Day 15 4/12/16: Capacitors and Inductors

In lecture we started by looking at a "Ragebridge" motor controller. It consists of many FETs that are connected together using an "H" interface. When a battery is attached to it, the connector releases a large spark. This spark was found to be a result of the capacitor being charged. When it is uncharged at first, a large current runs through the circuit to charge the capacitor due to no resistance by previous charges on the current. As the capacitor charges further the current then drops due to repulsion of like charges. The amount of charge a capacitor can hold is proportional to the voltage across it, with a constant called capacitance that is unique to each resistor. Because a capacitor is two sheets separated by a distance in between, with usually a dielectric between the sheets, the capacitance is dependent on the area of the sheets and the distance between. A sheet with a larger area can hold more charge, which therefore increases the capacitance. On the other hand, a larger distance between the two sheets causes the attractive forces of the charges on each plate to be weaker, resulting in the charges being less easily packed, lowering the capacitance. Lastly, the capacitance is proportional to the permittivity of the dielectric, or how good it is at storing electrical energy in an electric field.

We then learned about different types of capacitors, such as electrolytic, ceramic, silver-mica, polyester film and super conductors. We then got more into the math of capacitors in a DC circuit. We learned that the current across a capacitor is equal to the product of its capacitance and the time rate of change of the voltage (dv/dt). By integrating current with respect to time we can then find voltage. Lastly, the energy of the capacitor is found by integrating power (voltage times current) with respect to time, which provides us that the energy is equivalent to half of the product of the capacitance and the squared voltage.

We then looked at more qualitative properties of a capacitor. For example, in DC a capacitor acts as an open circuit (i.e. it prevents current from running through the branch it is attached to. If a battery is connected to the capacitor, though, it will charge. In addition, the voltage of the capacitor cannot instantaneously be altered because it would require an infinite current, which is not possible. This is due to the nature of the capacitor to resist changes in voltage. Lastly, we learned that an ideal capacitor doesn't dissipate energy, but an ideal one does because it has a resistor connected in parallel. We then did a problem involving finding the energy of capacitors in a circuit.

After, we did a lab titled "Capacitor Voltage-Current Relations", where we looked at the behavior of the voltage of a capacitor in ac in relation with the current through the circuit. Both of these were controlled using time-varying signals. A sinusoidal wave and a triangular wave signals were used. We then did a problem about equivalent capacitance.

Next, we learned about inductors, which are coils of wire that store energy in a magnetic field. The voltage is found by multiplying inductance and the derivative of current. Inductance is like capacitance; it is the opposition to change in current. This leads to the fact that inductors cannot have an immediate change in current. Back to inductance, it depends on the square of the number of coils, permeability of the core, the inverse of the length of the wire, and cross-sectional area. The current in the inductor is 1/L times integral of vdt, and the energy is 1/2 times L times squared current. In dc, the inductor acts like a wire, so the voltage across it is zero.  In addition, an ideal inductor does not leake energy, but a real one does because it has some resistance in series.

LECTURE:

In the above picture we derived the energy of a capacitor using the power across a capacitor, which is the product of voltage and current.  Knowing that the energy is the integral of power with respect to time it was found to be half of the product of capacitance and the square of voltage.

In this problem we determined the voltage of a capacitor over time using the graph of current over time. Knowing that voltage is the inverse of capacitance times the integral of current with respect to time, the voltage graph was found. The slopes of the current graph were used to find the voltage graph. As can be seen in the graph, the voltage of a capacitor cannot be instantaneously altered because a capacitor resists changes in voltage.

In this problem the maximum amount of energy stored in each capacitor was solved for. This was done by first finding the voltages across each capacitor. Knowing that at dc the capacitor prevents any current flow through its branch, the circuit was redrawn. Then, by knowing that parallel components have equivalent voltage, the voltages across the 2 kOhm and 4 kOhm resistors were found using KVL to determine the voltages across the capacitors. Then, knowing that the energy of a capacitor is equivalent to half the product of voltage squared and capacitance, the energy of each capacitor was determined.

In this problem the equivalent capacitance was found. The equivalent capacitance of parallel capacitors is obtained by obtaining the sum of the individual capacitance values. On the other hand, the equivalent capacitance of series capacitors is obtained by finding the reciprocal of the sum of the inverse capacitance of each individual capacitor.

LAB:

Capacitor Voltage-Current Relations:

Purpose:

The purpose of this experiment is to look at the relationship between the voltage across a capacitor and the current running through it in an ac circuit. This lab was performed by applying sinusoidal and triangular wave voltage signals to a capacitor and resistor in series. The resulting voltage across the capacitor was then measured using an oscilloscope and the current across the circuit was determined using the voltage across the resistor and Ohm's Law (i = v / R). Because the resistor and capacitor are in series, the current through the resistor is the same as the current through the capacitor.

Prelab:

In the prelab, we determined the graphs of current running through the capacitor over time using the voltage graphs over time. Knowing the current in a capacitor is equivalent to the product of the capacitance and the derivative of voltage with respect to time, the current graphs with their corresponding amplitudes and frequencies were determined. For the sinusoidal voltage wave, the
corresponding current wave is a simple phase shift of pi over 4, along with a change in the amplitude. The new amplitude would include the product of the original amplitude (A), the capacitance (C), and  the angular speed, or just 2 pi divided by the period (T).

For the triangular wave, on the other hand, it is a different graph for current as a function of time. The current of a triangular wave voltage signal can be found by looking at the slope of the wave and multiplying that by the capacitance. The slope is positive or negative 4A/T, simply based on rise and run of the graph.

Apparatus:

The apparatus of this experiment consisted of an analog discovery, a 1 uF electrolytic capacitor, a 100 ohm resistor, a laptop with Waveforms, a breadboard, wires, a DMM and an alligator clip. All of these electrical components were used to produce a circuit with the resistor in series to the capacitor. The Wavegen voltage source (yellow wire) was attached to the open end of the resistor. Channel 1 was connected across the resistor and channel 2 was connected across the capacitor.

Procedure:

The circuit described in the apparatus section was constructed, as shown in the picture above. Channel 1 was used with the oscilloscope software to measure the voltage across the resistor as a function of time. Channel 2 was used to measure the voltage across the capacitor as a function of time. Wavegen 1 (W1) was used to apply the voltage time-varying signals to the circuit. Lastly, a math channel (M1) was created to use the voltage across the resistor to find the current running through the capacitor, using Ohm's Law.

Once the preparation was complete, a sinusoidal function of 1 kHz and an amplitude of 2 V centered at 0 was applied using W1. The oscilloscope was then used to measure the voltages and currents. The graph of the voltage across the capacitor and the current when this sinusoidal signal was applied is shown below:

The voltage across the capacitor is the blue graph and the red graph is the current in the circuit.

The same was done with a sinusoidal wave of 2 kHz frequency and a 2 V amplitude centered at 0 V. The graph is shown below:

The blue graph is still the voltage across the capacitor and the red graph is the current in the circuit. Again, the same was done with a triangular wave that has a frequency of 100 Hz and an amplitude of 4 V with an offset of 0 V. This graph is shown below:

Again, the blue graph is still the voltage across the capacitor and the red graph is the current in the circuit.

Data:

Above is a picture of the data obtained in this experiment. The true value of the resistor was found to be 98.7 ohms. In addition, when the sinusoidal wave with a 1 kHz frequency was applied to the circuit, the amplitude of the voltage across the capacitor was 1.75 V and the maximum current was 10 mA. Looking at the sinusoidal voltage wave with a 2 kHz frequency, the amplitude of the voltage across the capacitor is 1.25 V and the max current in the circuit is 15 mA. Lastly, for the triangular wave, the amplitude of the voltage across the capacitor is 4 V and the max current in the circuit is 1.5 mA. 

Data Analysis/Conclusion:

Looking at the graph of the sinusoidal wave with a 1 kHz frequency, both the voltage and the current are sinusoidal, which makes sense based on our prelab since the derivative of a sinusoidal function is a sinusoidal function. In addition, the phase angle of the capacitor is a difference of pi over 4 compared to the current, which makes sense since a cosine function leads/lags the sine function by pi over 4. Calculating the expected amplitude of the current graph based on the value for voltage and the relationship seen in the prelab (2*pi*f*C*A), a value of 11.0 was obtained. Our experimental value is reasonably close to the expected ( -9.96%), and the difference must be due to incorrect reading of the graph because the intervals were large. In addition, it is also due to resistance not accounted for in the capacitor and wires.

Looking at the graph of the sinusoidal wave with a 2 kHz frequency, as seen in the 1 kHz graph, the voltage and current graphs are also sinusoidal which makes sense for the same reason as above. In addition, the capacitor voltage leads/lags the current by pi over 4, which also makes sense due to the relationship between sine and cosine. The expected value of the maximum current for this circuit was found to be 15.7 mA, which is close to the obtained value of 15 mA (a percent difference of -4.51%). The difference is a result of unaccounted resistance and error in reading.

Looking at the graph of the triangular wave, the obtained graph for the current in the circuit has the same shape/form as the expected in the prelab, since the derivative of a constant slope is simply a constant. However, the graph does have a small slope and not a zero slope as expected, and this is due to reasons not explored yet. In addition, the expected value of max current was found to be 1.6 mA (4*A*C*f). Our experimental value is also close to the expected (-6.25% difference). Again, the difference is due to unaccounted resistance and errors in measurements.