this robot was found.

In this simpler problem the load resistance needed to obtain max power was found, which was just using the addition of resistors to find the Thevenin resistance.

In the problem above, the voltage provided to the load resistor was found using source modeling. The above problem shows that a practical source does not actually provide the expected voltage in every circuit it is applied to; it only approaches its expected source values, but only when the load resistance is high enough. If not, it actually provides a lower source value depending on the resistance of the load resistor, as shown above.

In this problem, the resistance needed for R_s in the wheatstone bridge so that no current goes through the galvanometer was found (50 ohms). This type of circuit is used to measure resistance more accurately for medium levels.

Lastly, this problem was just review of the maximum power transfer theorem, along with Thevenin equivalents and source transformations.


LAB:

Maximum Power Transfer:

Purpose:

The purpose of this experiment was to test the max power transfer theorem by setting up a simple circuit with a 5V voltage source, a source resistor of 4.7 kOhms and a variable resistor all in series. The variable resistor was adjusted at resistance values of 1 kOhm to 10 kOhms with an  interval of 1 kOhm, and the power through the resistor at those resistances was found by first measuring the voltage. (NOTE: This lab was performed different than what the lab manual contains; we were proving the maximum power theorem instead of just showing that its conjugate (source resistance equal load resistance) is false).

Apparatus:

The apparatus in this experiment consisted of an analog discovery (5 V source), a source resistor (4.7 kOhm), a potentiometer, a breadboard, a laptop with Waveforms software, a digital multimeter, alligator clips and wires.

Procedure:

The schematic of the circuit in the apparatus section was built as shown below:

The voltages across the potentiometer for each resistance value were then measured with a voltmeter. The resistance in the potentiometer was measured with an ohmmeter. The voltages were then used to calculate the power at each resistance, and the graph below was made using Logger Pro.

The above data was used to make the graph of power versus load resistance below. A model fit was then manually made using the equation for power, which was seen in the introduction.

Data Analysis:

Looking at the graph, it peaks at about 4700 ohms, which is the resistance of our source resistor. This confirms the maximum power theorem, which states that max power is obtained when the load resistance is equal to the source resistance. The equation for the model fit is shown more closely below:

Looking at the equation, A is our thevenin voltage squared, which is 25 volts squared. In the graph, it is in millivolts squared (25000). In addition, B is the source resistance, which is 4700 ohms. The correlation obtained is perfectly 1, which shows our model is an excellent representation of  the maximum power transfer theorem.