Feb 10: The Capacitor

The Capacitor

Brief Overview
-stores charges and after waiting a long time, potential across C still V (V = Vmax). Thus, C = Q/V or amount of charge stored in the capacitor over the potential difference.

-In the limit in which Vout << Vin (i.e. the capacitor is not allowed to charge much), then I(t) is approximately equal to Vin(t)/R, (where delta V = Vin(t) - Vout(t) = Vint(t)), such that dVout(t)/dt = (1/RC)*Vin(t). This means in the case that a capacitor is not very charged, the rate of change of output voltage is proportional to the input voltage.

-For cases where  Vout is not << Vin, we considered a "step function" input (input voltage then no input voltage then input voltage...etc). Within that one "cycle" in which there is an input voltage, there is a time lag before the capacitor is fully charged, modeled by V-out = A(1-e^(-t/RC)), where A is the maximum output voltage (a capacitor has a limit on the amount of charge it can store). 

V-out = A(1-e^(-t/RC))

-In the equation, RC is the time constant = amount of time it takes for the output voltage to get about 2/3 of the way to its final value A. The time lag mentioned before can be characterized by this value RC or time constant.

-Similar to charging, the voltage across a discharging capacitor is Vout(t) = Ae^(-t/RC).

***More on Capacitors... (understanding how it works)

-no current flowing ACROSS capacitor 

-When input voltage is applied, positive charges accumulate on the top plate (actually, electrons accumulate on the other plate). And this accumulation of positive charges repels the positive charges on the other plate and creates an electric field. And the potential difference between these points is the voltage that we measure. This electric field becomes larger and larger until the voltage difference is equal to the voltage difference generated by the battery, which in that case,no more current is flowing as V becomes equal on both sides of the resistor.

An RC Circuit, Experimentally
Lab 2-1

Objective: We wanted to check that an RC circuit in fact behaves in the time domain as described above. Specifically, we checked if the time constant = RC was in fact the amount of time for the output voltage to reach about 2/3 of the final value.

After creating the circuit shown above, we measured the time constant by determining the time for the output to drop from 100% to 37%. Thus, we first looked at the falling portion of the "ocean wave-like" output signal.

Using the vertical cursors with one at the maximum voltage value, we determined delta t (time for the output to drop from 100% to 37%) to be 100 microseconds.

To check with the product RC:

R*C = (10k Ohms)(0.01E-6 F) = 100 microseconds

Using the cursors again, we then looked at the rising portion of the "ocean wave" and determined delta t (time for the output to climb from 0% to 63%) to be 100 microseconds (which made sense). 

Varying the frequency of the square wave
Observations: 
-The rise or fall times were not affected when the frequency was changed. And increasing the frequency increased the number of "cycles" of rising and falling "ocean waves" per given period of time. 
-However, when the frequency was increased significantly (from 500 Hz to 3 kHz), the output voltage looked like a sawtooth. This is because as the frequency increases to large frequencies, there is not enough time for the capacitors to fully charge (which means the maximum voltage is not reached), and the voltage pattern we see is only the "front" or rising portion of the voltage curve for a capacitor, which appeared to be rising almost linearly (at least locally). This is why with an even larger frequency, 10 kHz, we saw a triangle wave. In both cases, the voltages did not reach the original maximum value.

Frequencies of 3 kHz (left) and 10 kHz (right)

Arduino Capacitance Meter
-This circuit looked like a divider except the bottom of the divider was a capacitor (vs. resistor). The "resistance" of the capacitor depends on the frequency of the electrical signal (will be discussed later).
-We ran the capacitance meter by running this code:

Brief Summary of Program:

-Sets input/output pins, threshold values (what is the low vs. high threshold), and initial time values.

-Sets up by defining/declaring the digital pin (pin 2) as the output pin driving the RC circuit

-Opens serial communication (so we can see the output signals using the serial monitor)

-Loop: The input voltage starts off "low" (0 volts). Then, the input voltage is set to high (or 63% of 1023, which is the time constant = RC). From here, after setting the voltage to high, the start time (from an internal "clock" in Arduino) is recorded in microseconds and the system waits (for the capacitor to store charges) until the voltage output is equal to the high voltage threshold. While the voltage is less than the high voltage threshold, the endtime keeps being recorded and updated until the final value is recorded right before the voltage equals the upper threshold. Displayed on the serial monitor is the difference in time (delta t), or essentially the time constant (time needed for voltage to go from 0 to 2/3 of the maximum amount, in this case, 2/3 or 63% of 1023). Then, the program allows the capacitor to discharge by waiting until the voltage is less than 0.5% of the maximum voltage (small enough to start another "cycle").

Exercises:

1.) Tried using cap-meter for capacitor values for 0.1 uF, 0.47 uF, and 4.7 uF, expecting the numbers we get on the monitor (time constant) to increase with increasing capacitance because time constant = R*C. 

Capacitance (uF)	Readings (Time constant in us)	Calculated time constant (us)
0.01	228	100
0.47	~5000	4700
4.7	~46000	47000

As we expected, we saw the increase in time constants as capacitance increased.

2.) Calibration and display of capacitance (in uF)

-In order to calibrate the cap-meter, we created a calibration curve using the known capacitance values and the readings we took. 

Using the regression line equation, in which y is the capacitance and the value we want on the serial monitor and x being delta t (or end time - start time), we adjusted the last portion of the code:

This got us capacitance values that were very close to the theoretical values.

3.) Capacitors in Parallel vs. in Series (using 0.1 uF capacitors)
-In parallel, we read capacitance of 0.22 uF (which theoretically is 0.20 uF). Capacitance in parallel is calculated similarly to calculating resistance in a circuit with resistors in series.
-In series, we read C = 0.06 uF (which theoretically is 0.05 uF). Capacitance in series is calculated similarly to calculating resistance in a circuit with resistors in parallel.


Integrator
-Any time t is small enough (Vout << Vin), the output voltage proportional to integral of input voltage (the front portion of the curve which is locally linear in the case that Vout << Vin). 
-For example, if we apply a square wave input, we expect output to be ...: (in blue)

This makes sense because integrating a constant gives you a linear function.

Note that an RC circuit is a decent integrator ONLY when t is small enough (when the curve is locally linear). 

An RC Integrator, Experimentally

-Constructed an integrator and drove it with a 100 kHz square wave with 5Vpp amplitude:

We used a 100 kHz square wave. This means each period is 0.00001 seconds or 10 us. We saw from before that the time constant is 100 us. This means the period is 1/10 of the time constant, making this a decent integrator (because the curve is approximately linear in this region).

We then used a trial wave, and as we see in the photos above, the output waves are parabolas, which makes sense. When we integrate linear functions, we get quadratic functions, which are parabolas.

Next time....  things to consider:

-At low frequency (long periods), capacitor approaches infinite resistance (where no flow of current because Vin = V out, or Vbattery = Vcapacitor).

-At high frequency (short periods), capacitor approaches 0 resistance (the very front portion of the curve), in which Vout is approximately 0.

-Depending on the resistor and capacitor in this "divider" which appears similar to the voltage divider, we can create high pass or low pass filters.