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Department of Mechanical Engineering
MEMS 0031 Electric Circuits
Chapter 1 Circuit variables
Department of Mechanical Engineering
Chapter/Lecture Learning ObjectivesAt 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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A general two-terminal electric circuit element Circuit element
Circuit
Circuit variables
Resistor Switch capacitor Inductor Sources
o Voltage sourceo Current sources
Transducer
A simple circuitAn 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
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Circuit variables
dtdQi =
21 ii −=
Charge: the quantity of electrically responsible for electric phenomena
Current: time rate of flow of the electric charge past a given point
18
19
1 6.24 10 electron charge
1 e =- 1.602 10
C x
x C−
=
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Current: Net Positive Charge Flow
( ) ( )
et + charge;
Instantaneous current:
( )
R R L L
avg
Q Q Q Q QQN Q it
dqi tdt
+ − + −∆ = − − −∆
= =∆
=
LQ −RQ −
RQ +
LQ +
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 Ramp, slope=M
A Sinusoid, ω = angular frequency
An exponential, I, b= constant
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dtdqi =
( )∫∫ +==∝−
ttqiddiq
00ττ
Circuit variables
If the charge is known, the current can be find from
If the current is known, the charge can be find by
( )0q is the charge at t=0
2 1
2 1
,
( )
i islope Mt t
i t Mt b
−=
−= +
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Circuit variables
( ) 2
2 3 2 3 20 0 0
8 4 A8 8( ) (0) (8 4 ) 0 2 2 C3 3
tt t
i t t t
q t i d q d t tτ τ τ τ τ τ
= −
= + = − + = − = −∫ ∫
( )
( )
4sin 3 C
4sin 3 12cos3 A
q t tdq di t t tdt dt
=
= = =
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Example: Charge is given, q(t), find current, i(t)
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dtdqi = Divide q(t) into sections to get q(t)
expressions at each section
( )( 0) ( )
(1) 1, ( ) 0, ( ) 0(2 )1 3 , ( ) 1, ( ) 1(3)3 5 , ( ) 1.5 6.5, ( ) 1.5(4 )5 6 , ( ) 1, ( ) 0(5)6 7 , ( ) 7 ,( ) 1(6)7 , ( ) 0, ( ) 0
m slopeq t mt b
b q t q t mtt q t i t
s t s q t t i ts t s q t t i ts t s q t i ts t s q t t i ts t q t i t
== + = = = −
−∞ < ≤ = =< ≤ = + =< ≤ = − + = −< ≤ = − =< ≤ = − =≤ = =
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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
dtdqi = ( )∫∫ +==
∝−
ttqiddiq
00ττ
( )0
0
( ) 3 sin (5 + 30) 0
3 3 3 3cos(5 30) cos(5 30) .5 5 5 2
3 3cos(5 30)5 10
t t
t
q t id d q
t
t
τ τ τ
τ
−∝= = +
= − + = − + +
= − + +
∫ ∫
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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 variablesVoltage is the basic circuit variable describing energy (w) required to cause charge (q) to flow.
Voltage across an element is the work (energy) required to move a unit positive charge q from –to + terminal of the element dwv
dq=
Energy is capacity to perform work; Unit of voltage is the volt (V). 1V = 1J/C
voltage at b wrt aVoltage directions= terminal polarities (-,+)
ab bav v= −
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Power Absorbed by element
Power Supplied by the element
Circuit variables
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Circuit variables
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Power and EnergyPower is time rate of expending or absorbing
energy (w)
For an electric circuit element, power absorbed or supplied by the element is:
dwpdt
=
. .
. instantaneous power
dw dqp v idq dt
p v i
= =
= =
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The energy absorbed by an element for a given power:
If the element only receives power for t ≥ to, and Let to =0
t
dw pdt
w pdt−∞
=
= ∫
0 0
t
t
w pdt=
= ∫
Power and Energy
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+ -abv
( )i t→
- +bav
( )i t→
Power Absorbed by element
Power Supplied by the element
Passive Convention (Passive sign Convention):
Current enters terminal of higher voltage, element absorbed power
Passive Convention
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Passive Convention
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Lumped –Circuit elementsConsider 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 :
Department of Mechanical Engineering