Transcript Basic Concepts - Oakland University

Basic Laws
Discussion D2.1
Chapter 2
Sections 2-1 – 2-6, 2-10
1
Basic Laws
•
•
•
•
•
Ohm's Law
Kirchhoff's Laws
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Source Exchange
2
Georg Simon Ohm (1789 – 1854)
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.
http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Ohm.html
3
Resistance
  resistivity in Ohm-meters
Resistance
R  l A
l = length
Good conductors (low ): Copper, Gold
A
Good insulators (high ): Glass, Paper
4
v
R
i
+
v
i
R
v  iR
v
i1
-
R
+
-
i
v
R
-
+
Ohm's Law
v  i1R
(i  i1 )
Units of resistance, R, is Ohms (W)
R = 0: short circuit
R   : open circuit
5
+
1
G
R
G
v
-
i
-
+
Conductance, G
Unit of G is siemens (S),
1 S = 1 A/V
i
v
G
i  Gv
i
G
v
6
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
v
(t )
2
p(t )  v(t )i (t )  Ri (t ) 
0
R
2
i
(t )
2
p(t )  v(t )i (t )  Gv (t ) 
0
G
7
Basic Laws
•
•
•
•
•
Ohm's Law
Kirchhoff's Laws
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Source Exchange
8
Gustav Robert Kirchhoff (1824 – 1887)
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.
http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Kirchhoff.html
9
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.
10
Fundamental Theorem of Network Topology
For a network with b branches, n nodes
and l independent loops:
b  l  n 1
Example
b 9
7W
1W
DC
2W
3W
6W
4W
5W
2A
n 5
l 5
11
Elements in Series
Two or more elements are connected in series if they
carry the same current and are connected sequentially.
I
R1
V0
R2
12
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.
I
I1
V
R1
I2
R2
13
Kirchoff’s Current Law (KCL)
The algebraic sum of the currents entering a
node (or a closed boundary) is zero.
N
i
n 1
n
0
where N = the number of branches connected to
the node and in = the nth current entering
(leaving) the node.
14
Sign convention: Currents entering the node are positive,
currents leaving the node are negative.
N
i
n 1
n
0
i2
i1
i5
i3
i4
i1  i2  i3  i4  i5  0
15
Kirchoff’s Current Law (KCL)
The algebraic sum of the currents entering
(or leaving) a node is zero.
Entering:
i1  i2  i3  i4  i5  0
Leaving: i1
 i2  i3  i4  i5  0
i2
i1
i5
i3
i4
The sum of the currents entering a node is
equal to the sum of the currents leaving a node.
i1  i2  i4  i3  i5
16
Kirchoff’s Voltage Law (KVL)
The algebraic sum of the voltages around
any loop is zero.
M
v
m 1
m
0
where M = the number of voltages in the loop
and vm = the mth voltage in the loop.
17
Sign convention: The sign of each voltage is the polarity of the
terminal first encountered in traveling around the loop.
I
+
R1
V1
+
A
V0
R2
The direction of travel is arbitrary.
Clockwise:
V0  V1  V2  0
V2
-
Counter-clockwise:
V2  V1  V0  0
V0  V1  V2
18
Basic Laws
•
•
•
•
•
Ohm's Law
Kirchhoff's Laws
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Source Exchange
19
Series Resistors
I
+
R1
V1
 I  R1  R2 
+
A
V0
V0  V1  V2  IR1  IR2
R2
 IRs
V2
Rs  R1  R2
-
I
V
Rs
20
Voltage Divider
V0
V0
I

Rs R1  R2
I
R1
V1
R2
V2
A
V0
V0
V2  IR2 
R2
 R1  R2 
R2
V2 
V0
 R1  R2 
R1
Also V1 
V0
 R1  R2 
21
Basic Laws
•
•
•
•
•
Ohm's Law
Kirchhoff's Laws
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Source Exchange
22
Parallel Resistors
I
V V
I  I1  I 2  
R1 R2
I1
R1
V
R2
1
1
1
 
Rp R1 R2
I
V
I2
Rp
1 1 
V   
 R1 R2 
V

Rp
R1R2
Rp 
R1  R2
23
Current Division
i
+
i1
i(t)
R1
i2
R2 v(t)
-
R2
v(t )
i1 (t ) 

i(t )
R1
R1  R2
R1
v(t )
i2 (t ) 

i(t )
R2
R1  R2
R1R2
v(t )  Rpi(t ) 
i(t )
R1  R2
Current divides in inverse proportion to the resistances
24
Current Division
N resistors in parallel
1
1 1
1
 
  
Rp R1 R2
Rn
Current in
jth
branch is
v(t )  Rpi(t )
v(t ) Rp
i j (t ) 

i(t )
Rj
Rj
25
Basic Laws
•
•
•
•
•
Ohm's Law
Kirchhoff's Laws
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Source Exchange
26
Source Exchange
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
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
27
original source current.
Source Exchange Proof
ia '
ia
+
Rs
DC
RL vL
vs
+
vs
Rs
Rs
RL vL
-
-
RL
vL 
vs
 Rs  RL 
Rs
vs
ia ' 
 ia
 Rs  RL  Rs
vs
ia 
 Rs  RL 
RL
vL  ia ' RL 
vs
 Rs  RL 
Voltage across and current through any load are the same28