1

Basic Laws

Discussion D2.1Chapter 2

Sections 2-1 – 2-6, 2-10

2

Basic Laws

• Ohm's Law• Kirchhoff's Laws• Series Resistors and Voltage Division• Parallel Resistors and Current Division• Source Exchange

3

Georg Simon Ohm (1789 – 1854)

http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Ohm.html

German professor who publishes a book in 1827 that includes what is now known as Ohm's law.

Ohm's Law: The voltage across a resistor is directly proportional to the currect flowing through it.

4

Resistance

A

l = length

Resistance R l A

resistivity in Ohm-meters

Good conductors (low ): Copper, Gold

Good insulators (high ): Glass, Paper

5

Ohm's Law

v iR viR

Units of resistance, R, is Ohms ()

vRi

R = 0: short circuit :R open circuit

1v i R 1( )i i

Ri

+ -

v+ -

R i

v+ -

1

6

Unit of G is siemens (S),

Conductance, G

1GR

ivG

i GviGv

1 S = 1 A/V

Gi

+ -

v+ -

7

Power

A resistor always dissipates energy; it transforms electrical energy, and dissipates it in the form of heat.

Rate of energy dissipation is the instantaneous power2

2 ( )( ) ( ) ( ) ( ) 0v tp t v t i t Ri tR

22 ( )( ) ( ) ( ) ( ) 0i tp t v t i t Gv t

G

8

Basic Laws

• Ohm's Law• Kirchhoff's Laws• Series Resistors and Voltage Division• Parallel Resistors and Current Division• Source Exchange

9

Gustav Robert Kirchhoff (1824 – 1887)

http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Kirchhoff.html

Born in Prussia (now Russia), Kirchhoff developed his "laws" while a student in 1845. These laws allowed him to calculate the voltages and currents in multiple loop circuits.

10

CIRCUIT TOPOLOGY • Topology: How a circuit is laid out.• A branch represents a single circuit (network)

element; that is, any two terminal element. • A node is the point of connection between two or

more branches.• A loop is any closed path in a circuit (network).• A loop is said to be independent if it contains a

branch which is not in any other loop.

11

Fundamental Theorem of Network Topology

1b l n

For a network with b branches, n nodes and l independent loops:

DC

1 2

3 4 5

6

7

2A

Example

bn

l

95

5

12

Elements in Series

Two or more elements are connected in series if they carry the same current and are connected sequentially.

V0

I

R1

R2

13

Elements in Parallel

Two or more elements are connected in parallel if they are connected to the same two nodes & consequently have the same voltage across them.

VR1

I

R2I1 I2

14

Kirchoff’s Current Law (KCL)

The algebraic sum of the currents entering a node (or a closed boundary) is zero.

1

0N

nn

i

where N = the number of branches connected to the node and in = the nth current entering (leaving) the node.

15

1

0N

nn

i

1i

5i

2i3i4i

Sign convention: Currents entering the node are positive, currents leaving the node are negative.

1 2 3 4 5 0i i i i i

16

Kirchoff’s Current Law (KCL)

The algebraic sum of the currents entering (or leaving) a node is zero.

1i

5i

2i3i4i

1 2 3 4 5 0i i i i i

1 2 3 4 5 0i i i i i

The sum of the currents entering a node is equal to the sum of the currents leaving a node.

1 2 4 3 5i i i i i

Entering:

Leaving:

17

Kirchoff’s Voltage Law (KVL)

The algebraic sum of the voltages around any loop is zero.

1

0M

mm

v

where M = the number of voltages in the loop and vm = the mth voltage in the loop.

18

Sign convention: The sign of each voltage is the polarity of the terminal first encountered in traveling around the loop.

The direction of travel is arbitrary.

Clockwise:

Counter-clockwise:

0 1 2 0V V V

2 1 0 0V V V

0 1 2V V V

V0

I

R1

R2

V1

V2

A +

+

-

-

19

Basic Laws

• Ohm's Law• Kirchhoff's Laws• Series Resistors and Voltage Division• Parallel Resistors and Current Division• Source Exchange

20

0 1 2 1 2V V V IR IR

1 2I R R

sIR

1 2sR R R

Series Resistors

V0

I

R1

R2

V1

V2

A +

+

-

-

V R

I

s

21

V0

I

R1

R2

V1

V2

A

Voltage Divider0 0

1 2s

V VIR R R

0

2 2 21 2

VV IR RR R

2

2 01 2

RV VR R

1

1 01 2

Also RV VR R

22

Basic Laws

• Ohm's Law• Kirchhoff's Laws• Series Resistors and Voltage Division• Parallel Resistors and Current Division• Source Exchange

23

VR1

I

R2I1 I2

1 21 2

V VI I IR R

Parallel Resistors

1 2

1 1VR R

p

VR

1 2

1 1 1

pR R R

1 2

1 2p

R RRR R

V R

I

p

24

Current Division

i(t) R1

i

R2i1 i2 v(t)

+

-

1 2

1 2

( ) ( ) ( )pR Rv t R i t i tR R

21

1 1 2

( )( ) ( )Rv ti t i tR R R

12

2 1 2

( )( ) ( )Rv ti t i tR R R

Current divides in inverse proportion to the resistances

25

Current Division

N resistors in parallel

1 2

1 1 1 1

p nR R R R ( ) ( )pv t R i t

( )( ) ( )pj

j j

Rv ti t i tR R

Current in jth branch is

26

Basic Laws

• Ohm's Law• Kirchhoff's Laws• Series Resistors and Voltage Division• Parallel Resistors and Current Division• Source Exchange

27

Source Exchange

DC sv

sRabv

+

-

abv

+

-

sRs

s

vR

ai 'ai

We can always replace a voltage source in series with a resistor by a current source in parallel with the same resistor and vice-versa. Doing this, however, makes it impossible to directly find the original source current.

28

Source Exchange Proof

Voltage across and current through any load are the same

DC sv

sRLv

+

-

+

-

sRs

s

vR

ai 'ai

LRLR Lv

L

L ss L

Rv vR R

s

as L

vi

R R

' s s

a as L s

R vi i

R R R

' L

L a L ss L

Rv i R vR R