Download The RC circuit

Contents
Introduction
3
Charging of capacitors
5
The RC circuit
6
The time constant
7
Summary
10
Answers
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EEE042A: 18 Describe RC circuits
 NSW DET 2017 2006/060/04/2017 LRR 3675
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EEE042A: 18 Describe RC circuits
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Introduction
Here we examine the behaviour of dc circuits that contain both a resistor and
a capacitor. You will learn that the charging and discharging of any RC
circuit involves the same characteristically-shaped curve.
After completing this topic, you should be able to:

draw the shape of a charge or discharge curve for an RC circuit

calculate the time constant for a given resistance and capacitance

describe how the time constant relates to the shape of the charging curve

calculate the time required to effectively complete the charge or
discharge of a capacitor in an RC circuit.
EEE042A: 18 Describe RC circuits
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EEE042A: 18 Describe RC circuits
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Charging of capacitors
Because the dielectric between the plates of a capacitor is an insulator,
there is no passage of electrons between the plates. However, when a
capacitor is first connected into a circuit and a potential applied, a current
does flow into the capacitor. This is the charging current and it flows until
the capacitor voltage reaches the applied voltage.
Consider the circuit of figure 5, where a sensitive galvanometer detects the
flow of current. The capacitor is initially uncharged.
If a voltage of 100 V is applied, current will flow into the capacitor, moving
the needle to the right. After a very short time, the needle will return to the
centre, because the capacitor is fully charged to 100 V, and there can be no
more current flow.
Suppose now the supply is suddenly increased to 150 volts. More current
will flow, increasing the amount of charge in the capacitor, and increasing
its voltage to 150 V. Again, the needle will then return to zero, because the
capacitor voltage exactly matches the source voltage.
If the source voltage is again reduced to 100 volts, current will flow from
the capacitor back to the supply, deflecting the needle to the right. The total
charge on the capacitor decreases until its voltage is exactly 100 V, then the
needle returns to the centre position again.
Figure 5: Deflection of meter pointer as current charges and partially discharges
a capacitor
EEE042A: 18 Describe RC circuits
 NSW DET 2017 2006/060/04/2017 LRR 3675
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The RC circuit
Look at the circuit of Figure 6. Here we are charging a capacitor through a
resistor.
Figure 6: Charging a capacitor through a resistor
The charge in the capacitor (represented by the voltage across the capacitor,
Vc) takes a certain time to reach the final value. Note that the rate of
increase becomes less and less as the voltage across the capacitor
approaches the circuit voltage. If the resistance of the resistor R were
increased it would take longer to charge the uncharged capacitor, and if the
resistance were reduced, it would be shorter.
When discharging a capacitor though a resistor, the effect is similar. In the
circuit of Figure 7, after the switch is closed, the voltage across the capacitor
drops rapidly at first (as shown by the graph) then the voltage drops more
slowly to zero.
Figure 7: Discharging a capacitor through a resistor
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EEE042A: 18 Describe RC circuits
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This behaviour is easy to explain. The rate of change of the capacitor
voltage is proportional to the current flowing into or out of the capacitor. A
large current means that the curve will change rapidly. But the current
flowing is related to the voltage across the resistor. In the beginning, the
capacitor has zero charge, and the current is a maximum, because the full
source voltage appears across the resistor. As the capacitor charges, its
voltage opposes the source voltage, and so less current flows. At full charge,
the capacitor voltage is equal to and opposite to the source voltage, so no
current flows at all, and so there is no further change in voltage.
This situation is analogous to a pressure vessel being charged with gas from
a constant-pressure source, through a small hole. At first, the gas flows most
rapidly, because there is a large difference in pressure between the pressure
source and the vessel. As the pressure inside the vessel approaches the value
of the pressure source, the flow rate decreases because the difference in
pressure across the hole is smaller. We can make the following
comparisons:
pressure ↔ voltage
pressure source ↔ voltage source
size of pressure vessel ↔ capacitance
small hole size ↔ large resistance
Note that if we have a small hole, it will take longer for the pressure vessel
to fill. And if we use a larger vessel, it will take longer to fill also. Similarly
in our circuit, increasing the value of the resistance or capacitance will
increase the time to charge or discharge, and decreasing the resistance or
capacitance will decrease the charge/discharge time.
The time constant
We have deduced that the time for charging and discharging depends upon
both the capacitance and the resistance. We find that charge time is related
to the product of R and C, and this quantity is called the time constant. The
formula is:
  RC
where:
 = time constant (second)
Note:  is the Greek letter 'Tau'
R = resistance (ohm)
C = capacitance (farad)
EEE042A: 18 Describe RC circuits
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The significance of the time constant is this: in a period of one time
constant, the voltage will change by 63.2% of the difference between its
current and final values.
For example, let’s say a 1 mF capacitor is connected to a 10 V source via a
1 kΩ resistor. The time constant is:
10-3 F × 103 Ω = 1 s
So we have a one-second time constant in this case. The capacitor voltage
starts at zero and will increase (eventually) to 10 V. But how fast does it
change ?
After one time constant (ie one second), the amount of change will be 63.2%
of the difference, in other words the capacitor voltage will be 10 × 63.2% =
6.32 V
During the next time constant, the amount of change will again be 63.2% of
the difference between the current value and the final value. Note that the
current value is now 6.32 V, and the final value is still 10 V. So the change
is:
V  (10  6.32)  63.2%
 2.33 V
Therefore after two time constants, the voltage will be up to:
V = 6.32 + 2.33 = 8.65 V
We could continue this process to the third, fourth etc time constants. We
find that the changes are less each time, so the voltage approaches the final
value more and more slowly. In theory a capacitor is never fully charged or
completely discharged. However, we accept that a capacitor is fully charged
or discharged after about 5 time constants, when the voltage has undergone
99.3% of the total change.
Charge time is independent of source voltage
Notice that the time constant formula has no mention of the source voltage.
A large source voltage will indeed produce a greater absolute rate of change
of the capacitor voltage, but note that the capacitor voltage has ‘further to
go’ to reach the final voltage. The result is that the charge or discharge
curves always have the exactly the same basic shape, and the change in
voltage after 1, 2 or more time constants is always in the same proportion.
So for example, after five time constants, the charge or discharge is always
99.3% complete, whatever the actual change in voltage that represents.
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EEE042A: 18 Describe RC circuits
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Example 1
A 150 microfarad capacitor and a resistor of 1000 ohm are connected in
series. What time will it take to reach 63.2 per cent of the total charge that
can be applied to the capacitor?
R = 100 
C = 150 F
 =?
  RC
 1000 150 106
 0.15 s
The time constant of the RC circuit is 150 milliseconds.
If you have Hampson, read the section ‘The capacitor in a dc circuit’ on
pages 83 and 84 on page 125 for a brief coverage of this area.
If you have Jenneson, refer to Section 6.4 ‘Capacitors on direct current’ on
pages 125 and 126 to cover the effects of introducing a resistor into a
capacitor circuit, the time constant and the energy stored in the capacitor.
Notes:

Look again at the first diagram in this section, where we considered the basic
charging behaviour of a capacitor. Although no resistance is shown, the small
resistance of the wiring (and the voltage source) will limit the current, and the rate
of change of the capacitor voltage. So in this case the charge and discharge takes
place very rapidly, but it will still have the same shape as the other charging curves.
EEE042A: 18 Describe RC circuits
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Summary
10

Capacitors will charge or discharge until the voltage across the capacitor
is equal to the supply voltage

An RC circuit is similar to charging a pressure vessel with gas. The
resistance limits the rate of flow of charge, and therefore the rate of
change of capacitor voltage. Large capacitances also take longer to
charge.

The time constant τ = RC, is the time for the voltage to vary by 63.2%,
of the difference between the current and final values.

In theory, the charge or discharge continues indefinitely, but after five
time constants, the charge or discharge is over 99% complete, and this is
accepted as ‘fully’ charged or discharged for most purposes.

The magnitude of the source voltage will produce larger or smaller
versions of the charging curve, but they all have the same shape. The
time to charge to any given fraction of the source voltage depends only
upon the circuit time constant, not the source voltage itself.
EEE042A: 18 Describe RC circuits
 NSW DET 2017 2006/060/04/2017 LRR 3675
Check your progress
1
A circuit consisting of a capacitor and resistor in series is connected to a voltage
source. If the potential difference of the capacitor equals the potential difference
of the source, then the capacitor is:
(a) charging
(b) discharging
(c) fully charged
(d) fully discharged
2
( )
What capacitance capacitor would be used in combination with a 2 Megohm resistor
to give a time constant of 0.25 second?
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3
The time taken to charge a capacitor to a practical full charge:
(a) depends on the voltage supplied to the circuit
(b) is inversely proportional to the capacitance
(c) is independent of the voltage supplied to the circuit
(d) is inversely proportional to the series resistance
4
( )
If the voltage applied to a capacitor continually varies:
(a) current will be continual
(b) the capacitor will charge to the highest voltage and stay there
(c) no charge will be present in the capacitor
(d) there will be no current after the initial charging current
( )
Check your answer with those given at the end of the section.
EEE042A: 18 Describe RC circuits
 NSW DET 2017 2006/060/04/2017 LRR 3675
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Answers
Check your progress
12
1
(c)
2
0.125 F or 1.25 × 10-7 F
3
(c)
4
(a)
EEE042A: 18 Describe RC circuits
 NSW DET 2017 2006/060/04/2017 LRR 3675