source of electromotive force
A device that, by doing work on charge carriers, maintains a
potential difference between its terminals is called a source of
electromotive force (emf).
Other form of energy is converted into
electricity in a source of electromotive force:
battery
electric generator
solar cell
thermopile
living cell
-
chemical energy
mechanical energy
electromagnetic radiation
internal energy
chemical energy
electromotive force
The maximum electric potential difference that can exist
between the terminals of the voltage source is called the
electromotive force of that source.
Voltage produced by a real source of electromotive force:
r
+

-
I
V  V    Ir
direct and alternating current
+
++
+ +
+ +
+
+
+
+
+
+
+
+
+
++
+ ++
+ + ++ + +
+ ++ +
+ ++ +
++
+
+ ++
+ + ++
+ ++
+ ++ + + +
+ ++ +
+
+
+
+ + + ++
++
+
++ + + ++++ + +
+
++ + +
++ +
++ +
++ + +
If the charge moves in a circuit in the same direction at all times, the
current is said to be direct current (DC). Constant current (independent
of time) is a special case of direct current.
If the charges move (across a surface) changing their direction of
motion, the current is said to be alternating current (AC).
circuit analysis
Kirchhoff's Junction Rule:
The sum of all the currents
entering a junction is zero.
Kirchhoff's Loop Rule:
Around any closed circuit loop
the sum of potential differences
is zero.
V2
In
I1
Vn
I2
 Ii  0
i
V1
 Vi  0
i
electrical measurements
Current is measured with an ammeter, which must be inserted into the
circuit in series with the element in which the current is measured.
Voltage is measured with a voltmeter, which must be inserted into the
circuit in parallel to the elements across which the voltage is measured.
The resistance of passive elements can be measured with an ohmmeter.
V
-
?
V
A
+

electric current & the human body
Currents of 200 mA can be fatal. A current that strong can
affect the proper operation of the heart.
A current above 100 mA can cause muscle spasm.
A person can sense an AC with a current of 1mA.
NEVER TOUCH AN OPERATING CIRCUIT WITH
BOTH HANDS !!!
inductors
An inductor is an element of a circuit with two sides for which
(at any instant) the potential difference V between its terminals
is proportional to the rate of change in current I passing
through this element.
I
Va
Vb
dI
Va  Vb  L
dt
The proportionality coefficient L is called the inductance of
the inductor.
In SI the henry is the unit of inductance
Vs
H
A
Va(t)
sinusoidal
alternating current
Vb(t)
I (t)
For a sinusoidal alternating
current, both the voltage V(t)
across an element and the
current I(t) through this element
are sinusoidal functions of time.
V( t )  Vm sin ( t  v )
I( t )  Im sin ( t  I )
I
V
t
Vm, Im
- the peak value
(t+)
- the phase
 = 2f
- the angular frequency

- the initial phase
electric power
P
Pav
V
Electric power delivered to an
electrical element is a sinusoidal
function of time.
t
I
P(t )  It   Vt   I m Vm  sin( t   I )  sin( t   V ) 
1
 I m Vm  cos V   I   cos2t   I   V 
2
Pav  I rms Vrms cos 
where
  V  I

f rms  f t 2

av
1
2
AC in the US
standard one phase power line
breaker
120 V
“zero”
0V
“hot”
“ground”
0V
0V
The voltage oscillates with frequency f = 60 Hz.
AC in the US
three phase power line
2
Three "hot" wires with phases differing by
,
3
the "zero" and the "ground" wires are connected
to the outlet.
V
t
The rms voltage between any "hot" wire and the zero wire is 127 V.
The voltage between any two "hot" wires is 220 V.
complex voltage
The complex function V(t) such that the voltage across the element is
V(t) = Im V(t)
is called the complex voltage.
V
Im V
V (t)
Vm
t
t
t+V
Re V
-Vm
Sinusoidal voltage:


V  t   Vm e i V e it  V0  e it
Im V t   ImVm cost  V   i sin t  V   Vm sin t  V   V t 
complex current
The complex function I(t) such that the current through the element is
I(t) = Im I(t)
is called the complex current.
I
Im I
Im
I (t)
t
t
t+I
Re I
-Im
Sinusoidal current:


I t   ImeiV eit  I0  eit
Im I t   ImI m cost  V   i sin t  V   I m sin t  V   It 
relation between voltage and current
Vm
Vm = Im·Z
Im
t  V   t  I   

V
I
t
The coefficient Z relating the peak values of the voltage across the
system with the peak value of the current through the system is called
the impedance of the system.
The number  relating the phase of the voltage across the system
with the phase of the current through the system is called the phase
angle between the current and the voltage
note that: Vrms = Irms·Z
and
V = I + 
complex impedance
The complex coefficient Z, relating the complex voltage across
an element with the complex current through this element, is
called the complex impedance of this element at frequency :
V
Im
V t   I VttZ  I Vt 0,Z
 I0,   Z
I
Z





V  VmeiV  eit


V

  m ei   Imei I  eit  Z  I
 Im

Z

Re
Complex impedance Z includes information about both the
impedance Z as well as about the phase angle
Z Z
Im Z
tan  
Re Z
AC in a resistor
I
real analysis
I m sin t   I 
V t   R  It   I m  R
a

R
impedance and phase angle
b
ZR   R ,
0
average power:
V
Pav  I rms  Vrms cos 0
2
Vrms
2
 IrmsR 
R
I
t
AC in a resistor
complex
analysis
I
V t R RI I Im
t  R  I t 
Im V t   V
a
complex impedance

R
ZR() = R
b
ZR  Z  R ,
tan  
Im Z
0
Re Z
Im
I
ZR
V
V
Re
I
t
AC in an inductor
I
real analysis

d
d

Im sin t  I  =
It   L
dt
dt


 L  I m cost  I   L  I m   sin  t   I  
2

V t   L 
a
L
b
impedance and phase angle

ZL   L ,  
2
average power:

Pav  Irms  Vrms  cos  0
2
I
V
t
complex analysis
I
d
dI
dI 
dI
it 
t  VLt   V  L  I0eIm Li L  I t 
VIm
dt
dt
 dt 
a

complex impedance
L
ZL() = iL
b
ZL  Z  L ,
tan  
Im Z

Re Z
Im
V
ZL
I
V
I
Re
t
AC in an capacitor
real analysis
I
a

Q
C
-Q
b
d
t t CVm  sin t  V   C  Vm cost  V  
 V
ItQ
dt




 CVm  sin  t   V  
2

impedance and phase angle

1

ZC  
,
2
C
average power:
 
Pav  I rms  Vrms  cos    0
 2
V
I
t