Module 2: Capacitors, Inductors, RC and RL Circuits
I.
Introduction: Time Dependence and Circuits
So far, everything we have considered has been “static” in that we have ignored time dependence. We
solve for V’s, I’s and P’s assuming they are constant in time. Further, when circuits are changed (i.e.
“opened” or “closed” using a switch, or a component is replaced) we assume that the currents, voltages
and powers transform instantaneously from their “pre” values to their “post” values (i.e. when you close
a switch the current goes immediately from zero to its final value).
Well, we know that real life is much more complicated than that. Most interesting phenomena – and
circuits are no exception – have non-trivial time dependence. And when changes occur, they are never
instantaneous, and in fact take a finite time to occur. For example, in physics class a ball bouncing off a
wall is modeled as a projectile whose velocity goes from +v to –v instantaneously (and discontinuously).
But if v changes discontinuously, that implies that acceleration (a = dv/dt, and the applied force F = ma)
must be infinite! So a real physical model takes into account the fact that the velocity must gradually
transform from +v to –v, though in most cases this happens quickly enough that the assumption of
instantaneous velocity change gives effectively the same answer as you get with a more complex model.
However, this is by no means always the case, and it’s important to recognize when this is true.
In circuits, these non-instantaneous changes are due to the fact that circuits not only are resistive to the
instantaneous flow of current, but are also reactive to changes in current or voltage. In circuits, these
“reactive” devices are capacitors and inductors. (And remember the word “react,” as it’s going to come
back when we talk about AC circuits!)
II.
The Capacitor
We start from a simple model of a parallel-plate capacitor, which is
composed of two plates of area A, separated by distance d. On one
plate a charge +Q is stored, with an equal but opposite charge –Q on the
other. (Note that when we say “a capacitor has charge Q,” what we
actually mean is that the total charge is zero: +Q on one plate, and –Q
on the other!) Since each plate has area A, the charge density on each
plate is given by
σ = ±Q/A
where σ here is charge density (not to be confused with the conductivity σ introduced in Module 1A!).
We can use Gauss’ Law ∫∫E●dA = Qenc/∈0 to show that the electric field in the region between the
plates is uniform (provided A >> d, so that fringing effects are minimal) and given by
E = σ/∈0
where ∈0 = 8.85 x 10-12 C2m2/N is the permittivity of free space. It follows that the potential difference
between the plates is given by
V = ∫E●ds = E d = σ d / ∈0 = Qd/(∈0A)
Now, we may be tempted to write an “Ohm-like law” that removes all this “microscopic” stuff about
electric fields and charge densities (just as we removed all that “microscopic stuff” about collisions and
electron masses when we discussed resistivity) to give an expression which describes a relationship
between our circuit-analysis mainstays, V and I. Unfortunately, we cannot get an expression as simple
as V = I R, since the relevant quantities in the above expression are Q and V. So, we get
Q = (∈0A/d) V
→
Q=CV
Where C = ∈0A/d is defined as the capacitance. While you can use the above equation to determine
that the unit of capacitance is C/V, capacitance (like resistance and voltage) comes up frequently so it
has its own unit, the Farad (F), where 1 F = 1 C/V. However, the Farad is in practice an enormous unit,
so for the most part off-the-shelf capacitors are sized in µF or even pF. (While Farad-sized capacitors do
indeed exist, they’re either enormous – where’s your forklift? – or involve significant design trade-offs
which compromise other properties in favor of increased capacitance, so they have very limited uses.
At this point it’s worth taking the time derivative of Q = C V, to yield
I = C dV/dt
which is an equally good equation representing the behavior of a capacitor, so that we now have
expressions for both of our mainstays, V and I (and their friends Q and dV/dt), although there is not a
simple linear relationship as there was with the resistor. Given that V (and its time derivative), and I
(and its time integral) appear in the capacitor’s defining equations, it is reasonable to expect that V and I
may exhibit time dependence in a capacitive circuit.
Before moving on, let’s consider the energy stored in a capacitor. When a capacitor is charged, there is
an electric field between its plates which stores energy that can be harnessed later, when the capacitor
discharges. The energy dU required to add a charge dQ to a capacitor at a voltage V is dU = VdQ, where
V = Q/C. Of course, as you add more charge, V increases, so to calculate the total energy U, you must
integrate from the first charge added (i.e. Q = 0) up to the final charge Q:
U = ∫dU = ∫V(Q) dQ = ∫(Q/C) dQ = Q2/2C
We can also rewrite this, using the relationship Q = CV, to get
U = (1/2) CV2
As a capacitor charges (i.e. Q and V increase) an increasing amount of energy is stored in the electric
field between the capacitor’s plates. This energy is stored, and may be released later as the capacitor is
discharged.
This is an important concept, to which we will return later: while a resistor irreversibly converts
electrical energy into heat as current flows through it, a capacitor stores energy in its electric field,
which may be returned later to the circuit!
III.
The Simple RC Circuit
At right is a simple RC circuit, which consists of a resistor R
and a capacitor C connected in series. Since time
dependence is important, a switch is included, which can
be used to “open” and “close” the circuit on demand.
Let us consider that for t < 0, the circuit has been
disconnected for a long time, and that there is no charge
on the capacitor plates. Since the circuit is open, we know that for all t < 0, I = 0 in the circuit. Since the
capacitor is uncharged (Q = 0), it follows that the voltage between the capacitor plates VC = Q/C = 0.
A.
“Charging” RC Circuit
At t = 0, someone closes the switch. We can use KVL to write down an equation for the sum of the
voltage drops (let’s assume a clockwise-flowing current I, and traverse the loop in the clockwise
direction:
ε - VR – VC = 0
where VR = I R from Ohm’s Law, and VC = Q/C from the fundamental relation for capacitance. Therefore,
ε - I R – Q/C = 0
For reasons of curiosity, let’s take a time derivative of that equation. ε is constant in time, so its
derivative is zero. The derivative of I will be denoted as I’ = dI/dt, the derivative of Q is I, and R and C are
simple constant coefficients. I’ll also multiply through by a common minus sign to get:
I’R + I/C = 0
--or--
R(dI/dt) + I/C = 0
This is an equation that involves both I and its time derivative I’. You’ve probably never seen such an
equation (known as a differential equation) before, although you will soon enough! In your differential
equations class, you’ll spend a semester learning how to solve equations just like this. For now, I will
just show you a few simple cases.
B.
t = 0 behavior of Charging RC circuit
First, let’s consider what happens at exactly t = 0, right after the switch is closed. Since Q = 0, VC = 0, and
our KVL equation reduces to
ε-IR=0
→ I = ε/R
which is our familiar result from before. If we think about this a bit more carefully, this is the same
answer we’d get if the capacitor were replaced by a piece of wire (i.e. a short):
An uncharged capacitor in a circuit acts as if it were replaced by a short circuit.
Therefore, any problem (no matter how ugly!) involving only resistors and uncharged capacitors can be
reduced to one containing only resistors. And we know how to solve those, right?
We also find, by using the derivative of the fundamental capacitor equation, that
I = C dVC/dt
→
dVc/dt = I/C = ε /RC
which tells us that the voltage between the capacitor plates begins increasing with time. It also follows
that the charge on the capacitor’s plates increases with time, since C = QVC, and VC is increasing with
time.
But as VC increases, it follows that VR must decrease so that ε - VR – VC = 0, and this means that the
current I in the circuit must decrease, since Ohm’s Law tells us that the I through the resistor (which is
the only current in the circuit) is given by I = VR/R.
So the general picture is that I starts out at the value you get by short-circuiting the capacitor: I(t=0) =
ε/R. But as t increases, Q on the capacitor, and thus VC, begins increasing. As a result VR, and thus I,
must decrease.
C.
t → ∞ behavior of Charging RC circuit
If we allow this to continue for a “very long time” (which I’ll define more carefully in a moment!), the
circuit reaches a “steady-state” condition in which the V’s and I’s are constant in time. Well, if the V’s
(including VC) are constant in time, I = C (dVC/dt) implies that I = 0 (the current eventually stops flowing).
If I = 0, then VR = IR = 0, and thus VC = ε - VR = ε, and the capacitor’s charge Q = CVC = Cε. In this sense,
the “fully charged” capacitor acts like an ideal voltage source with “just the right amount” of voltage to
reduce I to zero. Equivalently,
A fully-charged capacitor (i.e. one present in a circuit in a steady-state equilibrium) acts as if it
were replaced by an open circuit.
No current flows through the capacitor, and the voltage between the terminals is equal to the value you
get from KVL by setting the current through the capacitor branch to zero.
D. What about intermediate times? Solution of the circuit differential equation
So at this point we’ve solved for the two “extreme cases:” at t = 0 the capacitor acts like a short, and at t
→ ∞ the capacitor acts like an open circuit. What about intermediate times? Here, unfortunately, we
have to get into some mathematics. Let’s return to our “circuit differential equation” obtained by
taking the derivative of our KVL loop equation:
R(dI/dt) + I/C = 0
I will rewrite this equation by doing something your calc prof probably told you never to do: I’m going
to “separate” the dI and dt in dI/dt:
dI/I = -(1/RC) dt
It may now occur to me to take a time integral of both sides:
∫dI/I = ln[I(t)] + C = (-1/RC) ∫dt = -t/RC + C’
Since there are two additive constants C’ and C that came from doing the integrals, I can replace them
both with a single additive constant C0 = C-C’:
ln[I(t)] + C0 = -t/RC
To solve for I(t), we raise e (the base of natural logarithms) to the power of both sides, and use the
familiar properties of exponents and logs:
exp (ln(I(t))+C0) = I(t)/I0 = exp(-t/RC)
→
I(t) = I0 exp(-t/τ)
where I0 = 1/exp(C0) and τ = RC represents the “time constant” of the circuit. I0 can be determined by
plugging in t = 0, in which case we see that I0 = I(t=0), which equals ε/R in our circuit. We also see that
I(t) starts at I0 = ε/R, and declines exponentially toward zero as t increases, which is exactly what we
expected from our simplistic circuit analysis.
We can now go back and solve for the rest of our circuit parameters using Ohm’s Law, the fundamental
relation for capacitance, and our KVL equation:
I(t) = (ε/R) exp(-t/τ)
(at t = 0, I(t) = ε/R, and declines toward zero with increasing t)
VR(t) = I(t) R = ε exp(-t/τ)
(at t = 0, VR(t) = ε, and declines toward zero with increasing t)
VC(t) = ε - VR(t) = ε(1 - exp(-t/τ)) (at t = 0, VC(t) = 0, but increases toward ε with increasing t)
Q(t) = C VC(t) = Cε (1 - exp(-t/τ)) (at t = 0, Q = 0, but increases toward Cε with increasing t)
An extremely useful property is that all of the behavior of this circuit can be described just using a single
number τ = RC, the “time constant.” All of the time dependence in the circuit “scales” with τ, which is
just the product of R and C. Thus we see that the “instantaneous” circuit behavior we previously
considered corresponds to the case that C = 0, so that τ = 0. But if C has a finite value (and more so if
either C or R is large), then the time dependence can be very significant!
E.
Behavior at intermediate times: Alternate derivation
Suppose we didn’t want to use the “brute force” method above of separating dI and dt? (In your
differential equations course, this is probably the first method you’ll learn, and is known as separation of
variables, because you separate your variables onto opposite sides of the equation – the I’s on one side,
the t’s on the other.)
Well, there is another, more widely-applicable method for solving differential equations. It’s known as
“guessing the answer,” or if you want to sound fancy, an “ansatz solution.” Let’s look at our original
differential equation from KVL:
R(dI/dt) + I/C = 0
If you look closely, you’ll notice that this equation has two terms – one involving a function I, and
another involving its derivative, dI/dt. Both I and dI/dt are, in principle, some horribly complex functions
of time, so both I and dI/dt can take on all sorts of values depending on what value of t you plug in.
Both terms are multiplied by only constant coefficients, yet somehow, miraculously, for any value of t
you plug in, the sum of those terms equals zero.
In other words, no matter what crazy time dependence I(t) may have, that crazy dependence is precisely
mirrored by what dI/dt does, so that the sum of the two terms is always exactly zero. This means that
both I and dI/dt must have the same functional dependence.
Well, we may ask, what types of functions return (essentially) themselves as their own derivatives?
Polynomials don’t do that. Logarithms don’t do that. Sines and cosines almost do, although you have to
take two derivatives of a sine to get back another sine (we’ll use this later when we talk about AC
circuits!). However, the derivative of an exponential is just another exponential; if we let f(t) = A exp(kt)
df/dt = d/dt ( A exp(kt) ) = A k exp(kt) = k f(t)
Let’s try plugging in this f(t) to our differential equation, using the fact that df/dt = kf:
R k f(t) + f(t) / C = 0
→
Rk + 1/C = 0
→
k = -1/RC
where the first step involved dividing by f(t) (which is okay as long as f(t) does not equal zero, also
known as the trivial solution), and the value of k represents the time constant τ we found above.
This leaves only the arbitrary constant A in f(t) = I(t) = A exp(-t/τ), which we can find using the “initial
condition” that I(t=0) = A = ε/R.
And we see that we get the exact same solutions as above.
F.
More about the time constant
One more thing we can do is inquire about what values I(t) and the other
variables take as a function of time. Since everything scales with τ, all we
need to do is compute values for one prototypical circuit for various values
of t/τ, and then “scale” the behavior for circuits with differing values of τ. I
present at right a table of values of exp(-t/τ) and 1-exp(-t/τ) for different
values of t/τ) (you can compute intermediate values using your calculator).
From this table we see, for example, that the current in a circuit drops to
0.37 of its initial value ε/R after one time constant, to 0.05ε/R after 3 time constants, and to less than
0.01ε/R after 5 time constants, for example. This gives rise to:
Rule of thumb: values may be considered to have reached their “steady state,” i.e. t → ∞ values,
after 5 time constants.
Of course, like all rules of thumb, this should be used with some discretion, since there may be cases
where that last 1% is significant! But the “5 time constant rule” is broadly applicable, and sufficient for
most purposes.
Circuit quantities that scale as 1-exp(-t/τ), such as VC(t) and Q(t), can be solved similarly. The charge on
the capacitor reaches 0.63 of its final value after 1 time constant, and the voltage across the capacitor
reaches 0.98 of its final value after 4 time constants
G. The Discharging RC circuit.
Let’s imagine that our charging RC circuit has been charging for a long time, so that all values in the
circuit reach their t → ∞ values:
I = 0, VR = 0, VC = ε, Q = Cε.
Now, the switch is opened. What happens to the charge on the capacitor?
Well, the charge has nowhere to go. The plates are, if you will, “floating” so there is no way for the
charge +Q to make its way to the opposite plate. So the capacitor will, if we ignore the effects of
leakage, remain charged with the same charge (and voltage between its plates), no matter how long we
leave the circuit there.
But I can go even further. I can remove the capacitor from the circuit, and it will continue to hold its
charge, as long as I take care not to short out the terminals. I can put the capacitor in a box somewhere,
come back later, and it will still have the same charge on its plates as it did when I put it there. (Again,
I’m neglecting the effects of leakage, which will be substantial for real capacitors.)
Or, I can take that charged capacitor and connect it up to yet another
resistor R’ (right).
As long as the switch remains open, there is nowhere for the charge on
the plates to go, so the capacitor remains charged with VC = ε (the voltage
it “inherited” from the previous circuit).
What happens when I close the switch? (For simplicity, let’s call this t = 0 yet again).
Well, once the circuit is closed, we can use KVL:
VC – VR = 0
With VC = Q/C and VR = IR’. Since at t = 0 the charge on the capacitor’s plates is not going to change
discontinuously (which would require an infinite value of I!), we can treat VC as a constant to get I =
VC/R’ = ε/R’.
As t increases and the current continues to flow, however, the capacitor begins discharging, and VC
decreases. Note that in this case dQ/dt = -I (where the minus sign represents the fact that a positive
current corresponds to a decreasing value of Q). Thus our KVL equation can be rewritten as
Q/C + (dQ/dt) R’ = 0
and we can take the time derivative of the whole equation to get (after multiplying through by a minus
sign):
I/C + (dI/dt) R’ = 0
Note that this is exactly the same equation that I solved in sections D and E, which has the solution
I(t) = I0 exp(-t/τ)
(i.e. has value I0 = ε/R’ at t=0 and declines exponentially toward zero)
Where τ is the time constant of the “new” circuit, τ = R’C.
We can now go back through, using Ohm’s Law, the fundamental relation for capacitors, and KVL to get:
VR’(t) = I(t) R’ = ε exp(-t/τ)
(has value ε at t=0 and declines exponentially toward zero)
VC(t) = VR(t) = ε exp(-t/τ)
(has value ε at t=0 and declines exponentially toward zero)
Q(t) = C VC(t) = C ε exp(-t/τ)
(has value Cε at t=0 and declines exponentially toward zero)
And this is the behavior of the discharging RC circuit.
IV.
Dealing with combinations of capacitors
So far our analysis has assumed that our RC circuit contains exactly one resistor and exactly one
capacitor in series with each other. Well, what if a circuit is more complicated than that?
If there are multiple resistors in series or parallel, we know how to deal with those -- series resistances
add, and parallel resistances combine using the “one over formula:”
Req = R1 + R2 + …
(series)
1/Req = 1/R1 + 1/R2 + …
(parallel)
Are there similar rules for combinations of capacitors? The answer is Yes.
Let’s consider parallel capacitors first. If three capacitors are connected in parallel, it follows from KVL
that all three must have the same voltage across them. Therefore, they will each carry a charge Qi which
can be calculated from
V = Q1/C1 = Q2/C2 = Q3/C3
The total charge Q carried by the three capacitors is simply the sum of the three individual charges:
Qtot = Q1 + Q2 + Q3
We can replace this network of three capacitors with one “equivalent” capacitance Ceq, with voltage V
across its plates and charges ±Qtot on its plates, with
Qtot = Q1 + Q2 + Q3 = Ceq V
→ Ceq = Qtot / V = Q1/V + Q2/V = Q3/V = C1 + C2 + C3
This can be generalized to any number of capacitors:
In parallel, capacitances add: Ceq = C1 + C2 + C3 + …
What about series combinations of capacitors? If we have three capacitors in series (depicted
at right), we know that the voltages across the three capacitors should add up:
Vtot = V1 + V2 + V3
What about the charges on the plates? There are two ways to think about this:
1.
The currents through all three capacitors must be the same at all times. If the currents are
the same at all times, then the total charge accumulated on the plates of all three must be
the same, since Q = ∫I dt.
2.
Notice that there are two “I-shaped” pieces of metal which effectively join the middle
capacitor to the two outer ones. These “I-shaped” pieces of metal are isolated from the
rest of the circuit. So if the lower plate of the upper capacitor has a charge of –Q1, and the
upper plate of the middle capacitor has a charge of +Q2, if the initial charge on that “Ishaped” piece of metal was zero, it must remain zero at later times, since there is no way for
charge to get on or off that isolated piece of conductor. Therefore, Q2 = Q1, and all
capacitors in series have the same charge on their plates.
I like to call the method in 2 “Carlo’s I-theorem,” because of the I-shaped pieces of metal. If the picture
is rotated 90 degrees, one can use its corollary, “Carlo’s H-theorem” (using the isolated H-shaped pieces
of metal that result) in the same way. As usual, the people responsible for naming things haven’t
responded to my application…
In any event, we agree that the charges stored on series capacitors are the same, and the total voltage
across the series combination is the sum of the individual voltages:
Vtot = V1 + V2 + V3 = Q/C1 + Q/C2 + Q/C3
We can define an equivalent capacitance Ceq = Q/Vtot, to get:
Vtot = Q/Ceq = Q/C1 + Q/C2 + Q/C3
→
1/Ceq = 1/C1 + 1/C2 + 1/C3
So, in parallel, you use the “one over formula” to compute Ceq.
Note that the formulas for combining capacitors are exactly the same as those for combining resistors,
except that they are reversed: you use the one over formula for capacitors in series, and the additive
formula for capacitors in parallel, rather than the obverse used for combining resistors.
Now, given some complex RC circuit, we can attempt to reduce it to the “simple RC circuit” we know
how to solve by (1) reducing our resistors down to a single Req using series and parallel combinations,
and (2) separately reducing our capacitors down to a single Ceq using series and parallel combinations.
In these cases, the circuit behavior is still the same as presented above in our “charging” and
“discharging” examples, but with τ = ReqCeq.
If, however, the circuit cannot be solved by this “decomposition” method we have to revert to our lesspreferred methods of circuit analysis.
In some cases (typically, when there is one capacitor connected to some horrid combination of power
sources and resistors) we can apply one of our linear network theorems (superposition, Thevenin or
Norton) to replace a horrible RC circuit with something less horrible, which can be solved in analogy to
our “simple RC circuit.”
In other cases, particularly where there are multiple capacitors connected in non-trivial ways, we have
to use our Kirchhoff-based methods (branch currents, mesh currents, or node voltages). In such cases,
the exponential solutions we derived above may not even be applicable, since different capacitors can
charge at different rates, and the behavior can depart significantly from the ideal exponential behavior
we found in the simple RC circuit.
However, if we do this, rather than having simultaneous equations to solve (which we know how to
solve, with some difficulty!), we end up with a system of simultaneous differential equations. These are
beyond the scope of this course since they require rather advanced mathematics (specifically, you can
use something called Laplace transforms to recast your simultaneous differential equations as good old
regular simultaneous equations), although if time permits at the end of the course I can show you how
to set these up, and perhaps also how to solve them using numerical methods!
V.
Inductors and Inductance
Consider a coil of wire in the form of a solenoid. The coil has a
radius r (and cross-sectional area A = πr2), length l and N turns
(usually expressed as n = N/l turns per unit length) carrying
current I. From Ampere’s Law we calculate the magnetic field B
inside the solenoid as
B = µ0In
where µ0 = 4π x 10-7 T●m/A is the permeability of the vacuum. Furthermore, if this current I changes at
a rate dI/dt, the magnetic field will have a rate of change given by the time derivative of the above:
dB/dt = µ0n dI/dt
Finally, we know from Faraday’s Law that a changing magnetic flux generates an induced EMF given by
εind = -dΦB/dt
where ΦB is the magnetic flux, and the minus sign denotes Lenz’s Law in that the induced EMF tends to
oppose the change in magnetic flux. In the case of the solenoid, ΦB = B (πr2) N, so
εind = -dΦB/dt = (πr2) N dB/dt = (πr2) N (µ0N/l) dI/dt = (µ0AN2/l) dI/dt
→
εind = -L dI/dt
Where L = µ0AN2/l is known as the “self-inductance,” usually referred to as simply “inductance” of the
solenoid. L is measured in units of V●s/A, but it comes up frequently enough that L is given its own unit,
the Henry, with 1 H = 1 V●s/A.
As with the Farad, the Henry turns out to be a rather large unit, so most practical inductors are
measured in mH or µH.
To increase inductance, inductors can be made with a larger number of coils (increase N), larger coils
(increase A) or make coils with thinner wires (so that n can be increased), but this can only be done to a
certain extent, as all will raise the internal resistance of the wires, giving rise to unwanted effects.
Instead, a piece of ferromagnetic or ferromagnetic material may be inserted into the core of the
inductor, which increases the strength of the magnetic field by a large factor. In this case µ0 is replaced
by µ, where µ represents the relative permeability of the core material, and is may be thousands of
times higher than µ0.
In circuits, the behavior of the inductor is described by its fundamental relation, εind = -L dI/dt, although
in practice the minus sign is usually omitted, and we’re forced to remember that the induced voltage is
induced in a direction that tends to oppose the change in current.
Finally, we may consider the energy stored in an inductor. There are several ways to derive this, of
which I pick what I consider to be the simplest. If we start with an inductor with no current going
through it, and “ramp up” the current toward a value I, there must be some ramping of the current
dI/dt, against which the inductor applies an induced EMF εind = L dI/dt (where I’m neglecting the minus
sign). The power (i.e. the rate at which work is done) at any moment is given by P = εindI, and the total
work done is:
U = ∫P dt = ∫εindI dt = ∫L I (dI/dt) dt = (1/2) LI2
That is, an inductor with a current I flowing through its coils stores a quantity of energy U = (1/2) LI2 in the
magnetic field the current generates. As I increases, the field strength (and thus the energy stored in
the field) increases, and as I decreases, the energy stored in the field decreases. Thus, like the capacitor,
the inductor stores electrical energy, which may be returned to the circuit some time later.
VI.
The Simple RL Circuit
A simple RL circuit is depicted at right. Since all of the functional
dependences in RL circuits are very similar to what we saw with RC
circuits, I will give a rather abbreviated discussion of the inductor’s
behavior.
We will assume that for t < 0 the circuit has sat silently with I = 0.
Therefore, for all this time, the inductor has been happy, as dI/dt was also zero, so VL = εind = 0.
At t = 0, someone closes the switch. At that moment, we can write KVL as:
ε - VR – VL = 0
where VR = I R, and VL = L dI/dt (where we’ve omitted the minus sign).
Note that the current in the circuit cannot change discontinuously: if this were the case, dI/dt would be
infinite, and the induced voltage would be infinite! That is unphysical. (This is analogous to the
capacitor disallowing any discontinuous changes in voltages; the inductor does the same with current.)
For this reason, at t=0, no current may flow. I = 0, so VR = I R = 0, and it follows that VL = ε. Using that,
we can state that
dI/dt = VL/L = ε/L
is the rate at which the current begins to increase at t = 0. (In analogy to what we saw with capacitors,
the limit in which the current instantaneously goes from zero to its final value corresponds to L = 0.)
Therefore, as time goes on, the current in the circuit gradually increases. If I is gradually increasing, then
VR = I R must be as well. Then VL = ε - VR must be gradually decreasing, and thus dI/dt must decrease
with time (although I itself is increasing!).
After a very long time, the circuit reaches a steady-state condition, in which all the V’s and I’s are
constant in time. If this is the case, dI/dt = 0, so VL = 0, and it follows that VR = ε, so I = ε/R is the longterm steady-state current that eventually flows.
Therefore the rules we can establish are:
When the switch is initially closed at t=0 and there is a “shock” of current that tries to flow, the inductor
acts like an open circuit in that it does not permit any current to flow.
After a very long time, the circuit reaches a steady-state condition, and the inductor acts like a short
circuit.
What about intermediate times? Here we will write down our KVL equation again:
ε - VR –VL = ε - IR – L dI/dt = 0
We note again that this is an equation involving only I, its derivative dI/dt, and a few constant values, yet
miraculously this linear combination of I, dI/dt and a constant value somehow always adds up to zero!
This implies that I and dI/dt have some functional form such that a derivative of I returns something
with the exact same functional form. Again, we will “guess” that the solution is something exponential,
but given that the current starts from zero and increases towards its max value (rather than vice versa),
we’ll use the “other” exponential formula we saw with RC circuits, and guess that I(t) = If(1-exp(-t/τ)),
where τ is the familiar time constant (to be solved in terms of L and R), and If represents the eventual
steady-state current that flows after a long time. If I(t) = If(1-exp(-t/τ)), then dI/dt = +If/τ exp(-t/τ), and
ε - VR – VL = ε - IR – L dI/dt = ε - RIf(1-exp(-t/τ) – LIf/τ exp(-t/τ)
Which we can re-arrange to group all the exponential terms, and all the constant terms, together:
ε - RIf - exp(-t/τ) (RIf – LIf/τ) = 0
Now I apply a little “trick” – I now have two terms, one of which is constant (ε - RIf), and one of which is
has a complex time dependence: exp(-t/τ) (RIf – LIf/τ). Since both terms have different time
dependence, but miraculously subtract out to zero at all values of t, this tells us that the coefficient of
the “exponential” term (RIf – LIf/τ) must be zero (so that the exponential time dependence is multiplied
by a coefficient of zero), and separately, then, the constant term ε - RIf must be zero (so that the overall
equation balances out). Therefore
ε - RIf = 0
--and--
RIf – LIf/τ = R – L/τ = 0
from which I determine that If = ε/R (which matches what I found earlier by “shorting” out the inductor),
and
τ = L/R
which you can compare to the τ = RC which I found for RC circuits.
With this in hand, I can go back and write down equations for all my LR circuit parameters:
I(t) = If (1-exp(-t/τ))
with If = ε/R and τ = L/R; I increases from zero toward If as t increases
t
VR(t) = R I(t) = ε (1-exp(- /τ))
VL(t) = ε - VR(t) = ε - ε (1-exp(-t/τ)) = ε exp(-t/τ)
dI/dt = VL(t)/L = ε/L exp(-t/τ)
And we’re done with the simple LR circuit!
One note is that the equations describing the LR circuit are essentially the same as those describing RC
circuits; this comes about because the fundamental relation for capacitors, I = C dV/dt, has such a similar
form to the defining relationship for inductors, V = L dI/dt.
VII.
Dealing with combinations of inductors
At this point we know how to solve the “simple RL circuit.” Now, if we were given a more complex
circuit, in analogy with what we did with RC circuits, a natural idea would be to try to reduce the circuit
to a “simple RL circuit.”
We now know how to reduce series and parallel combinations of both resistors and capacitors; do
similar relationships hold for inductors? The answer (as you may expect), is yes.
If three inductors are in series, they all have the same current (and thus dI/dt) through them. Since they
are in series, the total voltage across the three inductors Vtot is equal to the sum of the three individual
voltages. We can therefore replace the three inductors with a single Leq which obeys the fundamental
relation for inductors:
Vtot = Leq dI/dt = L1 dI/dt + L2 dI/dt + L3 dI/dt = (L1+L2+L3) dI/dt
Thus, for inductors in series,
Leq = L1 + L2 + L3 + …
which is the same formula we found for resistors in series, and for capacitors in parallel.
For inductors in parallel, we know that all must have the same voltage V across them, which means that
a changing current dI/dt will distribute itself among the inductors as
V = L1 dI1/dt = L2 dI2/dt = L3 dI3/dt = …
where I = I1 + I2 + I3 + …, so dI/dt = dI1/dt + dI2/dt + dI3/dt + …
So we can replace the parallel inductors with a single inductor with Leq, such that
V = Leff dI/dt = Leq (dI1/dt + dI2/dt + dI3/dt + …) = Leq (V/L1 + V/L2 + V/L3 + …)
After canceling the common factor of V, and dividing both sides by Leq, we find:
1/Leq = 1/L1 + 1/L2 + 1/L3 + …
for parallel inductors, which mirrors the familiar “one over” formula we found for resistors in parallel,
and capacitors in series.
Therefore, if we have an RL circuit, we may try to reduce it by (1) reducing all the resistors into a single
equivalent resistance Req; and (2) separately reducing all the inductors into a single equivalent
inductance Leq. If that is not possible, we can try using our network theorems (Thevenin, Norton,
superposition) to replace a nasty RL network with things we can solve.
Failing that, we have to revert to our Kirchhoff-based methods, which as with RC circuits, will generally
result in a system of simultaneous differential equations, which are beyond the scope of what we will do
in this course.
VIII.
Practical aspects of capacitors and inductors
So far I’ve concentrated on circuits composed of “ideal” capacitors and inductors.
A. Ideal and Real Capacitors
The ideal capacitor was modeled as a pair of parallel plates. The plates themselves are completely
without resistance, and there is zero leakage of current between the plates. Real capacitor plates have a
finite amount of resistance (which effectively is in series with the capacitance), and real capacitors have
leakage between the plates (which is effectively represented as a leakage resistance across the
capacitor.
As I mentioned earlier, the Farad is an enormous unit. Typically, a reasonably-sized and spaced pair of
parallel plates will have a capacitance on the order of pF (you can check this for yourself by plugging in
“reasonable” values to C = ∈0A/d; since ∈0 ~ 10-11 F/m, you need to plug in very “unrealistic” values for A
or d -- orders of magnitude away from the familiar 1-ish values we know and love -- to get a capacitance
that differs much from the pF regime.
So let’s think about how capacitance is affected by the geometric factors in C = ∈0A/d. Note that if the
plate area A increases, the capacitance increases. So one way to make a capacitor with larger
capacitance is to increase the plate area (usually, this is done by building a capacitor out of multiple
“layers” of small plates, or by having large-area plates rolled into a cylinder, rather than having huge flat
plates that take up a large amount of space).
Capacitance can also be increased by making the plate separation d as small as possible. Though this is a
bit counterintuitive, it makes sense. For a given value of ±Q on plates of area A, the electric field E
between the plates is fixed at Q/∈0A. If the distance between the plates is made smaller, then the
potential difference V = E*d between the plates decreases. Hence, for a given potential V applied
between the plates, a larger charge Q can be stored, and C = Q/V is larger.
A third way to increase C is to somehow decrease the E field between the plates, for a given value of Q.
Unfortunately, E = σ/∈0 comes from fundamental laws of physics (in particular, ∈0 is set by the
properties of the vacuum in our universe), so we can’t change that if there is only vacuum between the
plates without traveling to another universe. (Good luck with that!)
However, many insulating materials exhibit dielectric properties. This means that while the electrons in
the material cannot travel freely to conduct electricity (which is why they’re insulating), the charges can
displace a small amount from their equilibrium position, giving rise to an electric field which tends to
“screen out” the applied field – the negative charges lean slightly toward the positive plate, and the
positive charges displace a small amount toward the negative plate, giving the illusion that the actual
charges on the plates are less than ±Q. But the only issue that matters to us at present is that when a
dielectric material is present between the plates, ∈0 in the above formulae is replaced by ∈ = κ∈0, where
κ is known as the relative permittivity of the material. The vacuum has κ = 1 by definition, while air has
κ = 1.006. Paper typically has κ = 2-4, while mica has κ=5, and typical glasses have κ = 5-10. Some
ceramic materials, however, have κ ranging from 100 to over 10,000, so these can give a substantial
increase in the value of capacitance!
B.
Classification of Capacitors
As done with resistors, we should come up with a scheme to classify capacitors. The following
categories are useful:
-
Fixed or variable value of capacitance?
Breakdown voltage
Value and tolerance of C
Type of dielectric material
While the vast majority of capacitors in practical use have fixed values of capacitances, there are some
cases where a changing value of capacitance is required. In these cases, a variable capacitor is used.
Typically these have an array of air-spaced plates, which can be rotated relative to each other to vary
the effective plate area A by turning a knob. Variable capacitors can also be made by having two plates
whose separation can be adjusted by turning a screw; these are typically referred to as trimmer or
padder capacitors. While the total capacitance is typically only in the pF range, these are crucial in
tuning circuits.
Another consideration regards the presence of high voltage between closely-separated plates. If the
voltage across an insulator (such as the dielectric between the capacitor’s plates) becomes high enough,
breakdown occurs. This means that whatever charges have accumulated on the plates will short across
the insulating space between them: the device ceases to act as a capacitor, and in fact may be damaged!
For this reason, all capacitors have a maximum voltage rating, just as resistors have a maximum power
rating. Some dielectric materials have extremely high breakdown voltages, even for small thicknesses,
so these types of capacitors are usable at high applied voltages.
Most typical capacitors in use have thin layers of materials such as mica or various ceramics as the
dielectric, with the plate area increased either by rolling large plates into a cylindrical form, or by
stacking layers of plates. Ceramic and mica capacitors can be made with capacitance values from a few
pF up to roughly 0.05 µF, but have the advantage of rather high breakdown voltages (typically at the
very least several hundred, and usually several thousand, volts).
Some very old electronic devices use capacitors with paper as the dielectric material. Very frequently
such paper capacitors were hand-rolled, and used due to ease of manufacture and low cost, although
nowadays ceramic and mica capacitors are much more common. Since paper capacitors can be made
very cheaply, very large (in size) units can be made, and with rather small plate separations d (due only
to the thickness of the paper), potentially may have capacitances up to about 1 µF, though typically they
are much smaller than that.
Capacitors can be made with thin films such as mylar or polystyrene as the dielectric material. The
advantage here is that d can be extremely small, so film capacitors can have C up to several µF. They
have an advantage in that their capacitance is usually less sensitive to temperature changes than other
types, but a disadvantage in that their breakdown voltages are somewhat smaller than for mica and
ceramic capacitors (typically 50-1000 V).
One special type of capacitor is made by repeated deposition of alternating layers of metal film and
insulating materials. These are called electrolytic capacitors, and because they can have very small
effective plate separations d, and large effective plate areas A, can have large capacitances, on the order
of tens to hundreds (and sometimes thousands) of µF. However, this comes at a tradeoff. Firstly, since
the dielectric layers are so thin, breakdown voltages are often relatively small compared to those in
more “standard” capacitors (typically on the order of tens of volts, though some are on the order of
hundreds, and others are < 10 V). Secondly, because of the way the layers are applied, electrolytic
capacitors can generally only handle applied voltage of one polarity, and are clearly marked with “+”
and/or “-“ terminals to indicate how they should be wired into a circuit. If they are wired with reverse
polarity, they can be damaged! Electrolytic capacitors also are prone to drying out as they age, so over
time they are one of the most common components that needs to be replaced.
A special type of electrolytic capacitor known as the tantalum capacitor is also widely used. These can
also be manufactured with very tight tolerances on capacitance values and are less susceptible to drying
out. For these reasons and others, they are widely used in computers and related electronics. Tantalum
capacitors typically have capacitance values on the order of 1-50 µF with breakdown voltages on the
order of tens of volts.
With this deposition method taken to extremely thin layers, so-called supercapacitors can be made.
These typically have extremely small limiting voltages (typically a few volts), but can have capacitances
in the range of F in a relatively small package!
Finally, being that I am so enamored of quaint old electronics terms, you’ll sometimes hear capacitors
referred to as condensers.
C.
Ideal and real inductors
Earlier, we modeled inductors as resistanceless coils of wire obeying Faraday’s Law. In reality, these
coils have a non-zero resistance, which is effectively in series with the inductor. There is also a small
amount of “parasitic” capacitance between neighboring wires in the coil (since each turn is at a slightly
different voltage than its neighbors), which is effectively in parallel with the inductance. Typically, larger
inductors (which have larger coils with more windings) are more seriously affected by these stray
resistances and capacitances. And in AC circuits, inductors become increasingly badly-behaved as
frequency increases.
In addition, there can be significant energy losses due to resistive heating of the core material; the
induced electric fields not only generate EMFs in the inductor’s coil, but in the core as well. Since the
core material has nonzero resistance, these eddy currents can result in significant energy losses,
particularly in cases where the applied current changes rapidly (as in high-frequency applications). For
this reason, insulating materials, or thin laminated layers of conducting material, are often used as
cores, in order to reduce the losses due to eddy currents.
Another practical concern arises from the construction of inductors. While resistors and capacitors (as
well as diodes and transistors) can very easily be miniaturized, and fabricated using methods conducive
to construction on integrated circuit boards, inductors (like the vacuum tubes of old) actually do need to
be rather large in order to get a sizable inductance, and made in a particular physical way (due to the
need to generate a sizable magnetic field to induce back EMF) which does not lend itself readily to
integrated circuit construction. For this reason, inductors have become somewhat of a “fifth wheel” in
electronics. They’re inconvenient to deal with, so electronics engineers will typically look for ways to
avoid using inductors in circuits, usually using combinations of capacitors and active devices (usually
operational amplifiers, which are constructed from combinations of transistors). For this reason,
inductors are not used nearly as often as most of the other components we’ll study. That said, there are
some applications for which they are crucial, such as radio-frequency tuning applications, high-power
voltage transformers, and of course the Taser. So unlike the vacuum tube, whose applications were
nearly entirely supplanted by the vastly more desirable transistor, inductors have a long future of use in
their “killer apps,” although they are far less common than resistors, capacitors and the solid-state
devices we’ll study later in the semester.
Finally, we know that a changing current through an inductor induces a changing magnetic field in its
core. Some of that changing magnetic field escapes from the core, and can impinge on other inductors
that happen to be in the vicinity (even if they’re part of a separate circuit!), so that a changing current in
one inductor can actually induce a voltage in another! This is known as mutual inductance. In some
cases, mutual inductance is desired; the prototypical example of this is in a transformer. In other cases,
it is unwanted and unexpected, in which case it leads to the circuit behaving in undesirable ways. So
care must be taken to minimize these stray fields, and to ensure that the stray fields that do escape have
minimal effects on other inductors. And because these stray induced voltages are a function of
geometry and other aspects that are incidental to circuit design, this effect is generally impossible to
model from a schematic diagram. In other words, very bad.
D. Classification of inductors
Just as we did with resistors and capacitors, we can come up with some schemes to classify inductors.
However, inductors as a group tend to be a bit more homogeneous, in that they’re all more or less the
same: a coil of wire. But we can come up with a few classification schemes:
-
Fixed or variable?
Geometry and core material
Value of inductance
Current/voltage ratings
Just as with resistors and capacitors, most inductors you’ll run across have fixed values of inductance.
But on occasion a variable-value inductor is required (most often, in a tuning circuit). The typical way
this is accomplished is with an adjustable core. A ferromagnetic or ferromagnetic core material
substantially increases the magnetic field inside the coil, so an inductor with a core that can be partially
inserted or removed, so that some of the coils are wrapped around a magnetic core, and some enclose
only air, can have a widely-varying total effective inductance.
There are two principal geometries in use for inductors. There is the cylindrical solenoid shape, which
we considered in the above section on inductors. These are the simplest to construct, and are also the
most readily amenable to simple calculations in “textbook” problems. But toroidal, or donut-shaped,
inductors are common as well, in which a ring-shaped core is wrapped with wire. These fundamentally
work the same way as solenoid inductors, but with one significant advantage: since the core material is
in the shape of a ring, the “stray” fields escaping the core are much smaller, and the effects on other
inductors in the circuit should hopefully be smaller.
Like the Farad in capacitance, the Henry is a rather large unit. Most real inductors have inductance
values on the order of µH to mH. Typically, the inductance value, and tolerance, and any appropriate
limits on applied voltage or current, are stamped on the body of the inductor.
Finally, in continuing my tradition of mentioning old electronics terms, old-timers (who, sadly, are
becoming fewer in number as the years go by) will frequently refer to inductors as “coils” (should be
obvious why) or “chokes” (due to their effect of “choking” rapid changes of current).