EA3: Systems Dynamics
Electrical Systems
VIII. SYSTEMS DYNAMICS – Electrical Domain
Let us now delve into some simple electrical systems. Once again, we will adopt a
systems approach here and infer a lot about the behavior of electrical systems by analogy
with their counterparts in the mechanics domain.
VIII.1 Electrical Systems Basics
VIII.1.1 Dynamic Variables: The physical entities of interest in the electrical domain
are charge, current, and voltage. While a complete physical understanding of these
concepts must be deferred to another course (PhysA35-2, or other courses on
electromagnetism), I will attempt to briefly highlight some of the essential concepts here.
Charge: The building blocks of matter (elementary particles) have a property associated
with them called electric charge, symbolically denoted by q.
At one level of understanding, charge can be thought of as an analog of another
property of elementary particles -- mass.† Matter is discrete (quantized) and therefore the
mass of any measured amount of matter must be some multiple of the masses of all the
elementary particles making up that amount of matter. When we are concerned with
things at the atomic scale, it makes sense to consider the unit of mass to be the mass of an
electron at rest: me. The mass of a proton is 1836.15me, and that of a neutron is
1836.68me. However, when large amounts of matter are under consideration, it is simpler
to consider matter as being continuous and to use a better known (and larger) unit of
mass, the kilogram. (Note: 1me = 9.11x10-31kg.) This is what we have been doing all
along.
Analogous to mass, charges in nature are also quantized; that is, they occur only
in discrete multiples of the elementary charge, which is denoted by e. However, unlike
mass, charges come in two types: positive or negative. The charge of an electron is
labeled negative and the charge of a proton is labeled positive. Some elementary
particles such as the neutron are neutral with no charge. Once again, when large amounts
of charges are considered, it makes sense to ignore the fact that they come only in
discrete multiples, and to use a larger unit of charge, the Coulomb (C). The charge of an
electron is –e; the charge of a proton is +e, where e=1.60x10-19C. In nature, positive and
negative charges are found in great abundance, but they are almost always perfectly
†
This is not the primary analogy that we will use in this course. But it is important to realize that different
analogies can be drawn between different physical entities depending on the context. The purpose of an
analogy (or ‘similarity’ to use the standard terminology) is to leverage our understanding of one aspect of
nature to another area where we observe similar phenomena.
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balanced against each other so that the net charge of any macro-object is usually zero. Of
course, we (or nature) can at times do something to disturb this charge balance, and that
is when interesting things happen!
There are a couple of parallels between charge and mass.
(a) Charges are conserved (just like mass is conserved, …well at least ‘mass-energy’).
(b) Between two charged bodies that are held stationary, there is an electrostatic force
exerted by each on the other which is given by Coulomb’s law:
F=
1
4
o
q1q2
2 .
r
r
r
{
(8.1)
unit vector
where qi are the charges of the two particles, r is the vectorial distance between them, and
-12 2
2
o = 8.85x10 C /N.m is called the permittivity constant. Note how Coulomb’s law is
similar to Newton’s law of gravitational attraction between two bodies. The electrostatic
force and the gravitational force vary as the inverse square of the distance between the
two bodies, they are both proportional to the product of charges or masses respectively of
the two bodies, and they act along the radial line joining the two particles. There is one
noteworthy difference however: the electrostatic force can be attractive (for unlike
charges) or repulsive (for like charges). Gravitational force, on the other hand, is always
attractive since there are no negative masses in nature.
For a complete physical understanding of electromagnetic phenomena, one needs
to continue along these lines to define the electric field E, from which one can infer
quantities such as the electric potential and so on. Then as one considers charges that are
allowed to move, one would be led to the concepts of electric current. The phenomena
of magnetism would then have to be considered as one delves into time-varying currents
et cetera. The whole edifice of electromagnetic theory will then gradually fall into place.
That, however, is a story for another time and place (Phys A35-2 and elsewhere), and you
should listen to that story very, very carefully, as it is one of the most fascinating success
stories in scientific understanding of nature.
Here however, we need to take the analogy route, and we will see that we can go
quite far in understanding (and designing) electrical systems using this approach. The
analogy that I will draw upon here is the one between electric charge and mechanical
displacement (or angle of twist, for rotational systems). You will see how this works
shortly.
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Current: Current is a quantity defined to characterize the flow of electric charges
through any part of our electrical system (such as in wires or other elements of this
domain). Current is defined as the rate of flow of charge across any cross-section:
dq
(8.2)
i= .
dt
The unit of current is the ampere (or amp, A) which is the flow of one Coulomb of charge
in unit time.
Current is a flow variable because it characterizes the flow of something
(charges). The mechanical analog of current is velocity in the translational mechanics
domain, or angular velocity in the rotational domain.
Voltage: Now what causes charges to flow, ie what causes a current in our system? That
would be some kind of an “effort” variable, which is characterized by the electric
potential or voltage. Voltage is analogous to force in the translational mechanics domain
(or torque in the rotational domain). It turns out that positive charges would like to flow
from a point of higher potential to one that is lower. It is thus the potential difference (or
voltage difference) between two points which is important. It is usual to prescribe the
voltage at various points in an electrical circuit with respect to some common ground,
which all of us agree would be our reference level. (We have done something similar to
this when we looked into gravitational potential energy, as you should recall.)
The unit of voltage is the volt (V). The symbol for the potential difference
between any two points is also V.
If there is a potential difference V between two points in an electrical circuit, then if
possible an electric current will ensue. The direction of current flow will be taken to be
from the point of higher potential to the point of lower potential.* When we plug in an
electrical device (say a computer) to a wall outlet and switch it on, we make it possible
for a current to flow from the point of higher potential (the “hot” line of our wall outlet)
through the components that make up our device to the point of low potential (the
“ground” line of our wall outlet).
*
That is, we assume that the charges that flow are positive charges, even though in most conductors it is
electrons that flow (in the opposite direction). With a few exceptions, a flow of positive charges in one
direction is equivalent to the flow of negative charges in the opposite direction as far as the overall system
behavior is concerned.
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Let us now collect the dynamic variables of the electrical domain:
Dynamic Variable
Charge
Symbol (units)
q (C)
Current
i (A)
Voltage
V (V)
Mechanical Analogs
Displacement
Angle of twist
Velocity
Angular velocity
Force
Torque
Type
flow
flow
effort
Power: Since charges exert forces on each other, work is done in moving electric charges
from one location to another. The resulting rate of work or power through any circuit
element can by analogy be expressed as:
Power = (force analog)*(velocity analog)
P = Vi
(8.3)
The unit of power is the Watt (which is the same as joule/sec).
Similar to the energy storing or power-dissipating behavior of mechanical elements, there
are electrical circuit elements that can store energy or dissipate it.
VIII.1.2 Circuit Elements: When the two ends of an electrical component are somehow
put at different potentials, an electrical current ensues. Alternately, if there is an
electrical current flow ‘i’ through an electrical element, then there must be a potential
drop ‘V’ across the two ends of the element. What makes electrical systems interesting is
that different elements exhibit different q vs V or i vs V behavior.
Capacitors: The energy-storing element, which is the electrical analog to a spring
element, is the capacitor. The defining property of a capacitor element is that it relates
the voltage across it to the charge across it.
VC =
1
q
C C
(8.4)
where C is the capacitance of the capacitor. Capacitance is measured in farads (more
typically in micro- or pico-farads). Note the parallel between a spring, which relates a
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force to the relative displacement (stretch). Unfortunately, capacitance is defined
somewhat differently from the spring constant (recall: fsp = K rsp), and therefore the
analog of the spring constant K is 1/C.
The symbol for a capacitor is:
C
+
i
Figure 8.1: Capacitor element
Physically, a capacitor is made of two conductive plates separated by an insulating
dielectric such as air. Sometimes, the two conducting plates are rolled into sheets with a
flexible dielectric sheet of mylar between them. Here are a couple of real life capacitors:
Figure 8.2: Some capacitors (courtesy EA3 hyperbook)
When a potential difference is applied across a capacitor, negative charges
accumulate on one side of the plate, exactly balanced by a positive charge distribution on
the other. Work is done by whatever exerts the potential difference in moving these
electric charges around. Where does this work go? Is it dissipated? No, because if we
remove the externally applied potential difference across the capacitor, it is possible to
get these charges to move back yielding energy back . Actually, an electric field is set up
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between the two plates because of the charge distribution on them, and it is in this electric
field that a capacitor stores energy analogous to a spring storing energy. What is the
energy stored in a capacitor? For a linear spring, the potential energy stored was one-half
spring constant times the square of the stretch. By analogy, the energy stored in a
capacitor is:
analogy: potential energy stored = (one-half) (spring-constant ) (stretch) 2
1 2
q
2C C
1
= CVC2
2
ΦC =
(8.5)
(Note: K ßà1/C)
Resistors: The energy-dissipating element of the electrical domain is the resistor,
analogous to the damper. The defining characteristic of the resistor is that the voltage
across it is proportional to the current (or rate of charge flow) through it:
VR = R
dqR
= RiR
dt
(8.6)
where the proportionality constant R is the called the resistance of the element. The unit
of resistance is the Ohm (Ω). Note that the analog of resistance R is the damping
coefficient CD (Do not confuse this with capacitance C. We really do need a lot more
symbols to avoid repetition, but then I will have to draw upon not only Latin and Greek
but also Chinese or Hindi! Let us stick with re-using symbols; the domain should make it
clear what we are referring to.)
The symbol for a resistor element is:
R
+
i
Figure 8.3: Resistor element
Physically, a resistor is a piece of carbon or some other low conductive material which
offers resistance to the flow of charges. Here are a few real life resistors.
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Figure 8.4: Some resistors (courtesy EA3 hyperbook)
The work done in moving electric charges through a resistive element is not stored, but is
lost as heat. The power dissipated through a resistor is given by analogy to be:
analogy: Power dissipated in damper =(damping coefficient)(velocity) 2
P = Ri 2
(8.7)
Quite a bit of heat can be generated by current flow through a resistor.
Inductors: The electrical analog to the mechanical inertial elements (masses and rotary
inertias) is the inductor. The defining property of an inductor is that it relates the voltage
across it to the rate of change of current:
VL = L
diL
dt
(8.8)
where L is called the inductance (measured in henrys) of the inductor. Note the
mechanical analog where the force acting on an inertial element was related to the mass
times the rate of change of velocity. Note that when there is a steady current through an
inductor, the potential drop across it is zero. The inductor only plays a role in trying to
“oppose” any change in current, much like mechanical inertia that “resists” change in
velocity of a body.
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The symbol for an inductor is:
L
-
+
i
Figure 8.5: An inductor element
Physically, an inductor is just a loop of a conducting wire looped typically over a
ferromagnetic core.
Figure 8.6: Some inductors (courtesy EA3 hyperbook)
The reason why an inductor acts like an electrical inertia has to do with the fact that
changing currents induce a varying magnetic field and vice versa. Analogous to
mechanical inertial elements, inductors store “kinetic” energy (actually this energy is
stored in the magnetic field created).
analogy: kinetic energy = one-half mass time velocity squared
ΦL =
1 2
Li
2 L
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Miscellaneous things you will see in electrical circuits: Complex electrical circuits are
made up of several elements. We have seen three of these thus far, and they are all
passive devices in that they do not contain energy sources inside them. Really interesting
electronic devices have, in addition to the passive elements listed above, signal sources
(such as batteries) as well as active devices such as amplifiers. Also, most electrical
circuits are complex networks of circuit elements. In order to analyze such circuits, it is
essential to isolate parts of the circuit (think of it as the equivalent of a free-body diagram
in mechanics, even though this analogy does not really hold well) and to analyze each of
the parts separately. Often, we will assume that these parts are buffered from each other
in that the presence of other parts will not significantly draw current from or otherwise
alter the behavior of the part under consideration. For instance, here is a simple tuning
circuit that enables your radio to receive a radio station:
ground
Figure 8.7: A tuning circuit (courtesy EA3 hyperbook)
There are a few new symbols here that you might not be familiar with.
Ground:
Figure 8.8: Symbol for the voltage reference level known as ground.
This is the symbol for “ground” and what it means is that other voltage levels specified in
the circuit are with reference to this point in the circuit (ground is taken to be at zero
voltage). The points marked Vin and Vout therefore mean that the voltages at these points
in the ciruit are Vin and Vout with respect to the point marked ground.
Buffers: As mentioned, chunks of electrical circuits are usually analyzed in isolation.
There is something more fundamental here than just convenience of analysis. Turns out
chunks of electrical circuits are designed to accept an input signal, process it in some
fashion, and then pass on an output signal to another part of the circuit for further
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processing. You can think of chunks of circuits as meta-elements . Just like resistor,
capacitor and inductor elements can be thought of as taking an input (voltage difference
across them) and producing an output (current or charge), so too these meta-elements
take an input signal and produce an output signal. In the above, the radio signal that is
received by the tuning circuit may be passed on to another circuit which amplifies it prior
to in turn passing on the amplified signal to a speaker.
In order to make this whole thing work though, the presence of another meta-element
circuit should not change the performance of the circuit part under consideration. That is
the resistors and capacitors etc of the amplifier circuit should not change the tuning
frequency selected by the tuning circuit. While in reality it is never going to be possible
to obtain perfect isolation of the various meta-elements, we will assume that various
meta-elements are so isolated or buffered, and we will use a symbol like this to denote the
buffering.
Figure 8.9: A buffer
Buffers are complicated things, but for our purposes, this level of understanding would
do. The buffer at the input should be understood to mean that no matter what is to the left
of our tuning circuit, the voltage at the input to this tuning circuit will be Vin. That is, we
have a voltage source at this point in the circuit. Similarly, no matter what is connected
to the right of our tuning circuit, it will not affect the current through and the voltages at
any point in our circuit.
Switches and batteries: Let us now look at another circuit which contains a couple of
more symbols that may not be familiar.
Figure 8.10: An LC-oscillator circuit
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Switch: This just completes a circuit when turned on, and interrupts a circuit when it is
left open. The symbol for a switch is:
Figure 8.11: A switch
Batteries: A common voltage source is a battery which is represented as:
+
Vb
Figure 8.12: A battery
A battery is an active device which actually produces electrical energy by converting it
from chemical, for instance. Batteries keep the potential of one of their two terminals
(denoted by a long line) at a higher level than the other. In the circuit above, when the
switch is closed, a current will flow from the battery terminal marked ‘+’ through the
capacitor and inductor elements down into the battery terminal marked ‘-‘. Through each
passive circuit element, there will be voltage drop in the direction of the current arrow
shown. However, note that inside the battery itself, the current flow will actually be from
the low potential point to a higher potential point! Now, how does this happen?
Something must be doing work to get the charges to move against their natural desire.
That something is the chemical reaction taking place inside and this is why a battery is an
active device.
Voltage buses: Here is another electrical circuit fragment:
Figure 8.13: A circuit with a voltage bus (courtesy EA3 hyperbook)
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This circuit has two horizontal wires one connected to the ground, and the other marked
+12volts. Such wires which do not contain any circuit elements along it, but has circuit
elements hanging off it are called voltage buses. Just remember that every point on a
voltage bus is at the same potential (here: zero for the ground and +12V for the top
wire).
Measuring Instruments:
How do we measure the voltage across or the current through a circuit element? There
are devices called voltmeters and ammeters that do precisely this. You stick a voltmeter
across two points in a circuit, and it will read out the potential difference between these
points. Similarly, you stick an ammeter into any wire in the circuit (snip the wire and
insert the ammeter), it will read the current through that wire. It is important to realize
that when we connect such devices into a circuit, they may affect the circuit behavior by,
say, drawing current from the circuit itself, or by providing additional resistance to
current flow. We shall assume that all these observer instruments are all somehow
buffered, and that they do not impact the system performance at all. That is, when
connected to a circuit these instruments measure the values that would be obtained in the
circuit without these instruments connected.
VIII.1.3 Dynamic Rules: Now that we have all the dynamic variables of the electrical
domain in place, and we have looked at a small but significant cross-section of the
electrical circuit elements available, we are almost ready to analyze the behavior of
complex electrical circuits. In order to do so, however, we need some laws or rules of
dynamics that are analogous to Newton’s laws that related the effort and flow variables in
the mechanics domain. The ultimate laws of electrodynamics are what are called
Maxwell’s equations, but we really cannot go there in this course. Rather, we shall state
two rules (that are actually consequences of Maxwell’s laws) that must be obeyed by the
effort and flow variables of the electrical domain. These are:
Kirchhoff’s Current Law - junction rule:
The net current flowing into any circuit junction (or node which are points where
two or more circuit elements are connected) must be equal to the net current
flowing out of it.
∑i
in
= ∑ iout
This rule actually follows from conservation of charge.
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Kirchhoff’s Voltage Law - loop rule
The sum of the voltage drops across all the elements in any closed loop of a
circuit is zero. Equivalently, the potential difference between any two points in
a circuit is equal to the sum of the voltage drops across all the elements along
any circuit path connecting these two points.
When considering voltage drops across circuit elements in a loop, let us agree to always
go around loops clockwise.
VIII.2 Analysis of simple circuits:
A simple RC-circuit: Consider the circuit shown:
Fig. 8.14: Charging a capacitor
Suppose the switch is closed at time t=0. What charges, currents, and voltage drops
develop through or across the various elements in this circuit?
Step 1: Annotate the circuit:
This involves picking directions for the current flow. In this case, it is a simple matter
because we know that the current must flow from the positive terminal of the battery
through the circuit and down into the negative terminal. When there are many elements
connected to a junction or node in the circuit, we will have some leeway to pick the
currents in parts of the circuit. All we have to do is be consistent that current flows from
higher to lower potential points in a circuit. If the potentials are obvious (as in this case
because of the battery), then we might as well start off right. If not, just choose some
direction and assume that as the current flows through every element in the direction
chosen, there must be a potential drop of the farther end with respect to the near end. The
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algebra will then dictate if we have chosen correctly (if we get negative currents it simply
means that the current direction is actually reverse of what we assumed.)
Once the directions of the current have been picked, label all the voltage drops
across the various elements making sure that the points of higher and lower potential are
marked consistent with the current direction. I will use ‘+’ and ‘-‘ signs to indicate this.
Note that this simply means that the ‘+’ side is at a higher voltage than the ‘-‘ side, and
does not mean that these are positive and negative voltages with respect to ‘ground’. For
example, for the direction of the current chosen here, the top of the resistor must be at a
higher potential than the bottom, and the right side of the capacitor must be at a higher
potential than the left.
Step 2: Apply Kirchhoff’s current and voltage rules:
Kirchhoff’s current rule (KCL): In this particular example, we have only one current
(since there are no junctions where multiple circuit elements are connected), and
therefore KCL simply tells us that:
(i) iR = iC = i
Kirchhoff’s voltage rule (KVL): There is only one loop here. Let us sum all the potential
differences across all the elements traversing clockwise through this loop.
(ii) VR + VC − VB = 0
Now, why is there a minus sign in front of the voltage associated with the battery?
Remember that batteries are active devices, and that inside a battery current flows from a
point of lower potential to one that is higher. Here, I was summing the voltage drops
across each element, and I found that when I traverse a battery from its negative to
positive terminal, the voltage drop is –VB, that is the potential does not drop, but
increases. Another way to look at this is to apply Kirchhoff’s Voltage Law between the
two points given by the positive and negative terminals of the battery. I know that these
two points are kept at a potential difference of VB by the battery. Therefore, according to
KVL, if I connect these two points by now traversing through the circuit containing the
resistor and the capacitor, I must find that the potential drops across the resistor and the
capacitor must be equal to VB. That is, VR + VC = VB , which is exactly the same equation
as before. You can decide which way you want to handle KVL (traversing a whole
circuit and summing up the potentials, or picking two points in a circuit with known
potential difference, and then traversing any other circuit path connecting these two
points).
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Step 3: Invoke the constitutive relations of the various elements:
Here, for the resistor:
(iii) VR = RiR
and for the capacitor:
1
(iv) VC = qC
C
Step 4: Do lots of algebra to get the state equations
Here is where we pull together all the equations we have obtained for the system.
We should first decide what dynamic variables we are going to focus on. Analogous to
mechanical systems, we will pick dynamic variables associated with energy storing
(capacitor) and inertial (inductor) elements to focus on, and swap out dynamic variables
associated with dissipative elements (resistors)
Here we only have the capacitor to worry about then, and analogous to what we
did for mechanical systems, we should pick the corresponding flow variable (charge).
Turns out that folks dealing with electrical systems prefer to choose the corresponding
effort variable (voltage) for a capacitor. Note that it is a simple matter to go from voltage
to charge using (iv) the constitutive relation for a capacitor.
So our state variable vector is just:
X = {VC } ,
(not much of a vector, but we are just testing the waters here)
Now, we should try to get an expression for the time rate of change of the voltage drop
across the capacitor:
( iv)
(i)
(iii )
(ii )
} 1
dVC } 1 dqC 1 } 1 } 1
=
= iC = i R =
VR =
{V − V }
dt
C dt
C
C
RC
RC B C
We are happy with this last step because we now have only the state variables and circuit
parameters as well as the known voltage source (the battery) on the right side. Therefore
the state equation here is just:
dVC
1
1
+
VC =
V
dt
RC
RC B
(†)
This is an equation that you have almost seen before. It is clearly an Euler type
differential equation (with constant coefficients). If the right side had been equal to zero
(with no voltage source), you know the solution to this equation is just an exponential
decay:
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VC no −source = Ae−t / RC
where A is some constant to be determined from initial conditions. However, since we
now have something on the right side, we need to add a correction term to the above
solution, and the correction term is obvious here. It is just:
VC source = VB
The complete solution is therefore:
VC (t) = Ae− t / RC + VB
Make sure that you test this by substituting the above result back into the DE(†).
To obtain the constant A, we use the initial condition that VC(t=0)=0 at the time the
switch is closed. Therefore,
VC (t) = VB {1− e −t / RC }
We can readily calculate the charge across the capacitor through:
qC (t) = CVC (t) = CVB {1 − e −t / RC }
And the current through the capacitor (in fact, in this case this is the current through
every element of our circuit) is:
i(t) = iC (t) =
dqC VB −t / RC
=
e
dt
R
This last behavior is familiar to us from the ant problem of Pset 3 and from the springdamper mechanical system. As we turn the switch on, the current just starts at a value of
1
VB/R and subsequently decays to zero exponentially with a time constant t* =
.
RC
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The behavior of the voltage and charges across the capacitor is not familiar to us yet, and
so let us plot these (for the case: R=2Ω, C=10F, and VB=9V):
Voltage across the capacitor
10
5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Charge across the capacitor
0
0.2
0.4
0.6
0.8
1
Current
1.2
0
0.2
0.4
0.6
0.8
1
Time
1.2
1.6
1.8
2
1.4
1.6
1.8
2
1.4
1.6
1.8
2
100
50
0
6
4
2
0
The behavior of the voltage across the capacitor is now clear. It eventually reaches a
“steady state” value of VB , but it gets there exponentially slowly starting from zero
voltage. Similarly, at the time the switch is turned on, there is no charge across the
capacitor, but a steady-state charge distribution of CVB accumulates exponentially slowly
with time. Actually, as you can see from the plots, for all practical purposes, within five
time constants, the steady-state values are almost reached (to what percent of the final
values?)
The physical explanation for this behavior is as follows. When the circuit is
closed by the switch being turned on, because of the potential difference provided by the
battery, an electric current begins to flow through the components and a charge begins to
accumulate across the capacitor. The process slows down as the capacitor gets charged
resulting in a decreasing current till eventually (exponentially) there is no more current in
the circuit, and we have a battery and a charged capacitor (with a potential difference
balancing that of the battery).
Energy considerations: As the capacitor gets charged its stored energy increases
gradually:
2
1
1
Φ C = CVC2 = CVB2 {1− e − t / RC }
2
2
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In the process, power is also dissipated in the resistor:
VB2 −2t / RC
PR = Ri =
e
R
2
R
You have to integrate the power (remember power is the rate of work done) to calculate
the total work done in the process of current flowing through the resistor.
∞
WR = ∫ PR (t)dt = ?
0
which you can work out on your own. {What do you find?}
Note that the energy stored in the capacitor and the energy dissipated in the resistor both
come from the battery. It is the energy dissipated by resistors that causes our batteries
(except apparently the ones used by the Energizer bunny) to run out of juice.
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