Experiment Notes 2: Capacitors in DC Circuits
INTRODUCTION
CAPACITORS
Capacitors are circuit elements that store electric charge Q , and hence energy, according to the
expression
where V is the voltage across the capacitor and C is the constant of
proportionality called the capacitance. The SI unit of capacitance is the farad (after Michael
Faraday), 1 farad = (1 coulomb)/(1 volt). Capacitors come in many shapes and sizes, but the basic
idea is that a capacitor consists of two conductors separated by a spacing, which may be filled
with an insulating material (dielectric). One conductor has charge +Q and the other conductor has
charge −Q . The conductor with positive charge is at a higher voltage than the conductor with
negative charge. Most capacitors have capacitances in the range between picofarads (1pF =10 -12
F) and millifarads (1mF = 10-3 F = 1000 μ F) .
Note that we’ve also used the notation for a microfarad, 1μF=10-6 F =10-3 mF.
CHARGING A CAPACITOR
Consider the circuit shown in Figure 1. The capacitor is connected to a voltage source of constant
emf ε. At t = 0, the switch S is closed. The capacitor initially is uncharged, with
(In the following discussion, we’ll represent a time-varying charge as “q” instead of “Q”)
(a)
(b)
(c)
Figure 1 (a) RC circuit (b) Circuit diagram for t < 0 (c) Circuit diagram for t > 0
The expressions for the charge on, and hence voltage across, a charging capacitor, and the current
through the resistor, are derived in the Notes. This write-up will use the notation τ = RC for the
time constant of either a charging or discharging RC circuit.
The capacitor voltage as a function of time is given by
; a graph of this
function is given in Figure 2.
Figure 2 Voltage across capacitor as a function of time for a
charging capacitor
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The current that flows in the circuit is equal to the derivative with respect to time of the capacitor
charge,
where I is the initial current that flows in the circuit when the switch was closed at t =0. The graph of
current as a function of time is shown in Figure 3:
0
Figure 3 Current as a function of time for a charging capacitor
After one time constant τ has elapsed, the capacitor voltage has increased by a factor of
and the current has decreased by a factor of
DISCHARGING A CAPACITOR
Suppose we initially charge a capacitor to a charge Q through some charging circuit. At time t =0 the
switch is closed (Figure 4). The capacitor will begin to discharge.
The expressions for the charge on, and hence voltage across, a discharging capacitor, and the current
through the resistor, are derived in the Notes,
The voltage across the capacitor in a discharging RC circuit is given by
0
Figure 4 RC circuit with discharging capacitor
A graph of voltage across the capacitor as a function of time for the discharging capacitor is
shown in Figure 5:
Figure 5 Voltage as a function of time for a discharging capacitor
The current also exponentially decays in the circuit as can be seen by differentiating the charge on
the capacitor;
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This functional form is identical to the current found in Equation
and shown in Figure 3.
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RC Circuits
Electronic circuits often feature a resistor in series with a capacitor. Such circuits are often
referred to as resistor-capacitor circuits (or just RC circuits). The effect of the resistor is to limit
the rate at which current can flow onto the plates of the capacitor, thereby extending the time
required for the capacitor to become fully charged. Once the supply voltage is connected to the
circuit, the capacitor begins to charge. During the time in which the capacitor is charging, both the
current flowing in the circuit and the voltages across the capacitor and the resistor will be
constantly changing. These changing values are referred to as transients. The circuit diagram
below shows a simple series connected RC circuit.
A series connected RC circuit
When switch S is closed, then according to Kirchhoff's voltage law:
VSUPPLY = VC + VR
The terminal voltage supplied by the battery (VSUPPLY) is constant. The capacitor voltage (VC) is
given by Q/C where Q is the charge on the capacitor in coulombs and C is the capacitance value of
the capacitor in farads. The voltage drop across the resistor (VR) is given by IR, where I is the
current flowing in the circuit in amperes, and R is the resistance value of the resistor. The
following relationship is therefore true at all times:
VSUPPLY = VC + VR = (Q / C) + IR
At the instant when switch S is closed, the circuit will be in its initial state. The charge on the
capacitor will be zero, the voltage across the capacitor will be zero, and the voltage drop across the
resistor will be equal to the supply voltage. The current I flowing at this point in time will be equal
to VSUPPLY/R, since the resistance to current flow is solely due to the resistance R of the resistor.
As current continues to flow, the capacitor will become partially charged and the voltage across it
will increase as the voltage across the resistor falls. At the same time, the flow of current will slow
as the build-up of charge on the plates of the capacitor provides increasing resistance to the
movement of electrons. At some period of time after closing the switch, the capacitor will be fully
charged and current no longer flows in the circuit. The voltage drop across the resistor will be
zero, and the voltage across the capacitor will be equal to the battery terminal voltage (VSUPPLY).
The initial rate of change for both the voltage across the circuit components and the current
flowing in the circuit is rapid, but the rate of change slows as the capacitor approaches its fully
charged state. Graphs showing the changes in VC, VR and I are provided below. The graph
showing the way that VC varies with time is called an exponential growth curve, while the graphs
depicting the variation of both VR and I over time are called exponential decay curves. The use of
the word exponential here indicates that the shape of the curve can be expressed mathematically
using an exponential equation.
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The exponential growth curve for capacitor voltage
The exponential decay curve for resistor voltage
The exponential decay curve for current
The diagram below shows a circuit in which a capacitor is charged by a battery when the switch is
in position A. When the switch is moved to position B, the charge stored on the capacitor will
dissipate and current will flow through the circuit for a time until the capacitor is completely
discharged. At the instant when the switch is moved to position B, and providing the capacitor is
fully charged, the voltage across the capacitor will be equal to the battery terminal voltage
(VSUPPLY). The voltage drop across the resistor will also be equal to VSUPPLY. As current continues
to flow, the capacitor will lose charge and the voltage across it will fall, as will both the voltage
drop across the resistor and the level of current flowing through the circuit. These transients will
decay exponentially, until at some time after moving the switch to position B the capacitor will be
fully discharged and current no longer flows in the circuit.
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The capacitor can be charged and discharged using the switch
Graphs showing the transient curves representing capacitor voltage, resistor voltage and circuit
current during capacitor discharge are shown below.
Transient curves for voltage and current during capacitor discharge
When a capacitor is charged or discharged through a series connected resistor, the charging or
discharging process will take significantly longer than if a resistor is not present. Consider a
capacitor with a capacitance C (in farads) charging through a resistor of resistance R (in ohms). It
can be shown that, if during charging the charging current remained constant at its initial value,
the capacitor would be charged after a time (T) equal to C x R seconds. In fact, as we can see from
looking at the transient curves, the current decreases throughout the charging process. Time T is
known as the time constant, and represents the time taken for a capacitor to reach 0.63 (nearly two
thirds) of its full charge. In fact, for each further time period T that elapses during charging, the
capacitor will increase its charge by 0.63 of the difference in charge between its existing state and
the fully charged state.
Once charging starts, the capacitor will be almost (99%+) fully charged after a period equivalent
to five time constants (T x 5). The same is true when discharging a capacitor, with the charge on
the capacitor falling to 0.37 of its starting value in one time constant. The time constant is a useful
measure of how long it takes to charge (or discharge) a capacitor in series with a resistor where
both the capacitance of the capacitor and the resistance value of the resistor are known. When
discharging the capacitor, the time constant represents the time taken for the capacitor to lose
roughly two thirds of its charge. The following graph shows the charge and discharge curves for a
capacitor over a period of five time constants.
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