NAMI @PPKEE,USM
EEE105: CI RCUI T THEORY
CHAPTER 5: CAPACITORS AND INDUCTORS 5.1 Introduction • Unlike resistors, which dissipate energy, capacitors and inductors store energy. • Thus, these passive elements are called storage elements.
5.2 Capacitors • Capacitor stores energy in its electric field. • A capacitor is typically constructed as shown in Figure 5.1.
Figure 5.1
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NAMI @PPKEE,USM
EEE105: CI RCUI T THEORY
A capacitor consists of two conducting plates separated by an insulator (or dielectric) • When a voltage v is applied, the source deposits a positive charge q on one plate and negative charge –q on the other.
Figure 5.2 • The charge stored is proportional to the applied voltage, v
q = Cv
(5.1)
where C is the constant of proportionality, which is known as the capacitance of the capacitor. • Unit for capacitance: farad (F).
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EEE105: CI RCUI T THEORY
• Definition of capacitance:
Capacitance is the ratio of the charge on one plate of a capacitor to the volatge difference between the two plates. • Capacitance is depends on the physical dimensions of the capacitor. • For parallel-plate capacitor, capacitance is given by
C=
∈A d
(5.2)
where A is the surface area of each plate d is the distance between the plates
∈ is the permittivity of the dielectric material between the plates • The symbol of capacitor:
Figure 5.3
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EEE105: CI RCUI T THEORY
• The current flows into the positive terminal when the capacitor is being charged. • The current flows out of the positive terminal when the capacitors is discharging. • Differentiating both sides of Equation 5.1,
dq dv =C dt dt Thus,
dv i =C dt
(5.3)
• Capacitors that satisfy Equation 5.3 are said to be linear. • The voltage-current relation:
1 t v = ∫−∞ i (t )dt C v=
1 t i (t )dt + v(t0 ) ∫ t0 C
(5.4)
where v (t0 ) = q (t0 ) C is the voltage across the capacitor at time to. • Thus, the capacitor voltage is depends on the past history of the capacitor current – has memory.
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NAMI @PPKEE,USM
EEE105: CI RCUI T THEORY
• The instantaneous power given by:
dq p = vi = Cv dt
(5.5)
• The energy stored given by: t
t
w = ∫−∞ pdt = C ∫−∞ v
t
dv 1 t dt = C ∫−∞ vdv = Cv 2 dt 2 t = −∞
Note that v ( −∞) = 0 because the capacitor was
uncharged at t = −∞ . Thus,
1 2 q2 w = Cv = 2 2C
(5.6)
• Four issues: (i) From Equation 5.3, when the voltage across a capacitor is not changing with time (i.e., dc voltage), the current through the capacitor is zero.
A capacitor is an open circuit to dc. (ii)
The voltage on the capacitor must be continuous. The capacitor resists an abruot change in the voltage across it. According to 106
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(iii) (iv)
EEE105: CI RCUI T THEORY
Equation 5.3, discontinuous change in voltage requires an infinite current, which is physically impossible. The ideal capacitor does not dissipate energy. A real, nonideal capacitor has a parallel-model linkage resistance.
Figure 5.4 • Example 1: The voltage across a 5µF capacitor is
v(t ) = 10 cos 6000t V Calculate the current through it.
dv −6 d i (t ) = C = 5 × 10 (10 cos 6000t ) dt dt i (t ) = −5 ×10 −6 × 6000 ×10 sin 6000t = −0.3 sin 6000t 107
NAMI @PPKEE,USM
EEE105: CI RCUI T THEORY
• Example 2: An initially charged 1-mF capacitor has the current as shown in Figure 5.5. Calculate the voltage across it at t = 2 ms and t = 5 ms.
Figure 5.5 The current waveform can mathematically as:
50t mV i (t ) = 100 mV
be
described
0
