Energy Storage Circuit Elements
Capacitance is a measure of the ability of a device to store electrical charge. Electrical
charge storage appears in many circumstances, some rather subtle and others less so.
Inductance is a measure of the ability of a device to store electrical current. Electrical
current storage also appears in many circumstances, some rather subtle and others less
so.
Although the physical phenomena involved appear quite different they are actually
different aspects of the same fundamental phenomena. This being so it should not be
too surprising to find a duality in the description of inductive and capacitive phenomena.
We describe basic circumstances involving inductance and capacitance, but in language
which encompasses more complex conditions.
Capacitance
Electrical circuits 'work' by separating + and - electricity, doing work to do so against the Coulomb forces
of attraction between the separated electricities, and then allowing the Coulomb forces to do work in a
controlled and constrained manner to recombine the separated charges. It is possible to separate
electricities and prevent the separated charges, at least temporarily, from recombining, but it is not easy to
do so. A charged battery, a car battery is a familiar example, can maintain separated electricities for a
relatively long period when disconnected. But the Coulomb attraction forces are present, these are very
strong forces, and the battery must provide counterbalancing forces by means of an electrochemical
reaction to avoid a rapid charge recombination. While this counterbalancing can be maintained over a
relatively long period of time eventually the chemicals supporting the reaction have reacted completely and
irrevocably, and the battery is 'dead'.
A battery is an active device however, i.e., there are work-doing reactions going on. It is possible also to
store separated charges passively, i.e., using devices that have no inherent work doing ability. For example
suppose a battery is used to do the work needed to separate some charge, and the charge is placed on a pair
of electrodes insulated from one another. Thus in the figure to the left close the switch, wait a (short)
while, and then open the switch quickly. The battery has removed – charge from
one electrode leaving a residual + charge, and transferred – charge to the other
electrode. However the charges will not be able to cross over between the
insulated plates. Eventually (very quickly ordinarily) equilibrium is reached
where enough charge accumulates on the electrodes to prevent any further charge
accumulation. That is, the charge present on the electrodes opposes the transport
of any more like charge. (Physical experience indicates equilibrium occurs; after
all the continuing accumulation of charge means a corresponding continuing
increase in the force between the charges. Such an unstable condition is not observed.) The voltage
difference between the electrodes at this point is E.
Open the switch to trap the charge on the electrodes, equal + and - charges on the respective electrodes of
course since charge is conserved. This trapped charge represents a certain amount of work done previously
by the battery to effect the separation. This work, E joules per coulomb (i.e., volts) of trapped charge can
be recovered if a suitable conducting path between the electrodes is provided; the Coulomb force between
the separated electricities will do work.
For a wide variety of devices, in particular the devices considered in this course, the relationship between the
magnitude of the charge stored (Q) and the work per unit charge stored in the device (V) is linear, i.e., Q =
CV. C is a constant of proportionality called the capacitance which
represents the influence of the geometry and material properties of the device.
The device itself is called a capacitor. You should be careful to distinguish
the two words; one refers to the device and the other to a parameter
characterizing the device. Also note the voltage and current polarity
assignments; for this particular assignment C is a positive constant.
Circuits Energy Storage Elements
1
M H Miller
While this form of the relationship is very useful for many purposes it is generally more convenient for
analysis of circuits to differentiate both sides to obtain a dynamic relationship between the current through
the capacitor and the voltage across the capacitor, i.e.
dQ/dt = I = C dV/dt.
One might ask parenthetically what 'current' means here, since no charge flows between the electrodes.
What follows is a qualitative meaning; a quantitative explanation would take us into realm of
electromagnetic theory, too far afield of the present topic. If by whatever means one causes the charge to
change at one electrode the coulomb forces extending across the insulating region between the electrodes
and acting on the charge at the other electrode change as well. Because of this force there are consequent
changes on the charge at the other electrode, and this works out to satisfy KCL as long as the rate of
change of charge is not exceptionally fast. Since the electric force emanates from charge and acts on
charge it is possible mathematically to follow the electric force from one electrode to the other and in this
way to relate the charge changes. That tracking involves a concept called 'displacement' current and is the
basis for radio and TV broadcasting among other applications. For our purposes all this internal detail is
'lumped' into the terminal volt-ampere relations, and we are assured that whatever the internal details for
current associated with the charge transport at one terminal there is an equal current associated with the
charge at the other terminal so as to satisfy KCL.
Suppose that by some means (which we need not be concerned with here) a current flows 'through' an
initially uncharged capacitor. As charge accumulates the voltage across the capacitor changes, reflecting the
additional work involved in the charge separation. The power brought to the capacitor is IV= CV dV/dt,
and the energy change involved over a period of time is ∫CV dV/dt =(C/2) ∫ d(V2). Since the integrand is a
complete differential the energy expended depends on the difference in the square of the voltage only at the
end and at the beginning of the time interval considered. But this means that if the beginning and ending
voltages are the same no net energy is expended during the interim. Note carefully that there is nothing
that can be inferred about the details of what happened between the beginning and end points without an
examination of the voltage as a function of time during the interval.
If the beginning voltage is greater than the end voltage the capacitor has lost energy; it has less charge at the
end and the charge removed has carried off some of the stored energy. Conversely if the end voltage is
greater than the beginning voltage work is done to store energy in the capacitor (and increase the charge
stored). The point is that a capacitor stores rather than dissipates energy. What energy is put into the
capacitor can be removed at a later time (ideally that is; a real device approximating the theoretical capacitor
inevitably has losses and 'leaks' charge over a period of time). A capacitor can act as a sort of temporary
battery, i.e. as it discharges it does work. Unlike a battery however there is no internal mechanism to
replenish the charge continually, and so if the charge is not replenished eventually the capacitor becomes
inert. (The battery analogy should not be pressed too hard.)
There is another important aspect of capacitor circuit behavior also associated with time. There is a duality
in nature between energy and; it takes a finite time to make a finite change in the energy of a physical
system. A qualitative appreciation of this follows from the recognition that to increase/decrease the energy
stored in a capacitor the charge stored must be increased/decreased. But a change in charge supposes of a
charge transport, i.e., a current, and since infinite currents are not common in nature a finite change of
charge requires a finite time for the charge transport. Hence a capacitor may be anticipated to introduce
circuit time delays. If there is an attempt to change a voltage in a circuit involving capacitors a finite time is
needed for the charge transport to enable the change in capacitor voltage.
Consider an initially uncharged capacitor as in the circuit to the
left; the switch is to be closed at t = 0. (It would be a cruel world
indeed if a physical phenomena depended, all other things being
the same, on the particular time an experiment is started. One
expects physical laws to be less capricious. Thus a description of
the circuit behavior should depend on time intervals, i.e., changes in
time, and not on an absolute time. (There are some philosophical
problems with this view. Still the assumption of a relative
dependence of physical laws on time works well for the
Circuits Energy Storage Elements
2
M H Miller
circumstances encountered in this course, and we assume it. This means that a time origin t = 0 can be
selected arbitrarily from which to measure time changes. Given the same initial state the circuit
performance will be the same when described relative to any other origin in time.)
Before formally analyzing the circuit we make a qualitative evaluation of what to expect. When the switch
is closed current starts to flow. Since the capacitor is initially uncharged, and since it takes a finite amount
of time for the capacitor to charge, closing the switch 'instantaneously' has no immediate effect on the
capacitor voltage. Hence we conclude that initially, i.e., at t=0+, even with the switch closed, the capacitor
remains momentarily uncharged, and V=0. It follows then from KVL that the initial voltage across the
resistor is E, and so the initial current 'instantaneously' becomes E/R.
As this current flows it will charge the capacitor, and as a consequence V becomes greater than 0. But then
the current through the resistor is (E-V)/R, i.e., less than at first when V was 0. As the capacitor continues
to charge the voltage across the capacitor increases, and correspondingly the voltage across the resistor E V decreases. The current (E -V)/R also decreases, and we conclude that the capacitor continually charges,
but at an ever decreasing rate. Eventually that rate will become zero, and charging will stop. If the charging
rate is zero the charging current is zero, and so the voltage drop across the resistor is zero. Hence the
steady-state condition must be one in which the capacitor is charged to V=E. Note a time delay is involved
implicitly in charging the capacitor from its initial uncharged condition to the final steady-state condition.
A formal analysis proceeds in the same way as for a resistive circuit, except of course that one of the voltampere relations to be used is that for the capacitor. A related consequence is that the equations obtained
are differential not algebraic equations in general. In any event
and the solution to this first order equation with a constant driving force is
The RC product in the exponent has units of time, and commonly is called the circuit 'time constant'; it
provides a convenient measure of how quickly voltages and currents change in the circuit. The exponential,
for example, is e -5 after a time interval of 5 time-constants; this is less than 0.01 and so is very close to the
final steady-state condition. It is interesting to verify qualitatively the timing dependence on the RC
product. The larger C the more the final charge has to be, and so the longer it takes to transport the charge.
Similarly the larger R the greater the reduction in the charging current for an increase in capacitor voltage,
slowing the charging.
Consider (briefly) the discharge of an initially charged capacitor, as in the
circuit drawn to the left; the switch is closed at t=0. The charged
capacitor causes a voltage to appear across the resistor, and so current will
flow. The charge supporting the current flow comes from the capacitor,
reducing the energy stored in the capacitor to provide for the dissipation
in the resistor. The capacitor acts as would a battery, except of course the
capacitor has only a finite energy available that eventually runs out (an
idealized battery never runs down, by definition!). In the circuit
illustrated the capacitor discharges at a decreasing rate, since the capacitor
voltage decreases as it discharges and so the current through the resistor decreases.
The circuit equation is (note the polarity convention for I and consider the capacitor terminal volt-ampere
relation)
and
Circuits Energy Storage Elements
3
M H Miller
where Vo is the initial capacitor voltage. Note the appearance of the characteristic time constant again.
Graphs of the normalized charge/discharge expressions are drawn to the below. It can be shown easily that
the magnitude of the initial slope of the curves is 1, i.e., the intercept with the steady-state asymptote is at
t/RC = 1.
Circuits Energy Storage Elements
4
M H Miller
Inductance
Inductance is a measure of the ability of a device to 'store' electrical current. Electrical current storage
appears in many circumstances, some rather subtle and others less so. Although the physical phenomena
involved appear quite different charge storage and current storage actually are different aspects of the same
phenomena. This being so it should not be too surprising to find a duality in the description of inductive
and capacitive phenomena.
An electrical current, i.e., charge in motion, exerts a 'magnetic' inverse square law force on other currents
(Ampere law of magnetic force). The magnetic force is considerably weaker comparatively than the
Coulomb force between electrical charges, but nevertheless is very important in many applications. Work
is necessary to establish an electrical current flow; the higher the current the more the work necessary.
Changing the current then involves additional work.
The icon representing a 'lumped' inductor is drawn to the right. The coil represents
a means of concentrating a current flow in a small volume so as to enhance the
magnetic effect. If the current through the inductor is changed a voltage E is
'induced' across the inductor reflecting the necessity that work be done. The work
done in changing the current through an inductor over a period of time is
∫ VIdt = ∫ LI dI/dt =(L/2) ∫ dI2
Since the integrand is a complete differential the energy expended depends on the
difference in the square of the current only at the end and at the beginning of the
time interval considered. But this means that if the beginning and ending currents
are the same no net energy is expended during the interim. Note carefully that
there is nothing that can be inferred about the details of what happened between the
beginning and end points without an examination of the current as a function of time during the interval.
The inductor volt-ampere relation is the dual of that for the capacitor, i.e., the mathematical roles of V and I
are interchanged. One important difference to keep in mind is that a capacitor can store energy statically,
i.e., maintain a charge in the absence of a completed electrical circuit. The inductor requires a current flow
to maintain its energy storage, i.e., a closed current path. This is often a source of surprises to the unwary.
If an inductive circuit is open-circuited, just switched off for example, current flow is stopped and any
stored energy must be released; it does not simply disappear spontaneously. The inductor volt-ampere
relation indicates that a sudden change in current can produce a very large change in induced voltage, with
potentially destructive consequences if suitable precautions are not taken.
The circuit drawn to the right is similar to one used in the discussion of the capacitor. (To pursue duality
strictly the voltage source and series resistor should be replaced by a current source in parallel with a
resistor.) At t=0 the switch is closed and current begins to flow. Before the switch is closed there is no
current through the inductor; since a finite amount of time is needed to produce a charge transport the
current remains zero initially when the switch is closed. There is however a finite rate of change of current,
reflecting the initiation of the charge transport process. Ohm's
Law thus requires the voltage drop across the resistor initially be
zero, and this means the initial induced voltage is E. As current
flows the voltage drop across the resistor increases, necessarily
causing the induced voltage to decrease, i.e., dI/dt becomes
smaller. Hence current increases continually but at a decreasing
rate. Eventually a steady state will be reached in which the rate
of change of current has decreased to zero, and the current
produces a voltage drop E across the resistor; the induced
voltage is zero.
A formal solution is obtained from the equation
Circuits Energy Storage Elements
5
M H Miller
where (E-V)/R is the current through the series combination of the resistor and inductor.
The mathematics to solve the equation is the same as that described for the capacitor calculation; the initial
condition here is V (t=0) = E. Note: since I = (E-V)/R the equation can easily be rewritten in terms of the
current:
and solved with the initial condition I = 0. (Alternatively solve for V and use the inductor volt-ampere
relation to calculate I.) In any event the solutions are:
Note that L/R plays the same role as the capacitive time constant RC.
The circuit diagram drawn to the right is used both as
an additional illustration of an inductive transient
calculation, and also to suggest the curious behavior
inductive transients can display. Initially the switch
S1 is open and switch S2 is closed. Eventually a
steady-state condition (d/dt = 0) will be reached in
which V=E and E/R2 is the Ohm's Law current
through the R2 (and the inductor). This is the initial
state of the circuit when the two switch states are
changed concurrently.
When S2 opens the current through R2 becomes zero;
an open switch by definition carries no current.
However the current through the inductor cannot change instantaneously, but instead must be diverted into
the current path through R provided by the closing of S1. This causes an immediate change in voltage V to
IR1 + E = E (R1/R2) + E.
The value of V at t = 0+ then is E(1+(R1/R2)); there is an
initial voltage 'spike' whose amplitude is limited by the value
used for R2; adding a bypass resistor across an inductor is a
common precaution against excessive voltage induction in an
inductive circuit. The current circulating around L and R1
then decays exponentially with a time constant R1, and V
returns to a steady-state value of E.
Circuits Energy Storage Elements
6
M H Miller