1 basic laws discussion d2.1 chapter 2 sections 2-1 – 2-6, 2-10
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1
Basic Laws
Discussion D2.1
Chapter 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 ()
vR
i
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
1G
R
ivG
i Gvi
Gv
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 ( )( ) ( ) ( ) ( ) 0
v tp t v t i t Ri t
R
22 ( )
( ) ( ) ( ) ( ) 0i t
p t v t i t Gv tG
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
9
5
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
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
Kirchoff’s Current Law (KCL)
The algebraic sum of the currents entering (or leaving) a node is zero.
1i
5i
2i
3i
4i1 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 +
+
-
-
VR
I
s
21
V0
I
R1
R2
V1
V2
A
Voltage Divider0 0
1 2s
V VI
R R R
0
2 2 21 2
VV IR R
R R
2
2 01 2
RV V
R R
1
1 01 2
Also R
V 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 I
R R
Parallel Resistors
1 2
1 1VR R
p
V
R
1 2
1 1 1
pR R R
1 2
1 2p
R RR
R R
V
R
I
p
24
Current Division
i(t) R1
i
R2i1 i2 v(t)
+
-
1 2
1 2
( ) ( ) ( )p
R Rv t R i t i t
R R
21
1 1 2
( )( ) ( )
Rv ti t i t
R R R
12
2 1 2
( )( ) ( )
Rv ti t i t
R 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 t
R 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
DCsv
sRabv
+
-
abv
+
-
sRs
s
v
R
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.
28
Source Exchange Proof
Voltage across and current through any load are the same
DCsv
sRLv
+
-
+
-
sRs
s
v
R
ai 'ai
LRLR Lv
L
L ss L
Rv v
R 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 v
R R