Electric Circuit Analysis

Circuit elements

Ahsan KhawajaAhsan_khawaja@comsats.edu.pkLecturerRoom 102Department of Electrical Engineering

Base Quantity

SI base units

Name Symbol

length meter m

mass kilogram kg

time second s

electric current ampere Atemperature kelvin K

amount of substance mole mol

luminous intensity candela cd

Prefixes•Used to emphasize the significantly large or small numerical figures.•Some common prefixes…..

10-24 yocto y

10-21 zepto z

10-18 atto a

10-15 femto f

10-12 pico p

10-9 nano n

10-6 micro m

10-3 milli m

10-2 centi c

10-1 deci d

10 Deka da

102 hecto h

103 kilo k

106 mega M

109 giga G

1012 tera T

1015 peta P

1018 exa E

1021 zetta Z

1024 yotta Y

Charge

• In SI system, charge is measured in coulombs (C).

• The charge on an electron is -1.602 x 10-19 C.• Hence 1 C of charge, there are

1/(1.602 x 10-19) = 6.24 x 1018 electrons.• Denoted by ‘ q ’.

Current

• ‘I’, is the rate at which charge is transferred or "flows" through a device.

• Mathematically,dq(t)

dti(t) =

where i = the current in amperes (A) q = the charge in coulombs (C)

the amount of charge passing through a device per unit of time.

1 ampere is 1 coulomb per second.

i

t

0 5 10 15 20-1

-0.5

0

0.5

1

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20-3

-2

-1

0

1

2

3

4

5

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Direct Current (dc)

Alternating/SinusoidalCurrent (ac)

Exponential Current

Damped Sinusoidal Current

Transient Current

Types of current

Current sign conventionProper definition of current vector requires:•direction of the arrow(polarity).•value (magnitude).

A negative current of -3 A flowing in one direction is the same as a current of +3 A flowing in the opposite direction.

Electronic VS conventional current

Electrons flow from negative terminal to positive terminal but conventional current flows in the opposite direction.

Example DC current.

Find the charge passing through a point in a conductor if the current passing through the conductor has the waveform indicated.

i

t

2 A

Solution:The current does not change with time. Hence, a charge of 2 C is passing through the point each second.

Example

For the given current, which is given in graphical form, find the charge passing through a given point in the time interval 0-4 s.

t (sec)

10 A

10 2 43

i(t)

t (sec)

10 A

10 2 43

i(t)

Solution:4

0

q(t) i(t)dt1 2

0 1

10tdt 10dt 3 4

2 3

( 10t 30)dt (10t 30)dt

Ans.= 5 + 10 -25 +35 C

= 25 C

Voltage

• Recognized by the 18th century Italian physicist Allessandro Antonio Volta.

• Voltage" or "potential difference” is defined as the energy required to move a unit charge through an element.

• Measured in volts (V).• Mathematically,

dwdq

V = w = the energy in joules (J)

1 volt is 1 joule per coulomb.

Voltage sign convention

DC and AC voltage

• Like electric current, a constant voltage is called a “dc voltage” and is represented by “V”, whereas a sinusoidally time-varying voltage is called an “ac voltage” and is represented by “v”.

• A dc voltage is usually produced by a battery and an ac voltage is produced by an electric generator.

Passive Sign Convention• The passive sign convention is satisfied when

the current enters through the positive(plus-marked) terminal of an element as shown.

circuitelementv

–

+i

i

. . .

The passive sign convention.

Example

Solution: p2 = (240)(2) = 480 W

p3 = (60)(1.5) = 90 Wp4 = (60)(0.5) = 30 W

Total power = 480 W + 90 W + 30 W= 600 W

+300 V –

+60 V –

+60 V –

+ 240 V –

2 A2 A

1.5 A

0.5 A

3 4

2

Find the power dissipated for the following circuit. Hint: P = V x I

What is the power generated by the source?

Voltage VS Current

The voltage across an element can be • linearly proportional to the current through it

(a resistor) • proportional to derivative (an inductor) of the

current with respect to time. • proportional to integral (a capacitor) of the

current with respect to time.

Voltage and Current Sources

Two types of basic electrical elements:

• passive elements.(dissipate power) • active elements. (generate power)

Active elements such as voltage and current sources can be classified as either independent or dependent sources.

Independent Voltage Sources

An ideal independent source is an active element that can supply a specified voltage or current that is completely independent of a current or voltage elsewhere in the circuit.

vsVs

(a) and (b) DC independent voltage source symbols. (c) AC independent voltage source symbol.

(a) (b) (c)

V

Independent Current Sources

(a) (b)

(a) A time-varying independent current source. (b) A constant (DC) independent current source.

i I

Dependent Sources• Dependent, or controlled, source is a source in which the

source voltage or current is determined by another voltage or current elsewhere in the circuit being analyzed.

• It is usually represented by diamond-shaped symbol.

(a) (b)

(a) A dependent voltage source. (b) A dependent current source.

v i

Dependent sources of four types

1. A voltage-controlled voltage source (VCVS)

2. A current-controlled voltage source (CCVS).

3. A voltage-controlled current source (VCCS).

4. A current-controlled current source (CCCS).

VCVS

i

v = bvx

v = bvx

vx is somewhere (not shown) and b is a constant.

i = whatever

CCVS

i

v = bix

v = bix

i = whateverix is somewhere (not shown) and b is a constant.

VCCS

i = bvx

i = bvx

vx is somewhere (not shown) and b is a constant.

v = whatever

+

_v

CCCS

i = bix

i = bix

v = whateverix is somewhere (not shown) and b is a constant.

+

_v

Resistance ‘R’

• Unit of resistance is ohms (Ω)1 Ω = 1 V / A

• Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor.

• Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents.

Ohm’s Law

• Ohm’s Law is an empirical relationship that is valid only for certain materials– Materials that obey Ohm’s Law are said to be ohmic

I

V V = R I

Ohm’s law

Parallel connection Series connection

R1

R2

R3

R1R2 R3

Req = R1 + R2 + R3

1/Req = 1/R1+1/R2+1/R3

Q2. What is the ratio of the current flowing through each resistor (I1:I2) in the circuit?

1. 1:12. 3:13. 1:44. Need more info.

6 V

R1 = 10

R2 = 30

Ra

Rb

1/Req = 1/Ra + 1/Rb

Req is smaller than Ra and Rb

20

25

Req ≈ 10

1000 = 1k

2Req < 2Practically all the current flowsThough the bottom one!!

Practical considerations for ‘R’