Understanding Mathematical
Electrical/Electronic Relationships
Ohm’s Law
Power [Watt’s] Law
dB Conversion
V=R x I
P=I x E
dBgain=10log[Po/Pi]
I=V/R
P=I2 x R
dBm=10log[Po/1mW]
R=V/I
P=V2/R
Voltage: V = E
Unitless Measure
dBGAIN= 10 log
Power Out (W)
Power In (W)
“Figure of Merit”
V=R x I
V
R
I
I=V/R
R=V/I
Ohm’s Law
Voltage = Resistance x Current
V
R
Resistance = Voltage/Current
I
Current = Voltage/Resistance
Ohm’s Law
Units of Measure
Voltage (volts) = Resistance (ohms) x Current (amperes)
Current (amperes) = Voltage (volts)/Resistance (ohms)
Resistance (ohms) = Voltage (volts)/Current (amperes)
Voltage (Potential)
Voltage is actually a FORCE.
It is a special force: Electromotive Force (emf)
Force normally has a derived unit of measure: Newton (N)
In Electrical terms, we use the unit of measure: volt.
The symbol for voltage is the capital letter “V”.
Current
Current is the measure of electron flow thru a conductor.
Current is like the flow of water in a garden hose.
In Electrical terms, we use the unit of measure: amperes.
The symbol for amps is the capital letter “A”.
Resistance
Resistance is the measure of electron flow impedance thru
a conductor.
Resistance is like Friction in mechanical terms
In Electrical terms, we use the unit of measure: ohm.
The symbol for ohm is the Greek letter Ω (omega)
Resistor
A Resistor is an electronic component of a given value
identified by the color bands marked on the surface.
Resistance of a Conductor
By a derivation of Ohm’s Law, the resistance R of a conductor of
uniform cross section can be computed as
l
R =ρ
A
Where
l is the length of the conductor, measured in meters [m]
A is the cross-sectional area of the conductor, measured in square meters
[m²]
ρ (Greek: rho) is the electrical resistivity (also called specific electrical
resistance) of the material, measured in ohm-meters (Ω m).
Resistivity is a measure of the material's ability to oppose electric current.
Conductance
Conductance (G) is the inverse measure of electron flow
impedance thru a conductor for purely resistive circuits.
G=1/R
R=1/G
In Electrical terms, we use the unit of measure: Siemens.
The symbol for Siemens is the capital letter “S”.
Why the letter “G” for Conductance? There is no particular reason.
The closest explanation is that conductance is often referred to in “mhos” (ohms spelled backwards)…
the Greek letter for “g”: gamma kinda looks like an inverted gamma…. Best guess…
Conductance is not particularly used in DC Circuit analysis… more so in AC Circuit analysis where
Conductance is used in Complex terms as: Y=G+jB [Y and B are susceptance and admittance] in AC
Circuit Analysis.
Ohm’s Law in Circuit Analysis
VDC
I
Simple Series Circuit
R
Ohm’s Law in Circuit Analysis
V
I
Simple Series DC Circuit
Load (Ω)
Ohm’s Law in Circuit Analysis
V
I
Simple Series DC Circuit
Variable Resistor
Direct Current vs Alternating Current
+100 VDC
0 VDC
Direct Current
+110 vac
0 vac
-110 vac
Alternating Current
V=R x I=ohms x amps
V
R
I
I=V/R
R=V/I
Ohm’s Law
mks system (metric)
Fundamental Units
m: meters (distance)
k: kilogram (mass)
s: seconds (time)
All other units are Derived from these three…
This is nice-to-know information that helps fill in the gaps…
“mks” System (metric)
Derived Units
Voltage (Potential) is a derived unit… namely Force
Force in electrical terms is Electromotive Force (emf)
The metric derived unit of Force is the Newton (N).
In “mks” fundamental units: N = mass * acceleration
Or a kilogram (meter per second2) [kg*m/s2]
Voltage and Potential
An Analogy
Gravity
More height = more potential
Improves the current flow
Voltage (Force) and Energy (Work)
Voltage (emf) = N [kg m/s2]
Energy (work) = N*m [kg m2/s2]
Energy is not Power… Power is a rate of energy
consumption.
Energy is measured in Joules…
Power is measured in Joules per second… P = kg m2/s3
Engineering Notation
What is engineering notation?
Engineering notation is a method of writing really big and really small
numbers. In engineering notation, the exponent must always be a multiple of
three so that it is equivalent to a value represented by a metric prefix:
peta
tera
giga
mega
kilo
P
T
G
M
k
milli
micro
nano
pico
femto
m
µ
n
p
f
1,000,000,000,000,000
1,000,000,000,000
1,000,000,000
1,000,000
1,000
1
0.001
0.000001
0.000000001
0.000000000001
0.000000000000001
1015
1012
109
106
103
100
10-3
10-6
10-9
10-12
10-15
Scientific Notation
How is engineering notation different from scientific notation?
Engineering notation is written the same way as scientific notation, except
that the exponent is always a multiple of three. Because of this, in
engineering notation you have to move the decimal place at times.
So while in scientific notation you always put the decimal after the first digit
(such as 12300000 = 1.23 X 107), in engineering notation you move the
decimal to accommodate the exponent needing to be a multiple of three.
So the same number 12300000 in engineering notation would be displayed
12.3 X106. This is useful so we can attach a metric prefix to a quantity, such
as 12.3 MB (Mega Bytes) and put numbers into a language we can more
easily understand.
Typical Values for Electronics
Voltage
Current
Resistance
Mega Volts (MV)
Mega Ohms (MΩ)
Kilo Volts (kV)
Kilo Ohms (kΩ)
Volts (V)
Amperes (A)
Ohms (Ω)
milliVolts (mV)
milliAmps (mA)
micorVolts (μV) microAmps (μA)
Power
Mega Watts (MW)
Kilo Watts (kW)
Watts (W)
milliWatts (mW)
How much current can kill you?
As shown in the chart, shock is relatively more severe as the current rises.
For currents above 10 milliamps, muscular contractions are so strong that
the victim cannot let go of the wire that is shocking him. At values as low as
20 milliamps, breathing becomes labored, finally ceasing completely even at
values below 75 milliamps.
As the current approaches 100 milliamps, ventricular fibrillation of the heart
occurs - an uncoordinated twitching of the walls of the heart's ventricles
which results in death.
Above 200 milliamps, the muscular contractions are so severe that the heart
is forcibly clamped during the shock. This clamping protects the heart from
going into ventricular fibrillation, and the victim's chances for survival are
good.
Ohm’s Law in Circuit Analysis
Solve for Voltage
V=?
I=0.5mA
R=200kΩ
V=R x I
V=(200x103Ω) x (0.5x10-3A)= (200x0.5)x(103-3)
V=100x100 volts, recall 100 = 1
V=100 volts
Ohm’s Law in Circuit Analysis
Solve for Resistance
V=100V
I=0.5mA
R=V/I
R=100V/(0.5x10-3A)
R=2(100V)x103 A, k=103
R=200k Ω
R=?Ω
Ohm’s Law in Circuit Analysis
Solve for Current
V=100V
I=?A
R=200kΩ
I=V/R
I=100V/(200x103Ω) x (0.5x10-3A)= (200x0.5)x(103-3)
I=0.5x10-3 A, m=10-3
V=0.5mA
Watt’s Law in Circuit Analysis
Solve for Power
V=E=100V
I=0.5mA
P=IxE
P=(0.5x10-3A)x100V
P=50x10-3 W
P=50mW
NOT … P=50W
R=200kΩ
Watt’s Law in Circuit Analysis
Solve for Power
V=100V
I=0.5mA
P=I2xR
P=(0.5x10-3A)2x200kΩ
P=(0.25x200)(10-3-3+3)
P=50x10-3 W
P=50mW
R=200kΩ
Watt’s Law in Circuit Analysis
Solve for Power
V=100V
I=0.5mA
P=IxV=(V/R)xV
P=V2/R
P=(100V)2/200kΩ
P=(100)(100)/2x(100)(103)
P=50x10-3 W
P=50mW
R=200kΩ
Watt’s Law in Circuit Analysis
Solve for Power
V=100V
I=0.5mA
P=IxE
E=P/I
E=50mW/0.5mA
E=2(50)=100V
R=200kΩ
Another look at “mks” units
A logical progression
Fundamental Units
m: meters (distance)
k: kilogram (mass)
s: seconds (time)
Fundamental
Unit
m: (distance)
y(m)
0
x(m)
Derived Units
m/s: (velocity)
m/s2: (acceleration) kg*m/s2: (Force)
Voltage
EMF
Potential
kg*m2/s2: (Energy)
kg*m2/s3: (Power)
Work
Newton meter
Joule
Joule/second
Newton meter/sec
Power is a Rate.
Other terms to express the same thing!
Decibel
A decibel is one tenth of a bel.
The unit bel is too large for our purposes.
The decibel is a logarithmic function.
Used in audio measurements to rate the intensity of sound.
dB Conversion
Given:
Pout=35W
Pin=35mW
dBgain=10log[Po/Pi]
dBgain=10log[35W/35mW]
dBgain=10log[1000i]
dBgain=10 x 3
dBgain=30dB
The area of a typical eardrum is about 5.0×10-5 m2. Calculate the sound power (the energy per
second) incident on an eardrum at the threshold of hearing and at the threshold of pain.
The sound power incident on the eardrum is P = IA, where I is the intensity of the sound
and A = 5.0×10-5 m2 is the area of the eardrum.
At the threshold of hearing,=
I 1.0 ×10−12 W
℘
= IA
=
(1.0 ×10
−12
W
( 5.0 ×10
m )
2
−5
m2 )
℘hearing =5.0 ×10−17 W
At the threshold of pain, I = 1.0 W
℘
= IA
=
(1.0W m ) (5.0 ×10
2
℘pain = 5.0 ×10−5W
−5
m2 )
m2
, and
m2
, and
dB Conversion
Given:
Pout=35W
Pin=35mW
dBm=10log[Po/1mW]
dBm=10log[Pi/1mW]
dBm=10log[35W/1mW]
dBm=10log[35mW/1mW]
dBm=10log[35x103]
dBm=10log[35]
dBm=10 x 4.54
dBm=10 x 1.54
dBm= 45.4 dB output
dBm=15.4 dB input
Po-Pi = 45.4 dB -15.4 dB = 30dB