mems 0031 electric circuits - university of pittsburghpitt.edu/~qiw4/academic/mems0031/chapter1...

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

Department of Mechanical Engineering

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

Department of Mechanical Engineering

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−

=

Department of Mechanical Engineering

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

Department of Mechanical Engineering

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

Department of Mechanical Engineering

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

−=

−= +

Department of Mechanical Engineering

Department of Mechanical Engineering

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

=

= = =

Department of Mechanical Engineering

Example: Charge is given, q(t), find current, i(t)

Department of Mechanical Engineering

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

== + = = = −

−∞ < ≤ = =< ≤ = + =< ≤ = − + = −< ≤ = − =< ≤ = − =≤ = =

Department of Mechanical Engineering

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

−∝= = +

= − + = − + +

= − + +

∫ ∫

Department of Mechanical Engineering

Example : Find the charge and sketch its waveform for the given current entering a terminal of element as shown in P1.2-7

Department of Mechanical Engineering

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= −

Department of Mechanical Engineering

Power Absorbed by element

Power Supplied by the element

Circuit variables

Department of Mechanical Engineering

Circuit variables

Department of Mechanical Engineering

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

= =

= =

Department of Mechanical Engineering

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

Department of Mechanical Engineering

+ -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

Department of Mechanical Engineering

Passive Convention

Department of Mechanical Engineering

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

Department of Mechanical Engineering

Example :

Department of Mechanical Engineering

Example :

Department of Mechanical Engineering

Example :

Department of Mechanical Engineering