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Of course, there are other interesting properties of mechanical systems, such
as the mass (inertia), and it turns out that there is an electrical analog to inertia
also. It is possible to make something called an znmductor, having a property
called zmductance, such that a current, once started through the inductance, đoes
not tuant to stop. Tt requires a voltage in order to change the currentl TẾ the
curren£ is constant, there is no voltage across an inductance. DƠC circuits do not
know anything about inductance; it is only when we changøe the current that the
efects of inductance show up. The equation is
V = L(d1/dt) = L(d°q/dt), (25.12)
and the unit of inductance, called the henr, 1s such that one volt applied to
an inductance of one henry produces a change of one ampere per second in the
current. Equation (25.12) is the analog of Ñewton”s law for electricity, if we wish:
V corresponds to #', Ù corresponds to mm, and Ï corresponds to velocityl All of the
consequent equations for the two kinds of systems will have the same derivations
because, in all the equations, we can change any letter to its corresponding
analog letter and we get the same equation; everything we deduce will have a
correspondence in the two systems.
Now what electrical thing corresponds to the mechanical spring, in which
there was a force proportional to the stretch? If we start with # = kz and replace
†'— V and z — q, we get V = ag. lt turns out that there 7s such a thing, in fact
1t is the only one of the three circuit elements we can really understand, because
we did study a pair of parallel plates, and we found that If there were a charge of
certain equal, opposite amounts on each plate, the electric fñeld between them
would be proportional to the size of the charge. 5o the work done in moving a
unit charge across the gap from one plate to the other is precisely proportional to
the charge. This work is the definiiion of the voltage difference, and it is the line
integral of the electric field from one plate to another. It turns out, for historical
reasons, that the constant oŸ proportionality is not called Œ, but 1/Œ. It could
have been called Œ, but it was not. So we have
V =q/C. (25.13)
--- Trang 449 ---
'The unit of capacitance, Œ, is the farad; a charge of one coulomb on each plate
of a one-farad capacitor yields a voltage diference of one volt.
There are our analogies, and the equation corresponding to the oscillating
circuit becomes the following, by direct substitution of Ù for rn, q for ø, etc:
m(d®+/dt?) + ym(da/dE) + ka = F, (25.14)
L(d°q/dt2) + R(dq/dt) + q/C = V. (25.15)
Now everything we learned about (25.14) can be transformed to apply to (25.15).
lvery conseqguence is the same; so mụuch the same that there is a brilliant thing
we can do.
Suppose we have a mechanical system which is quite complicated, not Just
one mass on a spring, but several masses on several springs, all hooked together.
What do we do? Solve it? Perhaps; but look, we can make an clecfr?cal circuit
which will have the same equations as the thing we are trying to analyzel For
instance, iŸ we wanted to analyze a mass on a spring, why can we not build
an electrical circuit in which we use an inductance proportional to the mass, a
resistance proportional to the corresponding +, 1/C proportional to k, allin
the same ratio? 'Phen, of course, this electrical circuit will be the exact analog
of our mechanical one, in the sense that whatever g does, in response to V
(V also is made to correspond to the forces that are acting), so the # would
do in response to the forcel So if we have a complicated thing with a whole
lot of interconnecting elements, we can interconnect a whole lot of resistances,
inductaneces, and capacitances, to #mn‡øte the mechanically complicated system.
What is the advantage to that? One problem is jus6 as hard (or as easy) as
the other, because they are exactly equivalent. The advantage is not that it is
any easier to solve the rmmathematical equations after we discover that we have an
electrical circuit (although that ¿s the method used by electrical engineersl), but
instead, the real reason for looking at the analog is that it is easier to make the
electrical circuit, and to chønge something in the system.
Suppose we have desipgned an automobile, and want to know how much it
1s going to shake when iÈ goes over a certain kind of bumpy road. We build an
electrical eireuit with inductances to represent the inertia of the wheels, spring
constants as capacitances to represent the springs of the wheels, and resistors to
represent the shock absorbers, and so on for the other parts of the automobile.
Then we need a bumpy road. All right, we apply a 0ol#age from a generator,
which represents such and such a kind of bump, and then look at how the left
--- Trang 450 ---
wheel jiggles by measuring the charge on some capacitor. Having measured it
(it is easy to do), we fñnd that it is bumping too much. Do we need more shock
absorber, or less shock absorber? With a complicated thíng like an automobile,
do we actually change the shock absorber, and solve it all over again? Nol, we
simply turn a dial; dial number ten is shock absorber number three, so we put ín
more shock absorber. 'Phe bumps are worse—all right, we try less. The bumps
are sbill worse; we change the stifness of the spring (dial 17), and we adjust all
these things eleciricallu, with merely the turn of a knob.
This is called an ønalog compu£er. Tt is a device which imitates the problem
that we want to solve by making another problem, which has the same equation,
but in another circumstance of nature, and which is easier to build, to measure,
to adjust, and to destroyl
25-5 Series and parallel impedances
Finally, there is an important item which is not quite in the nature of review.
'This has to do with an electrical cireuit in which there is more than one circuit
element. Eor example, when we have an inductor, a resistor, and a capacitor
connected as in Eig. 24-2, we note that all the charge went through every one
of the three, so that the current in such a singly connected thing is the same at
all points along the wire. Since the current is the same in each one, the voltage
across Ÿ‡ is I, the voltage across Ù is E(đdTI/đf), and so on. So, the total voltage
drop is the sum of these, and this leads to Eq. (25.15). Using complex numbers,
we found that we could solve the equation for the steady-state motion in response
to a sinusoidal force. We thus found that Ÿ = ÊÂ. Now Z is called the ?mpcdance
of this particular circuit. It tells us that if we apply a sinusoidal voltage, £, Ww©
get a current Ỉ.
Now suppose we have a more complicated circuit which has two pieces,
which by themselves have certain impedances, ÊWŸ¡ and 22 and we put them in
1 [2] [Z:] 2 1 2
(a) Series (b) Parallel
Fig. 25-6. Two impedances, connected in series and ¡n parallel.
--- Trang 451 ---
series (Eig. 25-6a) and apply a voltage. What happens? It is now a little more
complicated, but if Ÿ is the current through VN the voltage diference across ôi,