Circuits

Circuits

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Circuits


There are two types of circuits that we dealt with in this lab. One type is a RC circuit, which has a power supply, resistor and a capacitor. The other type is a LR circuit, which has a power supply, inductor and a resistor.
The first one was a RC circuit. The capacitance (C) of a capacitor is equal to the charge (Q) on either plate divided by the potential difference (VC) between the plates.
C º Q / VC
The capacitance has the units of farads (F). Using Kirchoff's Loop Rule, we know that The voltage drop across the resistor (VR) plus the voltage drop across the capacitor (VC) equals the voltage rise across the battery (x). This equation looks like:
VR + VC = x
Using Ohm's law and the definition of current we get:
VR = IR I = DQ / Dt
Therefore: VR = (DQ / Dt )R
Using the above information, we can rewrite Kirchoff's Loop Rule, which looks like: R(DQ / Dt) + (Q / C) = x
Substituting the following variables x and t, we can look at the equation a different way. Here are the definitions of these variables:
x º Q - xC t º RC
To get the charge as a function of time, we use this equation
Dx / Dt = - x / t
By graphing x versus time, we get the following equation:
x = x0 e(-t / t)
x0 is the value of x at t = 0. If we replace the x in the above equation with the definition of x, we get:
Q = Q¥ (1 - e(-t / t))
Substituting the above equation into the equation for capacitance and resistance, we get: VC = x(1 - e(-t / t)) VR = xe(-t / t)
Since current (I) is VR / R, we can get the equation:
I = I0e(-t / t)
Discharging a capacitor in a RC circuit can be related to time also. This is seen in the following equation:
DQ / Dt = - Q / t
Relating this equation to a similar equation that we defined earlier, we can get:
Q = Q0e(-t / t)
Using the definition of capacitance and Kirchoff's Loop Rule, we can expand the above equation to these equations:
VC = xe(-t / t) VR = -xe(-t / t)
Using the definition of current, we get:
I = -I0e(-t / t)
The other type of circuit was a LR circuit. The back emf (xL) is equal to the rate of change of the current times the inductance of the coil (L).

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The inductance has units of henry (H), which is equal to the units Vs/A. Here is the equation:
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In this case, tau has the following equation:
t º L / R
Results
In this lab, we learned about RC and LR circuits. In part one of the lab, the percent error between our experimental tau versus the theoretical tau was very good. The average was 4.0 % and it ranged from 0 % to 12.0 %. In part two of the lab, we didn't do so well. I think the graph for this part was wrong or that we got very bad data because the percent error was 124% and we know that it physically can't be more than 100 %. In part three, we got two percent errors. The first one was 3.5 % and described the relationship between the actual value of tau, which was 33.33 microseconds versus the experimental tau, which was 34.5 microseconds. The second percent error described the relationship between the experimental tau, which was 34.5 microseconds, and another experimental tau, which was 26 microseconds. This percent error was 24.64 %. If you compared the second experimental tau to the actual value, you get a 22.0 % error. Some of the error sources were: the estimation of the divisions on the oscilloscope screen and especially human error.
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