Engineering Engineering FundamentalsFundamentalsEngineering Engineering

FundamentalsFundamentals

Session 4 (3 hours)Session 4 (3 hours)

Resistance

• Amount of current flows in an electric circuit depends on the resistance of the material through which it travels

I=V/R• A resistor is a device which provides resistance

in an electrical circuit

R=V/I

• SI unit is ohm, written as 1 =1 VA-1

• An electric circuit is a connection of sources and loads in any way which allow charge to flow.e.g. a simple circuit with a battery and

a light bulb

• Relationship between current direction and voltage polarity

V=____

V=_____

I

+ 10V -

• Consider the following circuit:

– What is the voltage VR through the 5 resistor, and its polarities?

– What is the current I, and its direction?

+

V

-

10+V-

5

VR

5

Measuring resistance

• Voltmeter-ammeter method

R=V/I

• Multimeter• What is the resistance of your body?

A

V

• An open circuit occurs when a conductor is broken between the points under test.

• A short circuit occurs when a low resistance conductor such as a wire is connected

R=____

R=____

Ohm’s Law • Voltage across two ends of a

resistor is directly proportional to the current flowing through it.

I

V

V-I graph for a resistor that obeys Ohm’s law

V=IRI

+ V -

Resistors in series

• A series circuit has only one path for current to flow– Example:

– Same current through each resistor I = I1 = I2

R2R1

+ V1- + V2 -

I

+ V -

I1 I2

– Applying Ohm’s law to each resistor,V1 = IR1, V2 = IR2

– Voltage of the source equals sum of V1 and V2

V = V1 + V2

= IR1 + IR2

= I (R1 + R2) = I R

– Equivalent resistance R = R1 + R2

– In general

R = R1 + R2 + R3 + • • • + Rn

R1 R2 R3

• • •Rn

• Example:

– What is the equivalent resistance?– What are the currents I1, I2 through the

resistors?– What are their directions?– What is the voltage across each

resistor?

Indicate the relative voltage polarities (+,-)

12V

2 4

V1 V2

I

I1 I2

• Solution:

– Equivalent resistance = 2 + 4 = 6– Current I = V/R = 12V/6 =2A

12V

2 4

12V

6

+ 12V -

2 4

I

I1 I2

• Solution:– V = IR

Voltage across 2 resistor = 2A*2 = 4V

Voltage across 4 resistor = 2A*4 = 8V

+ 12V -

2 4

+ 4V - + 8V -

I

I1 I2

• Consider the following circuit:

– What are the voltages V1, V2, V3, and V4, and their corresponding polarities 極性 ?

– What is the current? and its direction?

+

V

-

10

V1

5

V3

V2

V4

5

5 5

• Consider the following circuit:

– What are the voltages V1, V2, V3, and V4, and their corresponding polarities?

– What is the current, and its direction?

+

V

-

10

V1

5

V3

V2

V4

5

5 5

Resistors in Parallel

• A parallel circuit has more than one path for current to flow– Example:

– Same voltage across each resistorV = V1 = V2

R2

R1

+ V2 -

+ V1 -

I2

I1

+ V -I

– Applying Ohm’s law to each resistor,V = I1R1, V = I2R2

– Sum of current in branches equal to current I I = I1 + I2 = V/R1+V/R2

= V (1/R1+1/R2) = V 1/R

– Equivalent resistance 1/R = 1/R1+1/R2

– In general

1/R = 1/R1+1/R2+1/R3+ • • •+1/Rn

R1

R2

R3

Rn

...

• Consider the following with R1 > R2

What can be said about the equivalent resistance R ?

(a) R > R1

(b) R2 < R < R1

(c) R < R2

R1

R2

R

• Example:

– What is the equivalent resistance?– What are the currents through the resistors and

their directions?– What is the total current?

12V

2

4

• Solution:

– Equivalent resistance:1/R = 1/R1+1/R2 = 1/2 + 1/4 = 3/4

R = 4/3– Same voltage across each resistor = 12V

I1 =current through 2 =V/R1 =12V/2=6A

I2 =current through 4=V/R2=12V/4=3A

– Total current I = I1 + I2 = 6A + 3A = 9A

12V

2

4

12V

4/3

I

I1

I2

• Example:

– What is the equivalent resistance?

– What is the supply voltage (Vs)?

– What are the voltages across AB (VAB), BC (VBC)?

What are the currents through the resistors (I20, I30) ?

30unknown

voltage supply

10

20

B

C

A

2A+

V s

-

+ V AB -

+

V BC

-I20 I30

• Solution:

– Equivalent resistance

R = 22

30unknown

voltage supply

10

20

B

C

A

2A

unknown

voltage supply

10

12

B

C

A

2A

unknown

voltage supply

22

B

C

A

2A

• Solution:

– Supply voltage

Vs =IR=2A*22=44V– Voltage across AB,

VAB=2A*10 =20V

Voltage across BC, Vbc=V-VAB=44V-20V=24V

30unknown

voltage supply

10

20

B

C

A

2A

unknown

voltage supply

10

12

B

C

A

2A+

V s

-

+

V s

-

+ V AB - + V AB -

+

V BC

-

+

V BC

-

• Solution:

– Current through 20 resistor

I1 = VBC/ 20 = 1.2A

– Current through 30 resistor

I2 = VBC/ 30 = 0.8A

Check total current

I1 + I2 = 1.2A + 0.8A = 2A

30unknown

voltage supply

10

20

B

C

A

2A I1 I2

• Some problems to work in class:– Problem 1:

• What is the equivalent resistancebetween A and B?

3

9

9

B

A

9

3

3

– Problem 2:

• What is the equivalent resistance between A and B?

8

8

BA

8

8