Electrical Circuits I

Emam Fathy

Department of Electrical and Control Engineering

email: emfmz@aast.edu

http://www.aast.edu/cv.php?disp_unit=346&ser=68525

Lecture 1

Introduction; Circuit

Elements; Ohm's Law; KCL

Definition 1: An interconnection of electrical elements

linked together in a closed path so that an electric current

flow continuously

Battery Resistor

Wire

A Simple Circuit

Electric Circuit

Definition 2: A mathematical model that approximates the

behavior of an actual electrical system.

Definition 3: An interconnection between components or

electrical devices for the purpose of communicating or

transferring energy from one point to another. The

components of electric circuit are always referred to as

circuit elements.

Electric Circuit

Basic Electrical Quantities

• Basic quantities: current, voltage and

power

– Current: time rate of change of electric charge

I = dq/dt

1 Amp = 1 Coulomb/sec

– Voltage: electromotive force or potential, V

1 Volt = 1 Joule/Coulomb = 1 N·m/coulomb

– Power: P = I V

1 Watt = 1 Volt·Amp = 1 Joule/sec

Current, I

• The movement of positive charges although we know that, in general, in metallic conductors current results from electron motion (conventionally positive flow)

• Types of current:– direct current (dc): batteries and some special

generators

– alternating current (ac): household current which varies with time

Direct current (DC) is a current

that remains constant with

time

Alternating current (AC) is a

current that varies sinusoidally

with time

i

t

Direct current

(DC)

Alternating current

(AC)

Current, I

Voltage, V

Voltage is the difference in energy level of a

unit charge located at each of two points in a

circuit, and therefore, represents the energy

required to move the unit charge from one

point to the other

Circuit Element(s)

+ –V(t)

Active vs. Passive Elements

• Active elements can generate energy

– Voltage and current sources

– Batteries

• Passive elements cannot generate energy

– Resistors

– Capacitors and Inductors (but CAN store

energy)

Independent Sources

An independent source (voltage or current)

may be DC (constant) or time-varying (AC),

but does not depend on other voltages or

currents in the circuit

+

–

Voltage

Source

Current

Source

40

Ideal voltage source connected

in series

Independent Sources

41

Ideal current source connected

in parallel

Independent Sources

xs Vi xs iV

39

DEPENDENT SOURCES

voltage current

Active element in which the source quantity is

controlled by another voltage or current

Resistors

• A resistor is a circuit element that

dissipates electrical energy (usually as

heat)

• Real-world devices that are modeled by

resistors: incandescent light bulbs, heating

elements (stoves, heaters, etc.), long

wires

• Resistance is measured in Ohms (Ω)

43

Ohm’s Law

The current in a circuit is directly

proportional to the applied voltage and

inversely proportional to the resistance of

the circuit.”

IRV

44

8 V

100

Determine the current in figure below :

AV

R

VI

IRV

08.0100

8

Example

Ohm’s Law

The current in a circuit is directly

proportional to the applied voltage and

inversely proportional to the resistance of

the circuit.”

IRV

RIP 2

From Ohm’s Law, we can get:

R

VP

2

and

28

Power is the rate of doing work, or the rate of

transfer energy

Measured in Watts (W)

1 hp = 746 watts

1 W = 1 J/s

Power, P

t

WP

Energy (J)

Time (s)

RIP 2R

VP

2

and

Power is the rate of using energy or

doing work.

POWER

Work (W)

consists of a force

moving through a

distance.

Energy (W)

is the capacity to

do work.

Joule (J)is the base unit for both energy and work.

32

Energy, W Work consists of a force moving through a distance

Energy is the capacity to do work.

Energy = Power × time

Units are joules = watt-seconds, watt-hours, or

more commonly, kilowatt-hours.

The electric power utility companies measure

energy in watt-hours (Wh).

PtW t

WP

Energy (J)

Time (s)

from equation

of ‘Power’

46

Short Circuit

Short circuit is a circuit element with resistance

approaching zero

VIIRV

AV

R

VI

0)0(

0

47

Open Circuit

Open circuit is a circuit element with resistance

approaching infinity

AV

R

VI 0

Series

Two elements are in series if the current that

flows through one must also flow through

the other.

R1 R2

Series

Not SeriesR1 R2

R3

Parallel

Two elements are in parallel if they are

connected between (share) the same two

(distinct) end nodes.

(R1&R2) Parallel (R1&R2) Not Parallel

R1

R2

R3R1

R2

Kirchhoff’s Laws

• Kirchhoff’s Current Law (KCL)

– sum of all currents entering a node is zero

– sum of currents entering node is equal to sum

of currents leaving node

• Kirchhoff’s Voltage Law (KVL)

– sum of voltages around any loop in a circuit is

zero

KCL (Kirchhoff’s Current Law)

The sum of currents entering the node is

zero:

i1(t)

i2(t) i4(t)

i5(t)

i3(t)

n

j

j ti1

0)(

KVL (Kirchhoff’s Voltage Law)

The sum of voltages around a loop is zero:

0)(1

n

j

j tv

v1(t)

++

–

–

v2(t)

v3(t)+

–

KVL (Kirchhoff’s Voltage Law)

The sum of voltages around a loop is zero:

0)(1

n

j

j tv

v1(t)

++

–

–

v2(t)

v3(t)+

–

Single Loop Circuit

• The same current flows through each element of the circuit—the elements are in series

• We will consider circuits consisting of voltage sources and resistors

+

–VS

R

R

R

I

Solve for I

• In terms of I,

what is the

voltage across

each resistor?

Make sure you

get the polarity

right!

• To solve for I,

apply KVL

around the loop

+

–VS

R

R

R

I + –

I R

+

–

I R

I R

+

–

N Total

Resistors

IR + IR + … + IR – VS = 0

I = VS / (N R)

In General: Single Loop

• The current i(t) is:

• This approach works for any single loop

circuit with voltage sources and resistors

• Resistors in series

sresistanceofsum

sourcesvoltageofsum

R

Vti

j

Si

)(

jNseries RRRRR 21

Voltage Divider

Consider two resistors in series with a

voltage v(t) across them:

R1

R2

–

v1(t)

+

+

–

v2(t)

+

–

v(t)21

11 )()(

RR

Rtvtv

21

22 )()(

RR

Rtvtv

In General: Voltage Division

• Consider N resistors in series:

• Source voltage(s) are divided between the

resistors in direct proportion to their

resistances

j

iSR

R

RtVtV

ki)()(

Example

Example

Applying the KVL equation for the circuit of the figure below.

va-v1-vb-v2-v3 = 0

V1 = IR1 v2 = IR2

v3 = IR3

va-vb = I(R1 + R2 + R3)

321 RRR

vvI ba

Current Divider

1R 2R

1I

V

2I

_

I

21

212211

RR

RRIRIRIV

Current Divider

1R 2R

1I

V

2I

_

I

IRR

RI

IRR

RI

21

12

21

21

2121

21

11

RRV

R

V

R

VIII

I R1 R2 V

+

–

I1 I2

21

21

21

11

1

RR

RRI

RR

IV

21

2

1

21

21

1

1RR

RI

R

RR

RRI

R

VI

Current Divider

Three Resistors in Parallel

I= I1 + I2 + I3

1

1R

VI

2

2R

VI

3

3R

VI

I R2 V

+

–

R1

I1 I2

R3

I3

Solve for V

321321

111

RRRV

R

V

R

V

R

VI

eqRI

RRR

IV

321

111

1

j

par

SRR

RII

j

1

1R

VI

2

2R

VI

3

3R

VI

I1 R1= I Req

Three Resistors in Parallel

I= I1 + I2 + I3

1

1R

VI

2

2R

VI

3

3R

VI

I R2 V

+

–

R1

I1 I2

R3

I3

j

par

SRR

RII

j

2121

21

11

RRV

R

V

R

VIII

I R1 R2 V

+

–

I1 I2

21

21

21

11

1

RR

RRI

RR

IV

21

2

1

21

21

1

1RR

RI

R

RR

RRI

R

VI

Current Divider

j

par

SRR

RII

j

Example

Is1 Is2 VR1 R2

+

–

I1 I2

Example

21212121

11

RRV

R

V

R

VIIII ss

21

2121

RR

RRIIV ss

Is1 Is2 VR1 R2

+

–

I1 I2

End of Lec