basic law

1. 1
Basic Laws
Discussion D2.1
Chapter 2
Sections 2-1 – 2-6, 2-10

2. 2
Basic Laws
• Ohm's Law
• Kirchhoff's Laws
• Series Resistors and Voltage Division
• Parallel Resistors and Current Division
• Source Exchange

3. 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. 4
Resistance
A
l = length
Resistance R l Aρ=
ρ = resistivity in Ohm-meters
Good conductors (low ρ): Copper, Gold
Good insulators (high ρ): Glass, Paper

5. 5
Ohm's Law
v iR=
v
i
R
=
Units of resistance, R, is Ohms (Ω)
v
R
i
=
R = 0: short circuit :R = ∞ open circuit
1v i R= − 1( )i i= −

6. 6
Unit of G is siemens (S),
Conductance, G
1
G
R
=
i
v
G
= i Gv=
i
G
v
=
1 S = 1 A/V

7. 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 power
2
2 ( )
( ) ( ) ( ) ( ) 0
v t
p t v t i t Ri t
R
= = = ≥
2
2 ( )
( ) ( ) ( ) ( ) 0
i t
p t v t i t Gv t
G
= = = ≥

8. 8
Basic Laws
• Ohm's Law
• Kirchhoff's Laws
• Series Resistors and Voltage Division
• Parallel Resistors and Current Division
• Source Exchange

9. 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. 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. 11
Fundamental Theorem of Network Topology
1b l n= + −
For a network with b branches, n nodes
and l independent loops:
Example
b =
n =
l =
9
5
5

12. 12
Elements in Series
Two or more elements are connected in series if they
carry the same current and are connected sequentially.

13. 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.
V
R1
I
R2
I1 I2

14. 14
Kirchoff’s Current Law (KCL)
The algebraic sum of the currents entering a
node (or a closed boundary) is zero.
1
0
N
n
n
i
=
=∑
where N = the number of branches connected to
the node and in = the nth
current entering
(leaving) the node.

15. 15
1
0
N
n
n
i
=
=∑
1i
5i
2i
3i
4i
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. 16
Kirchoff’s Current Law (KCL)
The algebraic sum of the currents entering
(or leaving) a node is zero.
1i
5i
2i
3i
4i
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. 17
Kirchoff’s Voltage Law (KVL)
The algebraic sum of the voltages around
any loop is zero.
1
0
M
m
m
v
=
=∑
where M = the number of voltages in the loop
and vm = the mth
voltage in the loop.

18. 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= +

19. 19
Basic Laws
• Ohm's Law
• Kirchhoff's Laws
• Series Resistors and Voltage Division
• Parallel Resistors and Current Division
• Source Exchange

20. 20
0 1 2 1 2V V V IR IR= + = +
( )1 2I R R= +
sIR=
1 2sR R R= +
Series Resistors

21. 21
V0
I
R1
R2
V1
V2
A
Voltage Divider
0 0
1 2s
V V
I
R R R
= =
+
( )
0
2 2 2
1 2
V
V IR R
R R
= =
+
( )
2
2 0
1 2
R
V V
R R
=
+
( )
1
1 0
1 2
Also
R
V V
R R
=
+

22. 22
Basic Laws
• Ohm's Law
• Kirchhoff's Laws
• Series Resistors and Voltage Division
• Parallel Resistors and Current Division
• Source Exchange

23. 23
V
R1
I
R2
I1 I2
1 2
1 2
V V
I I I
R R
= + = +
Parallel Resistors
1 2
1 1
V
R R
 
= + ÷
 
p
V
R
=
1 2
1 1 1
pR R R
= +
1 2
1 2
p
R R
R
R R
=
+

24. 24
Current Division
1 2
1 2
( ) ( ) ( )p
R R
v t R i t i t
R R
= =
+
2
1
1 1 2
( )
( ) ( )
Rv t
i t i t
R R R
= =
+
1
2
2 1 2
( )
( ) ( )
Rv t
i t i t
R R R
= =
+
Current divides in inverse proportion to the resistances

25. 25
Current Division
N resistors in parallel
1 2
1 1 1 1
p nR R R R
= + +×××+ ( ) ( )pv t R i t=
( )
( ) ( )
p
j
j j
Rv t
i t i t
R R
= =Current in jth
branch is

26. 26
Basic Laws
• Ohm's Law
• Kirchhoff's Laws
• Series Resistors and Voltage Division
• Parallel Resistors and Current Division
• Source Exchange

27. 27
Source Exchange
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. 28
Source Exchange Proof
Voltage across and current through any load are the same
( )
L
L s
s L
R
v v
R R
=
+
( )
s
a
s L
v
i
R R
=
+
( )
' s s
a a
s L s
R v
i i
R R R
= =
+
( )
' L
L a L s
s L
R
v i R v
R R
= =
+