AC Circuits - Faculty.chemeketa.edu

piquantdistractedElectronics - Devices

Oct 5, 2013 (3 years and 6 months ago)

90 views

AC Circuits


An AC power supply connected to a resistor and capacitor in series can be analyzed using
Kirchoff’s voltage law:





V

= IR + Q/C




V =
potential difference

across entire circuit




I = current through entire circuit




R = resistance of resi
stor




Q = charge on one plate of a capacitor




C = capacitance of the capacitor


The
potential difference

across the power supply varies with time; ε = V
max
sin(ωt). The
current is dQ/dt o
r

ΔQ/Δt. This sets up a differential equation as follows:





V
max
sin(ωt) = dQ/dt*R + Q/C


This is a difficult differential equation to solve, and things only get worse with
additional
circuit elements. Instead of this approach, the concepts of imped
a
nce and reactance were
developed to greatly simplify the analysis. For a capacitor, the
capacitive

reactance, X
c
, is
1/(ωC)
, where ω=2πf.

The
equivalent
impedance of the
series
ci
rcuit in question can be
calculated with complex arithmetic (beyond the scope of this course) or taken from the text:





Z =

[R
2

+ X
c
2
]

Z =

[R
2

+ 1/(ωC)
2
]


Then,
the AC

version of Ohm’s law can be used to calculate the
rms
current:





V
input

= IZ for t
he power supply

I =
V
input
/Z

I =
V
input
/

[R
2

+ 1/(ωC)
2
]


This current will be the curren
t for all circuit elements.
Ohm’s law
can be used to
calculate
the
potential difference

across the
individual
capacitor

and resistor
:








V
c

=
IX
c

V
c

=
V
input
/

[R
2
+
1
/(ωC)
2
]*
1/ωC

V
c

=
V
input
/

[(ωRC)
2
+
1]



[1]





V
r

= IR

V
r

= V
input
/

[R
2

+ 1/(ωC)
2
]*R

V
r

=
V
input
/

[1+1/(ωRC)
2
]



[2]


These are

the equation
s

we will investigate as a test of this type of circuit analysis
.

Physics is
fun!


Experimental Procedures


AC RC
circuit


Note: Please be very careful in this lab because if you connect one output directly to the
other, you could damage
the
sound card.


1)

Disconnect the cable from the back of your computer speakers.

Do not disconnect
the cable from your computer.

2)

Set th
e volume on your computer to maximum.

3)

Download and install Audacity freeware from the following web site:
http://audacity.sourceforge.net/

4)

In the Audacity program, click on “Generate” and “Tone”. It should default to
Waveform “Sine” and amplitude of 1. For

frequency, choose 100 Hz. Hit “enter”.

5)

Configure a DMM as a voltmeter at the lowest setting for AC
potential difference

and connect it to the
cable from your computer

(or the wires from your
headphones)
.

There should be a small piece of insulation on the
cable separating the
+ and
-
. Place a probe on each side of the in
sulation.

6)

Click on the play button (the green triangle) and check if you o
btain a measurable
potential difference

(it should be at least 0.4 V).
Record this value

(V
input
)

and press
the stop

button on Audacity.

7)

Select a capacitor and resistor so that their time constant (R*C) is approximately 10
-
3

s.

8)

Calculate the
initial
frequency
using f = 0.1/(2πRC
).

9)

Connect the capacitor, resistor, and output of

the
computer

in a single loop.

10)

Connect the
voltmeter
s

so that
they

measure the
potential difference

across the
capacitor

and resistor
.

See figure at right.

11)

In the Audacity program, click on “Generate” and “Tone”. For frequency, choose
your calculated value. Hit “enter”.

12)

Measure the
potential differ
ence
s

across
the

capacitor

and resistor
.

13)

Repeat steps
1
1

and 1
2

for approximately
50% more than

your just measured
frequency until the
frequency is at least 100 times your initial frequency
.

14)

Calculate the theoretical
potential difference
s

across the capaci
tor

and resistor

for the
measured frequencies

using equations [1] and [2]
. You do not need to include an
error analysis in the
theoretical
potential difference
.

15)

Plot the experimental and theoretical
potential difference
s

(vertical axis) versus the
frequenc
y

on a single graph
. Use a logarithmic scale for the horizontal axis and a
linear scale for the vertical axis.
To use a logarithmic scale, make your graph
according to the general lab instructions. Once you have a graph, r
ight click on the
x
-
axis and selec
t “format axis”
.

16)

Compare the
theoretical and experimental

plots
.

In this case, since you do not have a
complete error analysis, there will be some subjectivity in the comparison and you
are allowed to use the word “close” in your report.

17)

Repeat steps
7

thr
ough
1
6

with a
different

resistor.