Chapter 2
Electric Circuit Elements
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2.1 Circuit Elements (1)
Active Elements Passive Elements
• A dependent source is an active
element in which the source quantity
is controlled by another voltage or
current.
• They have four different types: VCVS,
CCVS, VCCS, CCCS. Keep in minds the
Independent Dependant signs of dependent sources.
sources sources 2
2.1 Circuit Elements (2)
Example 1
Obtain the voltage v in the branch shown in Figure 2.1 for i2 = 1A.
Figure 2.1
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2.1 Circuit Elements (3)
Solution
Voltage v is the sum of the current-independent
10V source and the current-dependent voltage
source vx.
Note that the factor 15 multiplying the control
current carries the units Ω.
Therefore, v = 10 + vx = 10 + 15(1) = 25 V
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2.2 Capacitors and Inductors
2.2.1 Capacitors
2.2.2 Series and Parallel Capacitors
2.2.3 Inductors
2.2.4 Series and Parallel Inductors
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2.2.1 Capacitors (1)
• A capacitor is a passive element designed
to store energy in its electric field.
• A capacitor consists of two conducting plates
separated by an insulator (or dielectric).
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2.2.1 Capacitors (2)
• Capacitance C is the ratio of the charge q on one
plate of a capacitor to the voltage difference v
between the two plates, measured in farads (F).
q=C v A
and C=
d
• Where is the permittivity of the dielectric material
between the plates, A is the surface area of each
plate, d is the distance between the plates.
• Unit: F, pF (10–12), nF (10–9), and F (10–6)
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2.2.1 Capacitors (3)
• If i is flowing into the +ve
terminal of C
– Charging => i is +ve
– Discharging => i is –ve
• The current-voltage relationship of capacitor
according to above convention is
dv 1 t
i =C
dt
and v=
C
t0
i d t + v(t0 )
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2.2.1 Capacitors (4)
• The energy, w, stored in
the capacitor is
1
w= Cv 2
• A capacitor is
– an open circuit to dc (dv/dt = 0).
– its voltage cannot change abruptly.
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2.2.1 Capacitors (5)
Example 2
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
10
v(5ms) = 1.7361V
2.2.2 Series and Parallel
Capacitors (1)
• The equivalent capacitance of N parallel-
connected capacitors is the sum of the individual
capacitances.
Ceq = C1 + C2 + ... + CN
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2.2.2 Series and Parallel
Capacitors (2)
• The equivalent capacitance of N series-connected
capacitors is the reciprocal of the sum of the
reciprocals of the individual capacitances.
1 1 1 1
= + + ... +
C eq C1 C 2 CN
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2.2.2 Series and Parallel
Capacitors (3)
Example 3
Find the equivalent capacitance seen at the
terminals of the circuit in the circuit shown below:
Answer:
Ceq = 40F
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2.2.2 Series and Parallel
Capacitors (4)
Example 4
Find the voltage across each of the capacitors in
the circuit shown below:
Answer:
v1 = 30V
v2 = 30V
v3 = 10V
v4 = 20V
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2.2.3 Inductors (1)
• An inductor is a passive element designed
to store energy in its magnetic field.
• An inductor consists of a coil of conducting wire.
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2.2.3 Inductors (2)
• Inductance is the property whereby an inductor
exhibits opposition to the change of current
flowing through it, measured in henrys (H).
di N2 A
v=L and L=
dt l
• The unit of inductors is Henry (H), mH (10–3)
and H (10–6).
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2.2.3 Inductors (3)
• The current-voltage relationship of an inductor:
1 t
i=
L t0
v (t ) d t + i (t 0 )
• The power stored by an inductor:
1
w = L i2
2
• An inductor acts like a short circuit to dc (di/dt = 0)
and its current cannot change abruptly. 17
2.2.3 Inductors (4)
Example 5
The terminal voltage of a 2-H
inductor is
v = 10(1-t) V
Find the current flowing through it at
t = 4 s and the energy stored in it
within 0 < t < 4 s.
Answer:
Assume i(0) = 2 A. i(4s) = -18A
w(4s) = 324J
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2.2.3 Inductors (5)
Example 6
Determine vc, iL, and the energy stored in the
capacitor and inductor in the circuit of circuit shown
below under dc conditions.
Answer:
iL = 3A
vC = 3V
wL = 1.125J
wC = 9J
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2.2.4 Series and Parallel
Inductors (1)
• The equivalent inductance of series-connected
inductors is the sum of the individual
inductances.
Leq = L1 + L2 + ... + LN
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2.2.4 Series and Parallel
Inductors (2)
• The equivalent capacitance of parallel inductors
is the reciprocal of the sum of the reciprocals of
the individual inductances.
1 1 1 1
= + + ... +
Leq L1 L2 LN
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• Current and voltage relationship for R, L, C
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