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ELECTRICAL CIRCUITS
1. THE ORIGIN OF ELECTRICAL ENGINEERING
Introduction
The purpose of this section in The Book of John (BOJ) is to examine the genetics of Electrical
Engineering and it is the Physics of Electromagnetics. The language of Electrical Engineering is
the same as Physics: Mathematics. In this hand-out we will take a fast and abbreviated trip
through Electromagnetics to obtain some of the very basic concepts of electricity. We assume
that the reader has been exposed to these concepts and the purpose of this work is to give an
aerial view to tie all these concepts together. From the Physics of Electromagnetics we will
develop the basis for voltage, current, charge, loop equations, node equations, resistance, force,
energy, power, inductance, capacitance, motors, generators and transformers. We will start with
electric charge, electric field, electric forces, voltage, current, resistance and power. We then
move on to magnetism, related to current, magnetic field, magnetic forces, time varying
magnetic field and time varying electric field. The fundamental units of Electrical Engineering
(the MKS system: Meter, Kilogram, Second) are: length, the Meter; mass, the Kilogram; time,
the Second and current, the Ampere (Why was it not named the MKSA system? I believe the
new name of the “SI” system was an attempt to address this.) The first 3 units are obvious but
the last, current, is yet to be explained. Everything else is derived from these 4 units (force,
pressure, power, energy, velocity, acceleration, temperature, etc.). In general, Engineering
applies the laws of Physics to design, develop and build things for the benefit of humanity.
Electrical Engineering is the application of the laws of Physics, principally in the area of
Electromagnetics to do that same thing.
Electric Fields
Electric fields are caused in 2 ways:
1. Electric charge
2. Changing magnetic field (to be developed later)
Electric fields originate from electric charge as a gravitational field originates from mass.
Electric charge is what makes your clothes stick together when taken fresh out of a clothes dryer.
In order to understand the concept of charge we must engage in “circular reasoning”. An electron
has a quantity of charge of: 1.6 1019 coulombs. Electric current, I is the flow of
dQ 

charge, Q  I 
 . A coulomb is the negative of the cumulative flow of charge of
dt 

1.6 10 
19 1
electrons in one second for a current of 1 ampere or through some arbitrary cross
sectional area. In other words a coulomb is the cumulative charge for a flow of a current of 1
ampere for one second. Additionally, a proton has a positive charge of 1.6  1019 coulomb.
Charges apply a force between each other. Opposite charges attract like charges repel. That force
is given by Coulombs law:
2
F
Q1Q2
4 R122
(1)
Where: F is force with a direction between the charges, newtons
Q1,2 is charge assumed to be a point, coulombs
R1,2 is distance between charges, meters
2
4
 is a constant that is a property of the medium F Mt  Amp sec
(F is farads)
Mt 3 Kg
In air or a vacuum,  0  8.854 1012 F
. An electric field can originate from a charge. If it is a
Mt
positive charge the field shines out from the charge. If negative it shines in to the charge. If we
bring a 1 coulomb test charge into an electric field the test charge will experience a force acting
on it. That force is the direction of the electric field and that force normalized to the 1 coulomb
test charge is the magnitude of the electric field:

 F
(2)
E
Q

Where: F is force a vector, units are newtons
Q is a unit test charge, units coulomb

E is electric field a vector, units newton per coulomb
Figure 1 illustrates a point positive charge with the electric field shinning out from it.
Figure 1 Electric field emitting from a positive point charge
From Equation 2 we see that if test a charge is within an electric field it experience’s a force
acting on it. Equation 2 is re-arranged to illustrate.


F  QE
(2a)
3
If the charge is moved around work is performed:
 
 
W   F  ds   QE  ds
(3)
In Equation 3 we have the general physical concept of work, W a scalar with units in joules of
moving something with a force over a distance where the force F is a vector with units of
newtons and the differential distance ds is a vector with units in meters. If we plug in Equation 2
we obtain the relation in terms of charge, Q a scalar with units in coulombs and electric field, a
vector E with units of newtons per coulomb. In Equation 4 we have normalized the work to the
charge to obtain V, potential a scalar with units of voltage or joules per coulomb.
 
WA, B B  
W
V
 E  ds , VA, B 
 E  ds
Q 
Q A
(4)
Additionally, we see that the normalized work in moving a unit test charge from point A to point
B is as shown. Voltage is a derived unit but it is considered one of the fundamental units for
Electrical Engineering. For example E, electric field is typically expressed as in units of
Volts/Mt which is the same as Newtons/Coulomb. We also note that electric field is exactly like
a gravitational field and if we move our test charge right back to the starting point no work will
be done, AKA it is conservative. This is in fact why the sum of the voltage drops around any
closed loop in an electric circuit will always total to zero (Kirchhoff’s voltage law). A potential
source such as a battery is common and readily obtainable. An electric field across a distance can
easily be obtained using a potential source. If we apply a potential source such as a battery across
a length of a medium with a uniform cross sectional area that has a population of free charge
carriers an electric field will be established in the material. From Equation 2 a force will be
applied to the charge carriers in the medium and that force via Newton’s first law will accelerate
those carriers. They will collide with the material atoms and transfer their kinetic energy to the
atoms resulting in heat in the material. The acceleration process will start over again. The
physical property of the material that sets the degree of this process is conductivity it is a
measure of the density of charge carriers. Conductivity has units of Ampere/(Volt x Mt). The
complete description of the process includes the geometry of the material and is described in the
parameter of resistance, with units of Ohms or Volts/Ampere. The geometry relationship for
resistance is given by Figure 2 and the resistance is given by Equation 5.
Conductive Material
Test Leads
4
Figure 2 the geometry of resistance
R

A
(5)
Where: R is resistance, ohms or Volts/Ampere
A is cross sectional area, (Figure 2), Mt 2
 is length, (Figure 2) Mt
 is conductivity, Ampere / (Volt  Mt )
The schematic of Ohm’s Law is given in Figure 3 and the Equation of Ohm’s Law is given by
Equation 6.
Figure 3 Schematic of Ohm’s Law
V1  V2   IR
(6)
Where: V1  V2 is the potential difference, Volts
I is current, ampere
R is resistance, Ohms or Volts/Ampere
The Ohm is a derived unit but it also is considered one of the fundamental units for Electrical
Engineering. Recall from the description of the physical process of charge flowing through the
medium releases heat or power. Equation 7 gives that power relation
PI
2
V  V 
R 1 2
R
2
 V1  V2  I
(7)
Where: P is power, Joules/Sec, Watts
The Watt is a derived unit but it also is considered one of the fundamental units for Electrical
Engineering. It is left as an exercise to show that the units of Equation 7 are indeed that of
power. The unit for current, the Ampere is the only fundamental unit in Electrical Engineering
that is also a fundamental in the MKS system.
5
Magnetic Fields
Magnetic fields are caused in 2 ways:
1. Flowing electric charge, current.
2. Time changing Electric field
There is in nature no magnetic charge. An electric current will cause a magnetic field. A
magnetic field is solenoidal and unlike an electric field where the lines of force source from a
charge and shine out to infinity, the lines of force in the magnetic field loop to form a closed
path. Figure 4 illustrates the lines of force of a magnetic field from a long current carrying
conductor.
Figure 4 the Magnetic field caused by a current carrying conductor.
Ampere’s Law gives the relationship between a current carrying conductor and magnetic field:
 
m0 I = ò
(8)
 B⋅d
Where: I is current, Ampere

B is magnetic field a vector T, Tesla

d  is differential path length a vector, Meters
T ⋅ Mt
m0 is the magnetic permeability of air,
Ampere

The units of B are given as Tesla, a derived unit and not fundamental in the MKS system.
Additionally, the proportionality constant m0 is for air or a vacuum, in fact m0 = 4p ´10-7 in the
T ⋅ Mt
MKS system. Also the units of m0 ,
are not fundamental in the MKS system. The units
Ampere

for B and m0 will be decomposed into fundamental units later in this section. Equation 8 can be
solved for magnetic field if the geometry is limited to the simple case of Figure 3.
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Figure 3 Simplified geometry of magnetic field for a long current carrying conductor.

Observe in Figure 3 at a constant radial distance R from the conductor, the variable B is constant
and factors out of the integral of Equation 8:

mI
m0 I = B ò
 d  = B2p R, B = 2p0 R
(9)
Equation 9 gives the magnitude of the magnetic field and not the direction. The direction is
given by the “Right Hand Rule” as shown in Figure 4.
Fingers point in direction of magnetic field
Thumb points in direction of current
Figure 4 The “Right Hand Rule”
With the right hand, the thumb points in the direction of current flow in the conductor with the
fingers wrapping around the conductor giving the direction of the magnetic field.
MAGNETIC FORCES
As seen previously, an electric field imposes a force on an electric charge as per Equation 2a and
will accelerate it in exactly the same or opposite direction as the electric field. A magnetic field
will also impose a force on an electric charge and can accelerate it but in a totally different way.
First the charge will only experience the force if it is moving while in the magnetic field.
Additionally, that force is perpendicular to both the direction of the magnetic field and the
direction of the velocity of the charge. That cross product relationship is given by Equation 10.

 
(10)
F = QU ´ B
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
Where: F is force a vector, Newtons
Q is charge, coulombs

U is charged particle velocity, Mt/sec

B is magnetic field, a vector, units Tesla
The geometry of Equation 10 is illustrated in Figure 5
Figure 5 the directional relationship of Equation 10
Equation 10 gives us the unit relationships necessary to obtain the fundamental MKS units for
Tesla, the units of magnetic field. From Equation 10 we can obtain Equation 11 that result for
Tesla.
Mt
Newtons  Ampere  sec
 Tesla
sec
From Newton’s first law: F  MA, Newtons  Kg
Kg
Mt
sec 2
Mt
Mt
 Ampere  sec
 Tesla
2
sec
sec
Tesla 
Kg
Ampere  sec 2
(11)
In Equation 10 assume for the geometry that everything is perpendicular and normalize the force
to the charge observe that we obtain an electric field:
F = QUB 
F
= UB  E = UB
Q
(12)
Now, consider that we have a conductor (lots of free charge carriers in it) of length D moving
through the magnetic field B where the conductor is perpendicular to the magnetic field and the
velocity of the conductor U is perpendicular to the magnetic field and the conductor as Figure 5.
Equation 11 defines an electric field over the length of the conductor and if we integrate over the
length of that conductor we obtain a potential difference between the 2 ends of the conductor and
this is how a generator works. Equation 13 gives the generator relationship.
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Figure 5 Moving conductor in a magnetic field for a generator
VA, B = UBD
(13)
Where: VA, B is the voltage developed across D , Volts
B is the magnetic field, Tesla
D is the conductor length, Mt
U is the conductor velocity, Mt/sec
In a similar way, if the conductor in the magnetic field is carrying a current with the orientation
of Figure 6 then a force will be exerted on the conductor and this is how a motor works. Equation
14 describes the vector relationship.
Figure 6 Current carrying conductor in a magnetic field for a motor

 
F = I D ´ B

Where: F is the force developed on D , Newtons

B is the magnetic field, Tesla

D is the conductor length, Mt
I is the current in the conductor, Ampere
(14)
9
Time Varying Electric and Magnetic Fields
We stated in previous sections that an electric field can be caused by a time varying magnetic
field and a magnetic field could be caused by a time varying electric field. We now examine
these situations. We start with the changing magnetic field. Consider the Faraday experiment: a
bar magnetic thrust into a loop conducting wire hooked to a galvanometer will cause the needle
to flick due to a current flow. Figure 7 illustrates a cartoon of the experiment.
Figure 7 The Faraday experiment
The polarity of the induced current flow is such that the magnetic field from that current is
opposite the magnetic field increase caused by the moving magnet. The results from this
experiment yielded Faraday’s Law given by Equation 15:
VInduced = -N
df
dt
(15)
Where: VInduced is the potential developed in the loop, Volts
f is the total magnetic flux shinning through the coil, Tesla ⋅ Mt 2
N is the number of turns in the loop
If we assume that the magnetic field is uniform across the area of the loop then the total flux and
the derivative of the flux are:
f = BA,
df
dB
=A
dt
dt
(16)
Where: B is the magnetic field in the loop, Tesla
f is the total magnetic flux shinning through the loop, Tesla ⋅ Mt 2
A is the loop area, Mt 2
The minus sign in Faraday’s Law is the “no free lunch” factor. Without that minus sign, the
induced current would cause a magnetic field that aids the existing magnetic field and the bar
magnet would be accelerated into the coil yielding “free” energy something that never happens
in nature and only occurs in the minds of the ignorant and politicians. Figure 8 illustrates
10
Faraday’s Law with the polarity of the induced current I opposing the changing magnetic
df
flux
along with a resultant voltage Vab developed across a load resistor RL .
dt
Figure 8 Faraday’s Law showing polarities
Now, assume we have a perfect coil with N turns, wound with a no resistance wire and a
potential V applied to it. Figure 9 illustrates this.
Figure 9 Potential applied to a perfect coil
When S1 closes the potential V will try to drive a current into the coil and this current will
generate a magnetic field B as per Ampere’s Law:
 
m0 NI = ò
 B⋅d
Figure 10 the geometry details of this coil:
(17)
11
Figure 10 Perfect coil details
The geometry is such that the magnetic field B is in the direction as shown where the coil has
cross section area A and length  . Additionally, the changing B will cause a potential that bucks
the applied potential V and if the switch S1 and the wire for the coil and connection are perfect,
then the buck voltage will exactly equal the applied potential V . If we assume that the magnetic
field is perfectly uniform and all contained within the coil then from Ampere’s Law:
 
m0 NI = ò
 B ⋅ d  = B
(18)
Additionally, the magnetic flux within the coil is given by: f = BA . Thus Equation 18 becomes:
  f
m0 NI = ò
 B⋅d = A 
f=
m0 NI
A

Now, from Faraday’s Law the buck voltage becomes:
m NA dI
df
VBuck = -N
= -N 0
dt
 dt
(19)
(20)
(21)
Now by Kirchhoff’s Law:
m0 N 2 A dI
=0

dt
m NA dI
m N2A
, L= 0
V= 0
 dt


V + VBuck = 0, V -
L
(22)
12
As seen in Equation 22 we have developed an expression for inductance and thus the basis for
the V , I relationship of the inductor Figure 11 and Equation 23
Figure 11 V , I relationship of the inductor
dI
1
V = L , I = ò Vdt
dt
L
(23)
We have derived the voltage, current relationship for the inductor. This development came out of
Faraday’s Law. If we have 2 coils aligned such that flux developed in one coil shines into the
second coil we have the basis of a transformer. The reader is referred to hand-out in the circuits
portion of the BOJ where the concept of the transformer is developed in detail.
We now develop the concept of the capacitor. Recall the electric field from a point charge as
seen in Figure 12 is given by Equation 24.
Figure 12 Electric field from a point charge
E=
Q
e0 4p R 2
(24)
We state without proof that the electric field from an infinite flat conductive surface with a
Coul
surface charge density of rs
is perpendicular to the surface and points out if a positive
Mt 2
charge:
r
(25)
E= s
2e
13
Where: e is the dielectric constant of the material
Figure 13 Electric field from a surface charge density
Notice that if the area of the plate is infinite that the field is independent of distance from the
plate. Now consider the geometry of a capacitor as seen in Figure 14: 2 flat parallel conductive
plates of area A separated by a distance d where d << A and the material between the plates has
a dielectric constant of e .
Figure 14 Geometry of a capacitor
If the capacitor has a charge of + Q on the top plate and - Q on the bottom plate the respective
+Q
-Q
charge densities are: rs =
on top and -rs =
on the bottom. The electric field is the sum
A
A
of the fields from both plates and goes from the top to the bottom:
r
Q
(26)
E= s =
e
eA
The potential between the plates is given by:
Top
VTop / Bottom =
ò
E ⋅ d  = Ed =
Bottom
Qd
eA
But Q = ò Idt and thus:
VCap =
d
1
eA
Idt = ò Idt and thus C =
ò
eA
C
d
(27)
14
And thus the voltage current relationship of the capacitor:
Figure 15 V , I characteristics of the capacitor
1
VCap = ò Idt
C
dV
I Cap = C
dt
Let us work backwards to electric and magnetic fields:
d
VCap = Ed =
Idt
eA ò
1
d
E=
Idt , now
both sides:
ò
eA
dt
 
dE
I
=
but m0 I = ò
B ⋅ d  and thus:

eA
dt
1  
 B⋅ d
dE m0 ò
=
eA
dt
(28)
(29)
Equation 29 shows that a time changing electric field causes a magnetic field. We had previously
shown that (Faraday’s Law) that a time changing magnetic field causes an electric field. In
summary:








We explained electric charge
We defined electric field
We explained the force on an electric charge in an electric field
We defined electric potential
We explained electric current
We explained magnetic field
We explained magnetic forces on an electric charge
We explained the generator action
15





We explained the motor action
We explained how a changing magnetic field can cause an electric field
We derived the concept of inductance
We derived the concept of capacitance
We explained how a changing electric field causes a magnetic field
In all this we have shown that the origin of Electrical Engineering is the Electromagnetic field of
Physics.