MEMS 0031 Electric Circuits
Chapter 1 Circuit variables
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Chapter/Lecture Learning
Objectives
At the end of this lecture and chapter, you should able to:
 Represent the current and voltage of an electric circuit element,
paying particular attention to the reference direction of the current
and to the reference direction or polarity of the voltage
 Calculate the power and energy supplied or received by a circuit
element
 Use the passive convention to determine whether the product of
the current and voltage of a circuit element is the power supplied
by that element or the power received by the element
 Use scientific notation to represent electrical quantities with a
wide range of magnitudes
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Circuit variables
 Circuit
element
A general two-terminal electric
circuit element
 Circuit
A simple circuit
An electric circuit or electric network
is an interconnection of electrical
elements linked together in a closed
path so that an electric current may
flow continuously






Resistor
Switch
capacitor
Inductor
Sources
o Voltage source
o Current sources
Transducer
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Circuit variables
 Charge:
the quantity of electrically responsible for
electric phenomena
 Current: time rate of flow of the electric charge
past a given point
dQ
i=
dt
1C = 6.24 x10 electron charge
18
−19
1 e =- 1.602x10 C
i1 = −i2
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Current: Net Positive Charge Flow
QL +
QR +
QL −
QR −
∆Q
=
(QR + − QR − ) − (QL + − QL − )
=
N
et +Q charge; iavg
∆Q
∆t
Instantaneous current:
dq
i (t ) =
dt
Unit of Current : Ampere (A);
Unit of Charge: Coulomb (C);
1 A = 1 Coulomb per second
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Circuit variables
A
direct current (DC) of magnitude of I
 Time-varying
current i
A Sinusoid, ω = angular frequency
A Ramp, slope=M
An exponential,
constantEngineering
Department I,of b=
Mechanical
Circuit variables
 If
the charge is known, the current can be find from
dq
i=
dt
 If
the current is known, the
charge can be find by
q = ∫ idτ = ∫ idτ + q(0 )
t
t
−∝
0
q(0 ) is the charge at t=0
i2 − i1
slope, M =
t2 − t1
i (=
t ) Mt + b
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Circuit variables
i (=
t ) 8t2 − 4t A
8 3
8 3 2
2t
q(t ) = ∫ 0 i dτ + q(0)= ∫ 0 (8τ − 4τ ) dτ + 0= τ −2τ = t − 2 t C
0
3
3
t
t
2
q ( t ) = 4 sin 3t C
dq d
i (=
t)
4 sin =
3t 12 cos 3t
=
dt
dt
A
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Example: Charge is given, q(t),
find current, i(t)
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dq
i=
dt
Divide q(t) into sections to get q(t)
expressions at each section
 m = slope
) mt + b 
q (t=
= q (=
t 0)
= q (t ) − mt
b
(1) − ∞ < t ≤ 1,=
q (t ) 0,=
i (t ) 0
(2 1) s < t ≤ 3s, q (t ) = t + 1, i (t ) = 1
(3)3s < t ≤ 5s, q (t ) =−1.5t + 6.5, i (t ) =−1.5
(4 5
) s < t ≤ 6 s, q (t ) =−1, i (t ) =0
(5)6 s < t ≤ 7 s, q (t ) = t − 7 ,(
i t) =1
(6)7 s ≤ t , q (t ) = 0, i (t ) = 0
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Example : From known current, find charge q(t)
The current in a circuit element is i(t) = 3 sin (5t + 30)
when t ≥ 0 and i(t) = 0 when t < 0. Determine the total
charge that entered a circuit element for t ≥ 0
dq
i=
dt
=
q (t )
t
idτ
∫=
∫
−∝
t
0
q = ∫ idτ = ∫ idτ + q(0 )
t
t
−∝
0
3 sin (5τ + 30)dτ + q ( 0 )
t
3
3
3
3
=
− cos(5τ + 30) =
− cos(5t + 30) + .
5
5
5 2
0
3
3
=
− cos(5t + 30) +
5
10
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Example : Find the charge and sketch its waveform for
the given current entering a terminal of element as
shown in P1.2-7
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Circuit variables
Voltage is the basic circuit variable
describing energy (w) required to cause
charge (q) to flow.
Energy is capacity to perform work;
Voltage across an element Unit of voltage is the volt (V). 1V =
1J/C
is the work (energy)
required to move a unit
vab = −vba
positive charge q from –
voltage at b wrt a
to + terminal of the
element
Voltage directions
dw
v=
dq
= terminal polarities (-,+)
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Circuit variables
Power Absorbed by element
Power Supplied by the element
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Circuit variables
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Power and Energy
 Power
is time rate of expending or absorbing
energy (w)
dw
p=

dt
 For
an electric circuit element, power
absorbed or supplied by the element is:
dw dq
=
p =
.
v.i
dq dt
=
p v=
.i instantaneous power
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Power and Energy
 The
energy absorbed by an element for a given
dw = pdt
power:
t
w=
∫
pdt
−∞
the element only receives power for t ≥ to,
and Let to =0
 If
t
w=
∫
pdt
t0 = 0
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Passive Convention
→ i(t )
Power Absorbed by element
+ vab
-
- vba
+
Power Supplied by the element → i(t )
Passive Convention (Passive sign Convention):
Current enters terminal of higher
voltage, element absorbed power
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Passive Convention
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Lumped –Circuit elements
 Consider
current i(t) and v(t) of a circuit
element as shown:
– Passive Convention
– Current enters terminal of higher voltage
– Element absorbed power
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Example :
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Example :
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Example :
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