RC Circuits –

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BioE298 Lab

Session II

20
-
22 Feb

2007


Spring 2007

Page
1

of
5

RC Circuits



A resistor
-
capacitor circuit (RC circuit) is a fundamental electronic circuit that represents a
simple analog filter. Consider the following series RC circuit:













Ca
pacitor voltage Step response


Resistor voltage Step response


An uncharged capacitor behaves as a short circuit. The graph on the left indicates that with
time, the voltage across the capacitor, V
c

tends towards V=V
in
, while the voltage across the
resistor, V
R

tends towards zero. Intuitively, this makes sense, since
the capacitor charges
from the supply voltage (V
in
= V) over time, and ultimately becomes fully charged and
behaves as an open circuit. Meanwhile the voltage across the resistor, V
R
drops and tends
towards zero, since by conservation of energy:

Studying

the functions of time above indicates that a series RC circuit has a time constant
τ
=
RC
. This indicates that the time it takes the voltage across a component in an RC circuit
either rises (across C) or falls (across R) within 1/e of its final value. Therefore,
τ is the time
it takes V
c
to charge up to V(1
-
1/e) (=0.63Vin) and V
R

to reac
h V(1/e) (=0.63Vin).

Once the capacitor is fully charged, and the voltage supply is removed, the fully charged
capacitor begins to discharge, in which case, V
c
drops exponentially from V
in
towards 0.
After one time constant τ of discharging, V
c

= 0.37V
i
n
.



BioE298 Lab

Session II

20
-
22 Feb

2007


Spring 2007

Page
2

of
5

Low Pass Filters

A low pass filter is a filter that allows low frequencies through and attenuates frequencies
higher than the cutoff frequency. The cutoff frequency is determined by the R and C values
used to build the filter.

A low pass filter is
constructed by placing a resistor in
series with a load and a capacitor in parallel with the
load. At low frequencies, the capacitor has time to charge
up, and so the voltage across the capacitor is almost
equal to the input voltage, and the capacitor can
be
represented as an open circuit.

At high frequencies, the capacitor is only able to charge up to a small value before the input
switches direction, and the capacitor begins to discharge. With this, the output increases and
decreases only a fraction of t
he amount the input actually goes up and down. The faster the
input switches, the slower the output switches due to the time delay imparted by the
capacitor. Therefore, since only a small fraction of the input is able to flow through as Vout,
the remainder

of the input flows through the capacitor to ground. With this, the capacitor
behaves wire, short circuiting to ground.

Another way of considering the behavior of this circuit is to consider the reactance of the
circuit at low and high frequencies. When t
he input is approximated to be DC (i.e. low
frequency), the DC cannot flow through the capacitor, and thus the input must exit via the
V
out

path. This would be the same as treating the capacitor as an open circuit, or effectively
removing it from the circu
it.

With an AC input, it is able to flow very well through the capacitor, and the input flows
almost entirely through the capacitor, through to ground, and thus is analogous to replacing
the capacitor with a wire and effectively short circuiting to ground.


High Pass Filters

High pass filters attenuate frequencies that are lower than
the cutoff frequency that the filter is designed for. A high
pass filter with a very low cutoff frequency is especially
useful in building circuits because it can be used to
block
the DC component of a signal, which may be undesired,
and passes virtually all other components of the signal.
The simplest high pass filter is represented by the RC circuit to the right.





BioE298 Lab

Session II

20
-
22 Feb

2007


Spring 2007

Page
3

of
5

Time constant & Cutoff frequency

An important value to co
nsider when designing an RC circuit, is the value of the time
constant
τ =
R × C
. This is the time it takes the capacitor to charge through the resistor, to a
value of 63.2% of full charge, or to discharge the capacitor to 36.8% of its initial full charge.

The most critical design parameter in RC
circuits is the cutoff freq
uency f
c
.
A low pass
filter will attenuate frequencies f>f
c
,
high pass
filter will only pass frequencies f>f
c
, and a
bandpass filter has a lower cutoff frequency f
lo
and a higher cutoff frequency f
hi

and will only
pass frequencies f
lo
<f<f
hi
. The cutoff

frequency f
c

is the frequency above or below
which the power of the circuit, which is most
often a filter, is ½ the input power. Since P =
V
2
/R, half the power is proportional to
sqrt(1/2) the voltage. This is also referred to
as the =3dB point, or the k
nee frequency, due to the bending of the curve of the bode plot,
as s
een in the figure to the right.

dB (deciBel) is a way of comparing two signals. To compare two voltages with amplitude A
2

and A
1
:



Note that if A
2

is less tha
n A
1

their ratio will be less than 1 and less than 0 in dB.


The time constant



is related to the cutoff frequency by the following expression:




or














BioE298 Lab

Session II

20
-
22 Feb

2007


Spring 2007

Page
4

of
5







Experimental Objectives:

1.

How to design & build a low pass filter.

2.

How to use a function generator to drive
a circuit.

3.

How to use an oscilloscope to:

a.

Probe your circuit

b.

Measure frequency response

c.

Determine time constants.


Experiment:


Design a low pass filter with a cutoff frequency in the range of 1500Hz
-
10kHz.



What is the cutoff frequency of your filter?



c
=

______________




What R and C values are you using to achieve this cutoff?

Recall that at high
frequencies, the capacitor is a short and all of the input voltage will drop across the
resistor. Assume a peak input voltage amplitude of 10 V and also calcul
ate the peak
power dissipated by the resistor. If this value is over ¼ W, choose different
component values.


R = __________




P =





C = __________




Provide a circuit diagram with your chosen R & C values labeled.














Provide a screenshot of y
our
filter’s operation by finding where the output amplitude
is 0.707 (
-
3 dB) of the input amplitude.

BioE298 Lab

Session II

20
-
22 Feb

2007


Spring 2007

Page
5

of
5

o

Use a T
-
connector at the output of the function generator. Connect one
BNC cable directly to the oscilloscope input 1. Connect another BNC cable
to the
breadboard. This is the input voltage of your filter.

o

Connect the output voltage of the filter to the oscilloscope input 2.

o

Turn on the function generator. Use a sine wave with a frequency a few
orders of magnitude lower than your cutoff frequency as the

input voltage.
There should be no DC offset and the amplitude should be somewhere
between 5 and 10 V. (The exact value does not matter. Remember that you
are comparing the output to the input and their ratio should not change with
a change in input amp
litude.)

o

Use the oscilloscope cursors to confirm that the two voltage waveforms have
the same amplitude and are in phase (meaning they peak at the same time).

o

Use the voltage cursors to set the voltage amplitude to be 100% on the
oscilloscope. Then, keep
one cursor at 0 V and set the other at 70.7% of the
voltage amplitude.

o

Increase the frequency of the input voltage and watch the output voltage.
The input voltage amplitude should not change, but the output voltage
amplitude should decrease in value.

o

Find

the input frequency that results in an output voltage that is 70.7% of the
input voltage. This is the cutoff frequency of your filter. Does it match your
calculations above? If not, is it within the tolerance of the values of the
components that you us
ed?




Measure the time constant of your filter on the oscilloscope.

o

The time constant determines the transient behavior of the circuit. To
simulate a switch closing, change the function generator setting so that it
produces a square wave signal. The perio
d of the signal should be long
enough to allow the output voltage of your circuit to reach a steady state
value before the input voltage changes value.

o

Use the cursors set as 100% the entire voltage swing (from minimum to
maximum) of the output voltage.

o

If the input voltage is dropping from high to low on your oscilloscope
screen, you need to measure the amount of time it takes for the output
voltage to drop by 63%. If the input voltage is going from low to high on
your oscilloscope screen, you need to m
easure the amount of time it takes
for the output to increase from 0% to 63% of the final value.

o

Use the voltage cursors to find the 63% value, then use the time cursors to
measure the time between the input voltage changing values and the output
voltage r
eaching the 63% value. This is the measured time constant.




measured

= _________________




How does your measured time constant value compare with the theoretical value?



theoreticaol

= __________________