Page 260 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c05
246 MECHATRONICS Printer: Yet to Come October 28, 2014 11:15 254mm×178mm
R C L
1 i 2 1 i 2 1 i 2
=
V = Ri V = 1 idt V L di
12 12 12 dt
C
(a)

(b)
FIGURE 5.1: Basic components of electrical circuits: (a) symbols of resistance (R), capacitance
(C), inductance (L). (b) Pictures of resistors, capacitors, and inductors.

except that they are opposite signs, in units of Coulomb,
+
−
|e | = |p | = 1.60219 × 10 −19 C (5.2)
The passive components define relationships between current and voltage potential
difference between their two nodes. An ideal resistor has a potential difference between
its two ends proportional to the current passing through it. The proportionality constant is
called the resistance, R

V (t) = R ⋅ i(t) (5.3)
12
The resistance of a material is a function of the material property, that is the resistivity ( )
and its geometry. As the cross-section of the conductor increases, resistance to the flow of
charges reduces, as they have more room to move through. As the length of the conductor
increases, resistance increases. Hence, the resistance of a conductor is
l
R = (5.4)
A
where is the resistivity of the material ( = 1.7 × 10 −8 Ω ⋅ m for copper, = 2.82 ×
10 −8 Ω ⋅ m for aluminum, = 0.46 Ω ⋅ m for germanium, = 640 Ω ⋅ m for silicon, =
14
13
10 Ω ⋅ m for hard rubber, = 10 10 to 10 Ω ⋅ m for glass), l is the length, A is the
cross-sectional area of the conductor. It is also important to point out that the resistivity of
materials, and hence the resistance, is a function of temperature. The variation of resistance
as a function of temperature varies from material to material. In general, many materials,
but not all, have the following resistance and temperature relationship,
R(T) = R [1 + (T − T )] (5.5)
0 0
where R(T) is the resistance at temperature T, R is the resistance at temperature T , and
0 0
is the rate of change in resistance as a function of temperature. Superconductors are
materials that have almost zero resistance below a certain critical temperature, T . For each
c
superconductor, there is a critical temperature T below which the above relationship does
c
not hold. The resistance suddenly drops to almost zero at T ≤ T . This critical temperature
c
◦
for mercury is about T = 4.2 K.
c
There are two major types of resistors: wire wound and carbon types. Wire wound
resistors are used for high current applications such as regenerative power dumping as
heat in motor control. Carbon type resistors are used in low current signal processing