Pressure laws and fast Legendre transform - Philippe LeFloch
L2 basic laws - KMUTTstaff.kmutt.ac.th/~Anawach.San/eee105/L2.pdf · Kirchhoff’s Laws: KCL KCL...
Transcript of L2 basic laws - KMUTTstaff.kmutt.ac.th/~Anawach.San/eee105/L2.pdf · Kirchhoff’s Laws: KCL KCL...
Basic Laws
EEE105 Electric Circuits
Anawach Sangswang
Dept. of Electrical Engineering
KMUTT
Ohm’s Law Resistor
Materials with a characteristic
behavior of resisting the flow
of electric charge
Resistance: An ability to resist
the flow of electric current,
measured in ohm (Ω)
where ρ is the resistivity of the material in ohm-meters
l is the length in meters
A is the cross-sectional area in m2
2
lR
Aρ=
Ohm’s Law
Georg Simon Ohm (1787-1854: German)
Ohm’s law: the voltage across a resistor (R) is
directly proportional to the current (i) flowing
through the resistor
Mathematical expression:
or
Note: The direction of current (i) and the polarity
of voltage (v) must conform with the passive
sign convention
3
iRv =
v i∝v
Ri
=
Ohm’s Law
Two extreme possible values of R: 0 (zero)
and ∞∞∞∞ (infinite) are related with two basic
circuit concepts: short circuit and open circuit.
4
0v iR= = lim 0R
vi
R→∞= =
Short circuit Open circuit
Fixed resistors
5
Variable resistors Ohm’s Law
Linear and nonlinear resistors
Conductance is the ability of an element to conduct electric current; it is the reciprocal of resistance R and is measured in mhos or siemens.
6
v
i
RG ==
1
Ohm’s Law
The power dissipated by a resistor:
Note
The power dissipated in a resistor is a nonlinear
function of either current or voltage
The power dissipated in a resistor is always
positive
The resistor always absorbs power and is a passive
element (incapable of generating energy)
7
R
vRivip
22 ===
Example
Calculate the current i, the
conductance G and the
power p
The current
The conductance
8
3
306
5 10
vi mA
R= = =
×
3
1 10.2
5 10G mS
R= = =
×
330(6 10 ) 180p vi mW−= = × =
2 3 2 3(6 10 ) 5 10
180
p i R
mW
−= = × ⋅ ×
=
2 2 330 0.2 10
180
p v G
mW
−= = ⋅ ×
=
Power
or
or
Branch, Nodes A branch represents a single element such as
a voltage source or a resistor
A node is the point of connection between
two or more branches
9
Original circuit
Equivalent circuit
Loop, Series, Parallel A loop is a closed path in a circuit
An independent loop contains at least 1 branch
which is not a part of any other independent
loop or path sets of independent equations
A network with b branches, n nodes, and l
independent loops satisfies
Series: 2 or more elements share a single node
and carry the same current
Parallel: 2 or more elements are connected to
the same two nodes and have the same
voltage across them10
1−+= nlb
Example Number of branches, nodes, series and
parallel connection
11
Kirchhoff’s Laws: KCL
Kirchhoff’s current law (KCL) states that
“the algebraic sum of currents entering a
node is zero”
“The sum of the currents entering a node is equal to
the sum of the currents leaving the node”12
01
=∑=
N
nni
Applying KCL: 1 2 3 4 5( ) ( ) 0i i i i i+ − + + + − =
Rearranging the equation
1 3 4 2 5i i i i i+ + = +
Kirchhoff’s Laws: KCL
KCL also applies to a closed boundary
13
The total current
entering the closed
surface is equal to
the total current
leaving the surface
Kirchhoff’s Laws: KCL
14
I + 4-(-3)-2 = 0
⇒I = -5A
This indicates that the
actual current for I is
flowing in the
opposite direction.We can consider the whole
enclosed area as one “node”.
Example
Example: Determine the current I
Kirchhoff’s Laws: KVL
Kirchhoff’s voltage law (KVL) states that
“the algebraic sum of all voltages around a
closed path (or loop) is zero”
15
01
=∑=
M
mnv
1 2 3 4 5 0v v v v v− + + − + =
2 3 5 1 4v v v v v+ + = +or
Sum of voltage drops = Sum of voltage rises
Kirchhoff’s Laws: KVL
Series-connected voltage sources
Note:
2 different voltages cannot be connected in parallel
2 different currents cannot be connected in seriess16
1 2 3 0abV V V V− + + − = 1 2 3abV V V V= + −
Kirchhoff’s Laws: KVL
Example: Determine vo and i
Apply KVL around the loop
17
12 4 2 4 0o oi v v− + + − − =
The Ohm’s law at the 6-ohm
resistor gives 6ov i= −
12 4 2( 6 ) 4 ( 6 ) 0i i i− + + − − − − = 8i A= −
48ov V=
Example
Determine the current i
18
1 2 3 0a bV V V V V− + + + + =
1 1 2 2 3 3, , V IR V IR V IR= = =
The Ohm’s law at the each
resistor gives
KVL:
1 2 3 0a bV IR V IR IR− + + + + =
1 2 3
a bV VI
R R R
−=
+ +
Series Resistors
Two or more elements are in series if they are
cascaded or connected sequentially
and consequently carry the same current
19
1 2 1 2v v v iR iR= + = +1 2 eq
v vi
R R R= =
+ 1 2eqR R R= +
1 21 1 2 2
1 2 1 2
, R R
v iR v v iR vR R R R
= = = =+ + Voltage divider
Series Resistors, Voltage Division
The equivalent resistance of any number of
resistors connected in a series is the sum of
the individual resistances
The voltage divider can be expressed as
20
∑=
=+⋅⋅⋅++=N
nnNeq RRRRR
121
vRRR
Rv
N
nn +⋅⋅⋅++=
21
Parallel Resistors, Current Division Parallel connection: elements
are connected to the same
two nodes and consequently
have the same voltage across
them.
21
1 1 2 2v i R i R= = or 1 21 2
, v v
i iR R
= =
Applying KCL @ node a
1 21 2 1 2
1 1v vi i i v
R R R R
= + = + = +
1 2
1 1 1
eqR R R= + 1 2
1 2eq
R RR
R R=
+
Parallel Resistors, Current Division
The equivalent resistance of a circuit with
N resistors in parallel is:
Current
22
Neq RRRR
1111
21
+⋅⋅⋅++=
1 2
1 2eq
R Rv iR i
R R= =
+
21
1 1 2
Rvi i
R R R= =
+
12
2 1 2
Rvi i
R R R= =
+
Current
divider
Example 2.12
Find io, vo and calculate the power dissipated
in the 3-ohm resistor
23
2(12) 4
2 4ov V= =+
3 4o ov i V= =4
3oi A=
44 5.33
3o o op v i W = = =
Example 2.13
Find vo, power supplied/absorbed by each
element
24
1
1830 20
9 18
ki mA mA
k k= =
+
2
930 10
9 18
ki mA mA
k k= =
+
1 29 18 180ov k i k i V= ⋅ = ⋅ =
180 30
5.4source o op v i mA
W
= = ⋅
=2
9 (20 ) (9 ) 3.6kp mA k W= =2
6 (10 ) (6 ) 0.6kp mA k W= =
212 (10 ) (12 ) 1.2kp mA k W= =
Wye-Delta Transformation
When the resistors are
neither in parallel nor in
series
Delta to wye
25
12 1 3( )R Y R R= + 12 ( ) //( )b a cR R R R∆ = +
1 3
( )b a c
a b c
R R RR R
R R R
++ =
+ + 1 2
( )c a b
a b c
R R RR R
R R R
++ =
+ + 2 3
( )a b c
a b c
R R RR R
R R R
++ =
+ +
Wye-Delta Transformation
Delta-wye conversion
Wye-delta conversion
26
1b c
a b c
R RR
R R R=
+ +
2c a
a b c
R RR
R R R=
+ +
3a b
a b c
R RR
R R R=
+ +
11 2 2 3 3 1 2
( )
( )a b c b c a b c
a b c a b c
R R R R R R R R RR R R R R R
R R R R R R
+ ++ + = =
+ + + +
Wye-Delta Transformation
Wye-delta conversion
27
1 2 2 3 3 1
1a
R R R R R RR
R
+ +=
1 2 2 3 3 1
2b
R R R R R RR
R
+ +=
1 2 2 3 3 1
3c
R R R R R RR
R
+ +=
Example
Convert the delta network to wye network
28
1
10 25 2505
15 10 25 50b c
a b c
R RR
R R R
×= = = = Ω
+ + + +
2
25 157.5
50c a
a b c
R RR
R R R
×= = = Ω
+ + 3
15 103
50a b
a b c
R RR
R R R
×= = = Ω
+ +