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Experiment Guide for RC Circuits
I. Introduction
1.Capacitors
A capacitor is a passive electronic component that stores energy in the form of an
electrostatic field. The unit of capacitance is the farad (coulomb/volt). Practical capacitor
values usually lie in the picofarad (1 pF = 10
12
F) to microfarad (1
μ
F = 10
6
F) range.
Recall that a current is a flow of charges. When current flows into one plate of a
capacitor, the charges don't pass through (although to maintain local charge balance, an
equal number of the same polarity charges leave the other plate of the device) but instead
accumulate on that plate, increasing the voltage across the capacitor. The voltage V
across the capacitor (capacitance C) is directly proportional to the charge Q stored on the
plates:
Since Q is the integration of current over time, we can write:
Differentiating this equation, we obtain the
IV
characteristic equation for a capacitor:
(Eq. 1)
2.RC Circuits
An RC (resistor + capacitor) circuit will have an exponential voltage response of the
form
v
(t) = A + B
e
t/RC
where constant A is the final voltage and constant B is the
difference between the initial and the final voltages. (
e
x
is
e
to the x power, where
e
=
2.718, the base of the natural logarithm.) The product
RC
is called the
time constant
(whose units are seconds) and is usually represented by the Greek letter
τ
. When the time
has reached a value equal to the time constant,
τ
, then the voltage is B
e

τ
/RC
= B
e
1
=
0.368 * B volts away from the final value A, or about 5/8 of the way from the initial
value (A + B) to the final value (A).
The characteristic “exponential decay” associated with an RC circuit is important to
understand, because complicated circuits can oftentimes be modeled simply as resistor
and a capacitor. This is especially true in integrated circuits (ICs).
A simple RC circuit is drawn in Figure 1 with currents and voltages defined as
shown. Equation 2 is obtained from Kirchhoff’s Voltage Law, which states that the
algebraic sum of voltage drops around a closed loop is zero. Equation 1 (above) is the
defining
IV
characteristic equation for a capacitor, and Equation 3 is the defining
IV
characteristic equation for a resistor (Ohm’s Law).
C
dt
t
i
C
t
Q
t
v
∫
=
=
)
(
)
(
)
(
V
C
Q
=
dt
t
dV
C
t
i
)
(
)
(
=
1 of 10
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View Full Document(Eq. 2)
(Eq. 3)
Combining Equations 1 and 3 into 2, we obtain the following firstorder linear
differential equation:
(Eq. 4)
If
V
IN
is a step function at time
t
=0, then
V
C
and
V
R
are of the form:
(Eq. 5)
RC
t
R
e
B
A
V
/
'
'
−
+
=
If a voltage difference exists across the resistor (
i.e. V
R
< 0 or
V
R
> 0), then current will
flow (Eq. 3). This current flows through the capacitor and causes
V
C
to change (Eq. 1).
V
C
will increase (if
I
> 0) or decrease (if
I
< 0) exponentially with time, until it reaches the
value of
V
IN
at which time the current goes to zero (since
V
R
= 0). For the squarewave
function
V
IN
as shown in Figure 2(a), the responses
V
C
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 Spring '07
 Boser

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