# Lecture # 10

```Capacitor
ESE112
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Passive element designed to store energy
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Electric Circuit Theory: Capacitor
Stores energy in the form of an electrical charge
Like a battery but can’t produce new electrons
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Consists of two metal plates separated by a
non-conduction substance or dielectric
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Some common uses:
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Clocks (counter/timer)
X-rays and MRI (high-frequency applications)
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How it stores energy?
RC Circuit
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A resistor placed in series with capacitor will
control the rate at which it charges or
discharges
Charge flows up to the plates but not across the
insulator
Once it's charged, the capacitor has the same
voltage as the battery
Capacitor resists change in voltage
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Circuit Response (Discharge)
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Circuit Response (Discharge) contd..
Assuming that the capacitor is initially charged
The voltage across the capacitor is said to
decay over time and is given by:
v(t) = V0e-t/RC where V0 is voltage at t = 0
Rapidity at which the voltage decreases is
expressed in terms of time constant
denoted as τ (tau)
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τ = RC
v(t) = V0e-t/RC
V0
t
τ
V(t)/V0
0.36788
2τ
0.13534
3τ
0.04979
4τ
0 01832
0.01832
5τ
0.00674
Smaller the τ, the more rapidly the voltage decrease i.e.
the circuit response is faster.
rcTime in CPU class
int rcTime(int timeout, int portPin, boolean pinState)
 The method records the time it takes for the voltage at
the capacitor’s plate to cross 2.5V logic threshold
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The returned value (amount of time for the pin to reach
a desired state) and timeout parameter are in 8.68 us
units
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rcTime starts the counter that increments every 8
8.68us.
68us
It stops the counters as soon as the pin reaches
pinState
What does the rcTime method return ?
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v(t) = V0e-t/RC
&Icirc;v(t) = V0/et/RC
&Icirc; t/RC = V0/ v(t)
&Icirc;e
(t)
Take natural log (base e) on both sides
&Icirc; ln(et/RC) = ln(V0/ v(t) )
&Icirc; t/RC = ln(V0/ v(t) )
Therefore: t = R*C*ln(V0/ v(t) )
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Example
Consider a 10KOhm resistor and 1.0uF
cap
 t = (10 x 103) x (1
(1.0
0 x 10-66) x ln(5/2
ln(5/2.5)
5)
= 6.93 ms
In rcTime units of 8.68 us, 6.93 ms ~800
units
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Handy rule: 0.693 x R (kOhm) * C (in uF)
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