capacitors inductors
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Circuit Theory
Capacitors and Inductors
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Capacitors and Inductors
1. Capacitors
2. Series and Parallel Capacitors
3. Inductors
4. Series and Parallel Inductors
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1 Capacitors (1)
• A capacitor is a passive element designed
to store energy in its electric field.
• A capacitor consists of two conducting platesseparated by an insulator (or dielectric).
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1 Capacitors (2)
• Capacitance C is the ratio of the charge q on oneplate of a capacitor to the voltage difference v between the two plates, measured in farads (F).
• Where is the permittivity of the dielectric materialbetween the plates, A is the surface area of eachplate, d is the distance between the plates.
• Unit: F, pF (10–12), nF (10–9), and
vC q d
AC
and
F (10–6)
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Capacitors (4)
• The energy, w , stored inthe capacitor is
• A capacitor is
– an open circuit to dc (dv/dt = 0).
– its voltage cannot change abruptly.
2
21 vC w
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Example
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Example
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Example
The current through a 100-F capacitor is
i(t) = 50 sin(120 t) mA.
Calculate the voltage across it at t =1 ms and
t = 5 ms.
Take v (0) =0.
Answer:
v(1ms) = 93.14mV
v(5ms) = 1.7361V
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Example
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Example
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Example
An initially uncharged 1-mF capacitor has thecurrent shown below across it.
Calculate the voltage across it at t = 2 ms and
t = 5 ms.
Answer:
v(2ms) = 100 mV
v(5ms) = 500 mV
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Example – DC Conditions
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Example
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Capacitors (2)
• The equivalent capacitance of N series-connected capacitors is the reciprocal of the sum of thereciprocals of the individual capacitances.
N eq C C C C
1...
111
21
svC C
C v
21
2
1 sv
C C
C v
21
1
2
• Voltage Divider Rule
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Inductors (2)
• Inductance is the property whereby an inductorexhibits opposition to the change of currentflowing through it, measured in henrys (H).
• is the inductor core permeability• N: Number of turns
• L: Coil length
• A: Cross sectional area
• Unit: Henry (H).
t d
id Lv
l
A N L
2
and
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Example
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Inductors (4)
ExampleThe terminal voltage of a 2-Hinductor is
v = 10(1-t) V
Find the current flowing through it att = 4 s and the energy stored in itwithin 0 < t < 4 s.
Assume i(0) = 2 A. Answer:
i(4s) = -18V
w(4s) = 320J
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Series and Parallel Inductors (1)
• The equivalent inductance of series-connected inductors is the sum of the individualinductances.
N eq L L L L ...21
sv L L
Lv
21
1
1 sv
L L
Lv
21
2
2
• Voltage Divider Rule
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Series and Parallel Inductors (2)
• The equivalent capacitance of parallel inductorsis the reciprocal of the sum of the reciprocals of the individual inductances.
N eq L L L L
1...
111
21
si L L
Li
21
2
1 si
L L
Li
21
1
2
•Current Divider Rule
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Series and Parallel-Inductor (3)
Calculate the equivalent inductance for theinductive ladder network in the circuitshown below:
Answer:
Leq = 25mH
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