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make electric circuits. These 0dss?ue circu#t clements, as they are often called,
are of three main types, although each one has a little bit of the other wo mixed
in. Before describing them in greater detail, let us note that the whole idea of our
mnechanical oscillator beïng a mass on the end oŸa spring is only an approximation.
All the mass is not actually at the “mass”; some of the mass is in the inertia of
the spring. Similarly, all of the spring is not at the “spring”; the mass itself has a
little elasticity, and although it may appear so, it is not øbsolu‡elu rigid, and as it
goes up and down, i% fexes ever so slightly under the action of the spring pulling
it. The same thing is true in electricity. Phere is an approximation in which we
can lump things into “circuit elements” which are assumed to have pure, ideal
characteristics. It is not the proper time to discuss that approximation here, we
shall simply assume that it is true in the cireumstances.
The three main kinds of cireuit elements are the following. “The first is called a
capacitor (EFig. 23-4); an example is 0wo plane metallic plates spaced a very small
distance apart by an insulating material. When the plates are charged there is
a certain voltage diference, that is, a certain diference in potential, between
--- Trang 414 ---
ẠA C E
B D F
CAPACITOR RESISTOR _ INDUCTOR
Fig. 23-4. The three passive circuit elements.
them. "The same diference of potential appears bebween the terminals A4 and Ö,
because if there were any diference along the connecting wire, electricity would
fow right away. So there is a certain voltage diference V between the plates If
there is a certain electric charge +g and —q on them, respectively. Between the
plates there will be a certain electric field; we have even found a formula for 1%
(Chapters 13 and 14):
V = ơd/sạ = qd/eoA, (23.14)
where đ is the spacing and A is the area of the plates. Note that the potential
diference is a linear function of the charge. If we do not have parallel plates,
but insulated electrodes which are of any other shape, the diference in potential
1s still precisely proportional to the charge, but the constant of proportionality
may not be so easy to compute. However, all we need to know is that the
potential difference across a capacitor 2s proportional to the charge: V = q/C:
the proportionality constant is 1/Œ, where Œ is the capacitance oŸ the object.
'The second kind of circuit element is called a reszstor; 1t offers resistance to
the Ñow of electrical current. It turns out that metallic wires and many other
substances resist the fÑow of electricity in this manner: if there is a voltage
diference across a piece of some substance, there exists an electric current Ï =
dq/đt that is proportional to the electric voltage difference:
V = RÏI = hdaq/dt (23.15)
'The proportionality coefficient is called the resis‡tønece Rì. Thĩs relationship may
already be familiar to you; i% is Ohm”s law.
Tf we think of the charge g on a capacitor as being analogous to the displace-
ment # of a mechanical system, we see that the current, Ï = dg/dt, is analogous
to velocity, L/Πis analogous to a spring constant k, and ?# is analogous to the
resistive coefficlent e = zm+y in Eq. (23.6). Now it is very interesting that there
--- Trang 415 ---
exists another circuit element which is the analog of massl “This is a coil which
builds up a magnetic feld within itself when there is a current in it. A changing
magnetic feld develops in the coil a voltage that is proportional to đĨ/đf (this
is how a transformer works, in fact). The magnetic feld is proportional to a
current, and the induced voltage (so-called) in such a coil is proportional to the
rate of change of the current:
V = LdI/dt = Ld°q/dtẺ. (23.16)
The coeficlent Ù is the sejf-?nductfance, and is analogous to the mass in a
mmechanical oscillating circuit.
Fig. 23-5. An oscillatory electrical circuit with resistance, Iinductance,
and capacitance.
Suppose we make a circuit in which we have connected the three circuit
elements in series (Eig. 23-5); then the voltage across the whole thing from 1
to 2 is the work done in carrying a charge through, and it consists of the sum
of several pieces: across the induetor, Vy„ = Ld2q/di2; across the resistance,
Vn = Tìdq/dt; across the capacitor, Vơ = g/C. The sum of these is equal to the
applied voltage, V:
Ld?q/di° + Rdq/dt + q/C = VỆ). (23.17)
Now we see that this equation is exactly the same as the mechanical equa-
tion (23.6), and oŸ course it can be solved in exactly the same manner. WWe
suppose that V(£) is oscillatory: we are driving the circuit with a generator with
a pure sine wave oscillation. Then we can write our V(£) as a complex V with
the understanding that it must be ultimately multiplied by e”“, and the real part
taken in order to fnd the true W. Likewise, the charge g can thus be analyzed,
and then in exactly the same manner as in Eq. (23.8) we write the corresponding
equation: the second derivative of ậ is (2)2â; the fñrst derivative is (2)ệ. Thus
--- Trang 416 ---
Eq. (23.17) translates to
9 . 1Ì, _«
L()“ + T() + cl?Z V
L1(0u)2 + R() + —
which we can write in the form
ậ= Ÿ/L(uỆ — w2 + i2), (23.18)
where ϊ = 1/EΠand + = R/L. It is exactly the same denominator as we had
in the mechanical case, with exactly the same resonance propertiesl The corre-
spondence between the electrical and mechanical cases is outflined in Table 23-1.
Table 23-1
General Mechanical Eilectrical
characteristic property property
indep. variable time (#) time (?)
dep. variable position (z) charge (g)
inertia mass (mm) inductance (L)
resistance drag coef. (c = +m) resistance (lề = +yL)
stifness stifness (k) (capacitance)” (1/Œ)
resonant frequency u = k/m uạ = 1/LƠ
period to = 2mav/Èm/È to = 2xV ÙŒ
fgure of merit Q=,u0/^2 Q=uoL/R
We must mention a small technical point. In the electrical literature, a
diferent notation is used. (From one field to another, the subject is not really
any diferent, but the way of writing the notations is often different.) Eirst, 7
1s commonly used instead of ¿ in electrical engineering, to denote —1. (After
all, ¿ must be the currentl) Also, the engineers would rather have a relationship