3/15/16 Day 7: Superposition (II) and Linearity in Electric Circuits, Time-Varying Signals and Source Transformation

In class today we first did the lab called "Time-varying Signals", where we utilized common voltage functions and analyzed the linear relationship between the functions of the voltage across a circuit element and the voltage across the entire series circuit. After this lab we learned about the linearity of circuits and how it could be useful when solving for currents and/or voltages in linear circuits that are in multiples of each other. Then, a problem using linearity was performed in class. After learning about linearity we learned about superposition in electric circuits, where the current and/or voltage resulting from each voltage and current source can be found seperately, then added together to fnd the true voltage and/or current in the circuit. After, we performed another experiment titled "Superposition II", which involves finding the voltage across a resistor using superposition, then building and testing it to see if superposition in circuits holds true. After the lab we learned about source transformations, where a voltage source can be transformed into an equivalent current source and vice versa depending on the resistance in series to a voltage source or parallel to a current source. This powerful concept was then put to the test in a problem to find the current across a resistor.

LECTURE:

Above is a problem testing the linearity theory of linear circuits. The current Io across the dependent voltage source was found using mesh analysis, when the voltage of the independent source is 24 V. Half of the class solved the problem with Vs being 24 V and the other half with Vs being 12 V. As seen for 24 V, the current Io is 0.32 A. The part of the class using 12 V showed that the current Io is 0.16 A. As can be seen, when the voltage of Vs was doubled the current Io was also doubled, proving the linearity of these circuits.

In this problem, the superposition theory was put to the test. First, the partial current (1 A) across the 5 ohm resistor was found by solely analyzing the 30 V voltage source, making the 6 A current source an open wire. Then, the 6 A current source was only taken into account, leaving the voltage source as a wire. The partial current (2 A) was also solved using this method. Then, the actual current Ix (3 A) across the 5 ohm resistor was found by just summing the partial currents determined using superposition. To test the theory of superposition, this circuit was plugged into EveryCircuit to find the true current value. It was also found to be 3 A, which shows that the theory of superposition is in fact true.

In this last problem, the source transformation analysis was used to determine the current across the 4 ohm resistor adjacent to the 12 V supply. The two current sources were transformed to voltage sources to simplify the circuit. The current through the 4 ohm resistor was found to be 0.111 A.

LAB:

Time-Varying Signals:

Purpose:

The purpose of this experiment was to determine the relationship between the functions of the voltage across a circuit element and the voltage across the entire series circuit. It was found that this relationship is linear; the voltage function across the element is the same, but of a smaller amplitude depending on the ratio of the amount of voltage it uses or provides with the total amount of voltage in the circuit.

Prelab:

The prelab of this experiment was to sketch the voltage functions of a sinusoidal wave, a triangle wave, and a square wave, and to determine the relationship between the voltage as a function of time across a resistor and the total voltage as a function of time in the circuit. The circuit used in this experiment, as as shown in the drawing above, is very simple and consists of two resistors in series. The two resistors were assumed to be equal in resistance.  Because of this, the voltage drop is the same between the teo two, which helped us predict that half of the voltage function of the circuit would be the voltage across the resistor

Apparatus:

The apparatus of this experiment consisted of the usual materials needed, such as a breadboard, two 100 ohm resistors, an analog discovery, a laptop with Waveforms software, a digital multimeter, wires and alligator clips. The different voltage functions were applied to the circuit using the Waveforms oscilloscope.

Procedure:

The schematic of this circuit, seen in the prelab above, was constructed using two 100 ohm resistors, the analog discovery box, and a breadboard. The different wave voltage functions were applied to the circuit using the yellow (W1) wires, and ground was placed on the other end of the circuit. It is not displayed in this picture, but the two orange wires (1+ and 1-) were placed across the resistor with the alligator clips.

Using the Waveforms software, the sinusoidal wave was created and then run into the circuit. The oscilloscope was then used to graph the voltage as a function of time across the circuit and the resistor with channel 1 across it. This is seen below:

As can be seen, the voltage function across the reistor (channel 1) is exactly half of the voltage across the circuit. Therefore, the frequency (1 kHz) and the period (1 ms) of each function is the same; however the only difference is the amplitude, where the amplitude for the resistor voltage function (1 V) is half the amplitude for the circuit voltage function (2 V). The same was also performed with a triangular wave, as shown below.

As can be seen, the period (1 ms) and the frequency (1 kHz) is the same for both voltage functions; however the only difference is the amplitude, where it is 1.5 V for the function across the resistor and 3 V across the circuit. Again, the function across the resistor is half the function of the circuit. The same was then done again for the square wave:

As expected the frequency (500 Hz) and the period (2 ms) is the same for both voltage functions. All that differs is the amplitude; the amplitude (2.5 V) of the circuit voltage is twice the amplitude (1.25 V) of the resistor voltage.

Data Analysis and Conclusion:

Above is the measured resistances of the resistor, along with the amplitudes and frequencies for each wave used.

As determined experimentally, the relationship between the voltage across the resistor and the whole circuit is linear; the voltage across the circuit is twice the voltage across the resistor, due to the resistance of the circuit being twice the resistance of the resistor under inspection.

Superposition II:

Purpose:

The purpose of this experiment is to find the voltage V across the 6.8 kohm resistor by using superposition for the 3 V and 5 V voltage sources. Then, the value of voltage obtained from the superposition theory was tested by building the circuit and then measuring the value using a voltmeter. The values were then also measured for removing the 3 V supply and then the 5 V supply from the circuit to see if superposition holds true.

Prelab:

The prelab of this experiment was to use the theory of superposition in electrical circuits to find the voltage V across the 6.8k ohm resistor. In the above picture, first the 3 V supply was not considered in the circuit, and the voltage across the resistor resulting from only the 5 V supply was found. Nodal voltage analysis was used to solve for this partial voltage.

The same was also done in that the other part of the partial voltage across the 6.8k ohm resistor was found resulting from solely the 3 V power supply. It was found that the voltage given by the 3 V supply is 1.99 V, and the voltage given by the 5 V supply was 0.71 V. Then, the total voltage across the 6.8k ohm resistor resulting from both supplies was found by adding the two partial voltages resulting from each isolated supply together. The true voltage across the 6.8k ohm resistor was then found to be 2.70 V.

To check this answer, the circuit was built in EveryCircuit, as seen below:

As seen above, the voltage across the 6.8k ohm resistor was found to be 2.70 V as well, showing that the theory of superposition does hold to be true.

Apparatus:

The apparatus of this experiment consisted of the same equipment used in the "Time-varying Signals" lab, with the exception of more resistors of varying resistances.

Procedure:

To see if superposition works in electrical circuits, the circuit seen in the Every Circuit schematic above was built using a breadboard, the analog discovery, resistors and wires, as seen below:

The 3 V power supply was provided using the Waveforms channel and the W1 wire. It was supplied to the 10k ohm resistor. The 5 V, given by the red wire, was provided to the 1k ohm resistor. The ground (seen by the yellow wire going out of the picture) was placed to the 22k ohm resistor, textbook to the schematic in the Every Circuit picture. The voltage across the 6.8k ohm resistor was then measured using a voltmeter (2.66 V).

The circuit was then slightly changed so that the 3 V supply is no longer part of the circuit, being replaced with a wire. Only the 5 V supply was left in the circuit. The voltage across the .8k ohm resistor in this set up was then measured (1.96 V). This set up is seen below.

The same was performed again, with the exception that the 3 V supply was left in the circuit and the 5 V supply was removed from the circuit. The voltage across the 6.8k ohm resistor was then measured (0.70 V). This is seen below:

Data Analysis and Conclusion:

The picture above shows the measured resistances of each of the resistors used. It also shows the voltages measured in the circuits with both voltage sources connected and only one connected in separate circuits. A table summing all of the useful data obtained is shown below:

The above table contains the experimental and calculated voltages across the 6.8k ohm resistor in circuits with the 5 V supply connected, the 3 V supply connected, and both the 5 V and 3 V supplies connected. It also contains the percent differences between the calculated and experimental voltages.

As can be seen, the percent differences are very small, which shows the the variations are due to assumptions (eg. no resistance in wires) and experimental uncertainties, such as measurement errors. Overall, it can be concluded that the superposition theory can be correctly applied to electrical circuits.