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Lecture 1

Dr Kelvin Tan

Electrical Systems 100

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Contents• Resistance

• Ohm’s Law

• Series Circuits

• Parallel Circuits

• Voltage Divider Rule

• Current Divider Rule

• Kirchoff’s Voltage Law (KVL)

• Kirchoff’s Current Law

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ResistanceThe flow of charge (q) in a material per unit time is current (i)

I = dq/dt

The resistance of a material is the property to resist the flow of electrons when an external electric field is applied. The resistance converts the applied energy into heat much like the mechanical friction due to colliding electrons and collisions between electrons and other atoms. The unit of Resistance is known as Ohms ()

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Resistance The Resistance of a material depends on its length (l), area (A) and the resistivity ()

R = l/A ()

The Resistivity of some materials at 20 degree celsius. The unit is CM. /ft and in SI system is Ohm-meters

Resistivity (ρ) of various materials

Silver 9.9

Copper 10.37

Gold 14.7

Aluminium 17.0

Tungsten 33.0

Nickel 47.0

Iron 74.0

Nichrome 295.0

Chlorite 720.0

Carbon 21,000.0

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Resistance Resistors can be various types:

• Fixed Resistor

•Variable Resistors (Rheostat)

• Varistors

• Thermistors

• Photocells

•Photocells

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Resistance Fixed and Variable Resistors

The physical size (or shape) of a resistor is no clue to its resistance value, but can be a rough guide to its power rating.

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Variable Resistors (Rheostat) - resistance change by turning a knob

ResistanceThermistors - Resistance that varies rapidly and predictably with temperature

A Varistor - Voltage Dependent Resistor or VDR Photocells - resistance change in light level

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Color Coding of Resistors

ExampleFind the resistance of a semiconductor resistance if the bands are:

1st band is Gray, 2nd band is Red, 3rd band is Black, 4th band is Gold, and fifth band is Brown?

Solution:82*(10e0) () ± 5% (1% reliability)

77.9 to 86.1 and 1out of every 100 will fail to satisfy manufacturer’s range after 1000 hr of operation.

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Conductances Conductances are reciprocal of Resistances

G = 1/R (siemens, S) or Mho

G = A / l (S or Mho)

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A simple resistive circuit

Electron flow direction Conventional current direction

I I

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Ohm’s LawOhm’s Law states that

the current through an electric path is dependant on the resistance of the path and is given by:

R

V I

+

_VI

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V-I Characteristic of a Linear Resistor

I

V R

IRV

R

V I

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Power

SI unit is the Watt (I in Amperes, V in Volts and R in Ohms). 1 Watt is 1 Joules/Sec

1 hp = 746 Watts (the power of an average dray horse over a full working day)

Electrical Power

R

VRIP

VIP2

2

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Polarity references and the expression for

power

• If the power is Negative (P <0), power supplied by circuit

If the power is Positive (P > 0), power absorbed by the circuit

+_V

I

Absorbor supply ?

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Energy: Electric Energy is the Electric Power consumed over a period of time

SI unit is the Watt (I in Amperes, V in Volts and R in Ohms).

1 Watt is 1 Joules/Sec

1 hp = 746 Watts (the power of an average dray horse over a full working day)

Electrical Energy

t

PdttR

VtRItPW

0

22 ***

How to calculate the Electric Power consumed in 1 day ?

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The unit of Energy is Watt-hours (Wh), kWh, MWh or GWh

Energy (Wh) = Power (W) * Time (h)

1 kWh is 1000 Wh

1 MWh = 1000 kWh

1 GWh = 1000 MWh

We pay for electricity on the basis of kWh consumed in a specified period @ 12 cents (approx) per kWh

Electrical Energy

How many kWh is this family using ??

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Direct Current Circuits

• Analysis of simple circuits with batteries, resistors, and capacitors connected in various combinations (series/parallel).

• Kirchoff’s rules (based on law of conservation of energy and charge).

• Steady-state (current constant in magnitude and direction).

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A simple resistive circuit

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Circuit diagram

Vr = I × r

VR = I × R

I

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The battery has an internal resistance r and is connected in series with a resistor R.

IrVVV ab

Battery in Series with a Resistor

IrIR rR

EI

Battery

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Multiplying by the current yields:

The total power of the battery is converted to heat in the two resistors.

Battery in Series with a Resistor

rIRII 22

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Series connection of two resistors

Req = R1 + R2

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Two or more resistors are in series if connected together so that they have only one common point per pair.

The current is the same through each resistor because any charge flowing through one resistor must also flow through the other.

Two Resistors in Series

)( 2121 RRIIRIRV

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More than Two Resistors in Series

..........321 RRRReq

The equivalent resistance of a series connection of resistors is always greater than any individual resistance.

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Parallel connection of two resistors

21eq R

1

R

1

R

1

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Two Resistors in ParallelTwo or more resistors are in parallel if there is an equal potential difference across each resistor.

The current is not the same through each resistor because any charge flowing through one resistor cannot flow through the other. But the Potential difference (voltage) across each parallel branch is identical.

21 III 21

111

RRReq

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More than Two Resistors in Parallel

The equivalent resistance of a parallel connection of resistors is always less than the smallest individual resistance.

.......1111

321

RRRReq

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Kirchoff’s Laws

• The sum of the currents entering any junction must equal the sum of the currents leaving that junction. (KCL)

(Conservation of charge)

• The algebraic sum of the changes in potential across all of the elements around any closed circuit loop must be zero. (KVL)

(Conservation of energy)

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Kirchoff’s Voltage Law

The algebraic sum of the changes in potential across all of the elements around any closed circuit loop must be zero. (KVL)

0 riseordropsV

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Kirchoff’s Voltage Law

I

V1

V2K V L

0 riseordropsVKVL

0 21 VVE

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Kirchoff’s Current Law

Kirchoff’s Current Law states that the amount of current entering a junction must be equal to the current leaving that junction.

0

,

I leaving

I

or

I entering

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Kirchoff’s Current Law

321

leaving

0

,

I

III

I

or

I entering

I1

I2

I3

Example : I1 = 5A, I2 = 2A and I3 = 7A

Taking current entering the node as reference I1 +I2 - I3 = 0A

Taking current leaving the node as reference

I3 -I1 -I2 = 0A

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Kirchoff’s Current Law

Example :

I1=2A

I2=3A

I3=5A

I4=1A

I5= 6A

a b

Node (a) KCL

I1 + I2 + I3 = 0

Node (b) KCL

I3 + I4 + I5 = 0

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Voltage Divider Rule

In a Series Circuit, the Voltage across resistances divide according to the magnitude of the resistances

Consider a Series circuit with two elements R1 and R2 and the total voltage applied across them is V. Let the current flowing through each of the two elements (series) be “I”

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Voltage Divider Rule

E R

R

RR

E IR

E R

R

RR

E IR

R

E

21

22

221

22

21

1

121

1

21

2

1

11

RV

RVV

RV

RVV

RI

V

21VVV

+

_

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Current Divider Rule

The current in parallel circuit elements divide in ratios according to the inverse of their resistor values. Consider two parallel elements R1 and R2 and a total current I, then the current in two elements are:

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Current Divider Rule

21

2

1

21

1

21

2

R

RI I

III

R

RR

RII

I

I

I1

I1

I2

I2

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Single Subscript Voltage NotationVa means that the voltage of point “a” is Va

with respect to the ground.

Va Vb

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Double Subscript Voltage Notation

Vab means that the voltage of point “a” with respect to point “b” is Vab

Vab = Va – Vb

Vba = Vb – Va = -Vab

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If point “b” is at ground potential then Vab is simply Va

Vab = Va – 0 = Va

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What is Vab?

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Measurement of Voltage

Voltages are measured by an instrument called Voltmeter. They can be either analog (AVO) or digital (DMM).

Voltmeters are always connected in parallel with circuit where the voltage is to be measured. Voltmeters should have very high resistance, so that they do not allow current flowing through the circuit to enter in them due to their insertion.

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Measurement of Voltage

I + VR -

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Measurement of Current

Currents are measured by an instrument called Ammeter. An Ammeter can be analog or digital. An Ammeter is always connected in series with the circuit where current need to be measured.

An Ammeter should have almost zero resistance, so that they do not alter the magnitude the of the current due to their insertion.

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Measurement of Current

I

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An Open Circuit An Open circuit is one which have voltage across it but current through it is zero. An open circuit is represented by infinite resistance.

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A Short Circuit

A short circuit is one which allow current through it but voltage across it is zero. A short circuit is represented by zero resistance.