# DC circuits

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7 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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DC circuits
Basics of Electrical Engineering UNH
DC circuits 1/25
Electric current
analogy between water ow and current
The current is the rate at which charge ows through a
surface A
If Q is the amount of charge passes through the surface A
within t time,then the average current is
I
av
=
Q
A
If current varies in time,the instantaneous current can be
dened as I as the dierential limit of average current
I =
dQ
dt
SI unit of the current is [A]
It is conventional to assign to the current the same direction
as the ow of positive charge
Basics of Electrical Engineering UNH
DC circuits 2/25
Microscopic model of current
I
av
=
Q
t
=
nAv
d
tq
t
= nAv
d
q
v
d
is the so called drift speed
E = 0:electrons has a random motion that is analogous to
the motion of gas molecules
E 6= 0:the electrons do not move in straight lines along the
conductor.Instead,they collide repeatedly with the metal
atoms,and their resultant motion is complicated and zigzag
Basics of Electrical Engineering UNH
DC circuits 3/25
Resistance
electric eld inside a conductor is zero!only in static
equilibrium
What happens when the charges in the conductor are not in
equilibrium?
current density J (current per unit area)
J =
I
A
= nqv
d
in general it is a vector quantity
J = nqv
d
the current density is in the direction of charge motion for
positive charge carriers and opposite the direction of motion
for negative charge carriers
Basics of Electrical Engineering UNH
DC circuits 4/25
Resistance
A current density J and an electric eld E are established in a
conductor whenever a potential dierence is maintained across
the conductor
Ohm's law
J = E
where  is the conductivity of the conductor
Basics of Electrical Engineering UNH
DC circuits 5/25
Resistance
J = E = 
V
l
expressing V and utilizing that J = I =A
V =
l

J =

l
A

I = RI
R is called the resistance of the conductor,its unit is
=V=A
The inverse of conductivity is the resistivity  (units
m)
The resistance is R = 
l
A
Basics of Electrical Engineering UNH
DC circuits 6/25
Resistor
Most electric circuits use circuit elements called resistors to control
the current level in the various parts of the circuit.Two common
types of resistors are the composition resistor,which contains
carbon,and the wire-wound resistor,which consists of a coil of
wire.Values of resistors in ohms are normally indicated by
color-coding.
Basics of Electrical Engineering UNH
DC circuits 7/25
Temperature dependence
the resistance of a conductor varies approximately linearly
with temperature
 = 
0
[1 +(T T
0
)]
R = R
0
[1 +(T T
0
)]
where  temperature coecient of resistivity.Index 0 refers to
reference point (usually taken at 0

C)
this property enable to measure temperature by resistors:
resistance thermometer also called resistance temperature
detectors RTD
most common device used in industry Pt100 sensors
"Pt"is the symbol for platinum,100 for the resistance in ohm
at 0

C
the sensitivity of a standard Pt100 is nominal 0:385
=

C
termocouples are also used for temperature measurement,but
they based on a dierent principle (see later)
Basics of Electrical Engineering UNH
DC circuits 8/25
Superconductors
class of metals and compounds whose resistance decreases to
zero when they are below a certain (so-called critical)
temperature T
c
the resistance of superconductor below T
c
is 10
17
times
smaller than the resistance of the copper
the phenomenon was discovered in 1911 by the Dutch
physicist Heike Kamerlingh-Onnes (he worked with mercury
(T
c
= 4:15 K))
Basics of Electrical Engineering UNH
DC circuits 9/25
Superconductors
One of the truly remarkable features of superconductors is
that once a current is set up in them,it persists without any
applied potential dierence
Steady currents have been observed to persist in
superconducting loops for several years with no apparent decay
superconductors applied in the development of
superconducting magnets,in which the magnitudes of the
magnetic eld are about ten times greater than those
produced by the best normal electromagnets
it is used in medical magnetic resonance imaging (MRI) units
Basics of Electrical Engineering UNH
DC circuits 10/25
Power
electric power of the resistor
P = I V = I
2
R =
V
2
R
The power of the resistance R is often called joule heating
and it is also often referred to as an I
2
R loss
transporting energy by electricity through power lines we
cannot make the simplifying assumption that the lines have
zero resistance.Real power lines do indeed have resistance,
and power is delivered to the resistance of these wires
Utility companies transport energy at low currents and high
potential dierences
Copper wire is very expensive,and so it is cheaper to use
high-resistance wire (with small A).Thus R is xed to a high
value and the loss I
2
R can be reduced by keeping I as low as
possible
Basics of Electrical Engineering UNH
DC circuits 11/25
Electromotive force
The electromotive force (emf,denoted by ) of a power source
(eg.battery) is the maximum possible voltage that the power
source can provide between its terminals
You can think of a source of emf as a"charge pump".When
an electric potential dierence exists between two points,the
source moves charges"uphill"from the lower potential to the
higher
analogy (emf is the lifting height of the pump)
Basics of Electrical Engineering UNH
DC circuits 12/25
Electromotive force
r is the internal resistance
 is equivalent to the open-circuit voltage
the current when R = 0 is the short circuit current
Basics of Electrical Engineering UNH
DC circuits 13/25
Power curve tting
l
= V
ab
I = I
2
R =
V
2
ab
R
=

2
(R+r)
2
R
total power P
t
= I = I
2
(R +r) =

2
R+r
eciency  =
P
l
P
t
=
V
ab

=
R
R+r
= 1 
I
I
sc
Maximum Power:P
max
= I
sc
=

2
r
= I
2
sc
r
loss:
P
loss
P
max
=
IV
ab
I
sc

=
I
I
sc
(1 
I
I
sc
)
P
l
P
max
=
I
2
r
I
2
sc
r
=
I
2
I
2
sc
Basics of Electrical Engineering UNH
DC circuits 14/25
Resistors in series
the voltage drop from a to b is IR
1
,from b to c is IR
2
The voltage drop from a to c
V = IR
1
+IR
2
= I (R
1
+R
2
)
|
{z
}
R
eq
The equivalent resistance of three or more resistors connected
in series is
R
eq
= R
1
+R
2
+R
3
+:::+R
n
=
n
X
i =1
R
i
Basics of Electrical Engineering UNH
DC circuits 15/25
Resistors in parallel
the potential dierence across the resistance is the same,the
expression V = IR gives
I = I
1
+I
2
=
V
R
1
+
V
R
2
= V

1
R
1
+
1
R
1

=
V
R
eq
1
R
eq
=
1
R
1
+
1
R
2
R
eq
=
1
1
R
1
+
1
R
2
=
R
1
R
2
R
1
+R
2
= R
1
R
2
Basics of Electrical Engineering UNH
DC circuits 16/25
Delta - Star transform
between terminal 1 and 2
R
10
+R
20
= R
12
(R
31
+R
23
) =
R
12
(R
31
+R
23
)
R
12
+R
23
+R
31
between terminal 2 and 3 R
20
+R
30
= R
23
(R
12
+R
31
)
between terminal 3 and 1 R
30
+R
10
= R
31
(R
12
+R
23
)
Basics of Electrical Engineering UNH
DC circuits 17/25
Delta - Star transform
R
10
=
R
12
R
31
R
12
+R
23
+R
31
R
20
=
R
12
R
23
R
12
+R
23
+R
31
R
30
=
R
23
R
31
R
12
+R
23
+R
31
Basics of Electrical Engineering UNH
DC circuits 18/25
Star - Delta transform
by using the same contexture and the conductivity G instead
or resistance (G = 1=R) the star - delta conversion can be
done as
G
12
=
G
1
G
2
G
1
+G
2
+G
3
G
23
=
G
2
G
3
G
1
+G
2
+G
3
G
31
=
G
3
G
1
G
1
+G
2
+G
3
Basics of Electrical Engineering UNH
DC circuits 19/25
Kirchho's rules
Kirchho's Current Law (KCL or junction rule):the sum of
the currents entering any junction in a circuit must equal the
sum of the currents leaving that junction:
X
in
I =
X
out
it is the statement of conservation of electric charge
Kirchho's Voltage Law (KVL or loopr rule):The sum of the
potential dierences across all elements around any closed
circuit loop must be zero:
X
closedloop
V = 0
it follows from the law of conservation of energy
Basics of Electrical Engineering UNH
DC circuits 20/25
Voltage divider
V
2
= IR
2
= V
R
2
R
1
+R
2
=
R
2
R
V = V
 can be varied between 0 and 1
Basics of Electrical Engineering UNH
DC circuits 21/25
Current divider
I
2
=
V
R
2
= I
R
p
R
2
=
R
1
R
1
+R
2
I = I
where R
p
=
R
1
R
2
R
1
+R
2
Basics of Electrical Engineering UNH
DC circuits 22/25
Sources
Voltage source:two terminal device which can maintain a
xed voltage.An ideal voltage source can maintain the xed
voltage independent of the load resistance or the output
current.
Current source:delivers or absorbs electric current which is
independent of the voltage across it
Basics of Electrical Engineering UNH
DC circuits 23/25
Thevenin's theorem
any linear network with voltage and current sources and only
resistances can be replaced at terminals AB by an
equivalent voltage source V
th
in series connection with an
equivalent resistance R
th
This equivalent voltage V
th
is the voltage obtained at
terminals AB of the network with terminals AB open
circuited
The equivalent resistance R
th
is the resistance obtained at
terminals AB of the network with all its independent
current sources open circuited and all its independent voltage
source short circuited
Basics of Electrical Engineering UNH
DC circuits 24/25
Norton's theorem
any linear network with voltage and current sources and only
resistances can be replaced at terminals AB by an
equivalent current source I
NO
in parallel connection with an
equivalent resistance R
NO
This equivalent current I
NO
is the current obtained at
terminals AB of the network with terminals AB short
circuited
The equivalent resistance R
NO
is the resistance obtained at
terminals AB of the network with all its independent
current sources open circuited and all its independent voltage
source short circuited
Basics of Electrical Engineering UNH
DC circuits 25/25