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Rapid Practical Electronic Design

Learn How to Design Analog and Digital Electronic Circuits

Copyright 2019 Richard Walbaum

Create a project to design after every lesson. Grappling is your friend. Makes you strong.

Table of Contents - Resistors

Resistors...................................................................................................................................................................2
Chapter 1: Properties of Resistors...........................................................................................................................3
Resistor Color Code............................................................................................................................................4
Ohm's Law..........................................................................................................................................................6
Resistor Power Derating Curve...........................................................................................................................6
Difference Between Power and Energy..............................................................................................................7
Power in a Resistor..............................................................................................................................................7
Noise in a Resistor..............................................................................................................................................8
Resistor Types and Properties.............................................................................................................................9
Resistor Table....................................................................................................................................................10
Potentiometers...................................................................................................................................................11
Resistor Networks.............................................................................................................................................12
Batteries in Series and Parallel.....................................................................................................................12
Resistors in Series and Parallel....................................................................................................................13
Analyzing a Simple Battery and Resistor Circuit........................................................................................14
Analyzing a Simple Battery and Two Resistor Circuit................................................................................15
Solving Resistor Networks...........................................................................................................................16
Kirchhoff's Circuit Laws.........................................................................................................................16
Voltage Division......................................................................................................................................17
Source Transformation............................................................................................................................18
Thevenin's Theorem................................................................................................................................20
Kirchhoff”s Loop Equations....................................................................................................................21
Chapter 1 Summary...........................................................................................................................................23
Chapter 1 Problems...........................................................................................................................................25

1
 Resistors

General purpose ¼ watt carbon film1 Precision ¼ watt metal film2 Metal Oxide power3

Precision wirewound Aluminum housed Wirewound power

.005% wirewound power, 50 watt4 1,000 watt5

Wirewound power, adjustable versions6 Multiple resistors in one package7

Chip resistors,8 Chip resistor enlarged,9

have no leads surface mount device (“smd”)

2
Chapter 1: Properties of Resistors

“The simplest circuit might be a battery, wire, and a resistor. The battery is like a water pump and provides
electrical pressure; the wire is like a pipe, and the resistor is like a constriction in the pipe. The same way that
water flows in pipes to get from point A to point B, electricity flows through wires to get water from point A to
point B. Just as there is always water in the entire pipe, there are always electrons in the wire. Current doesn't
start at the pump, or at the battery; it is continuous throughout the pipe or wire; but this does not apply in really
high speed circuits where it takes time for a pulse to travel down the wire. The purpose of a resistor is to
constrict the flow of electrical current to a known desired value, just like a blockage or valve in a water pipe.”10:

10,000 (“10K ohms” or “10kΩ”)

Current flows
from + to -. Constriction
i = 1mA
(m = .001) Pump
“1 milliamp”

11
Resistors are the most common
component used in electronics, so you
need to master their use. They come in
different types (which emphasize some
properties), size (which determines their
wattage ratings), and resistance.

Georg Simon Ohm

1789 - 1854

3
Resistor Color Code

The value of a resistor is usually specified by a color code:

12 13
2.7M Resistor with 5% tolerance

To distinguish left from right there is sometimes a gap between the C and D bands.
Band A is the first significant figure of component value.
Band B is the second significant figure.
Some precision resistors have a third significant figure, and thus five bands.
Band C is the decimal multiplier, the number of trailing zeroes.
Band D indicates the tolerance of value in percent (no band means 20%).

Gold signifies that the tolerance is ±5%.

Silver signifies that the tolerance is ±10%
In the above example, a resistor with bands of red, violet, green, and brown has first digit 2 (red; see table
below), second digit 7 (violet), followed by 5 (green) zeroes: 2700000 ohms, or 2.7MΩ.
Tight tolerance resistors may have three bands for significant figures rather than two, or an additional band
indicating temperature coefficient, in units of ppm/K (parts per million per degree Kelvin).
Zero ohm resistors are marked with a single black band.
The standard color code per IEC 60062:2016 follows; you can also see a table in the footnote14:

4
5
Ohm's Law
“The most fundamental and important equation in electronics is Ohm's law. Ohm's law states that the voltage
across a resistor is equal to the current flowing through it, times its resistance:
voltage = current x resistance; V = iR
“Here is an easy way to remember the formula:” Memorize
This

Cover the one you are looking for with your finger;
the formula to find it remains. If you cover current,
you divide voltage by resistance. If you cover voltage,
multiply current by resistance.

V – Voltage or electric pressure (The letter e is often used. Lower case v often means AC (alternating current),
upper case V often means DC (direct current i.e. constant voltage.)
i – Current (engineers use j for the imaginary number to avoid confusion with i which is reserved for current).
R – Resistance.

Resistor Power Derating Curve

I put this slide on the projector:

Electron entering a resistor Exiting the resistor all hot

“Notice the symbol for resistance is a jagged line. When electrons bounce through those jagged lines, they get
hot. You must be careful not to overheat them or they will get a stroke.”

Then Shristi said “Now you're just being silly!”

I continued “You can operate the resistor at its rated power up to an ambient temperature of 70°C; then you
must reduce its power linearly according to the following chart. Derating is typical of many components
including transistors.”

“A resistor must be derated when operating at elevated temperatures; check its data sheet. Here is an example
for a Stackpole CF/CFM series carbon film resistor.” I put this slide on the projector:

6
Difference Between Power and Energy
“When current flows through a resistance it produces heat, and you have to select your resistor to be able to
handle it. Resistors are rated by the amount of power in watts that they can handle, so we need to understand
what power is and be able to calculate it. First lets look at the difference between power and energy.
“Energy, E, is like jewels in a bucket, its quantity measured in joules or watt-seconds, watt-hours (=3,600 watt-
sec), or kilowatt hours (=3.6M watt-sec). When it sits there doing nothing, it is still energy, it is potential energy.
When you pour the energy out, the flow of energy is measured in watts. If it contains 100 watt-seconds of
energy, you can pour it all out quickly, 100 watt-seconds in 1 second, or you can pour it all out slowly, 1 watt-
second per second for 100 seconds.

Energy is measured in
watt-seconds or joules.
Remember it as “jewels”
in a bucket. Energy is
the quantity contained
in a bucket.

Power is the flow of

energy measured in
joules per second, or
watts.
James Joule
1 joule/second = 1 watt
1818 - 1889

“To state it mathematically”:

Energy = watts x seconds = 1 joule15 Energy has been given its own special name
Power = watts (the rate of producing or
using energy; joules/second)

Power in a Resistor
“When current flows through a resistor, it gets hot.”

7
 16

The power rating of a

resistor (in watts) is the rate
of energy that a resistor can
safely dissipate as heat.

James Watt

1736 - 1819
“The power in a resistor is the product of the voltage times the current”:
P = Vi
“Since V = iR, and i = V/R, you can substitute for V and for i in the above equation and get other useful forms”:
P = (iR) x i = i2R
Power = Vi Memorize
P = V x (V/R) = V2/R Power = i2R
Power = V2/R This
“When you design a circuit, to be conservative, select a resistor
with a power rating that is about twice the operating power.
Shristi asked “What happens if you overload a resistor?”
I answered “A slightly overloaded resistor will change color as it slowly burns up, and its value will change
with time. Really overloaded, it can smoke or burst into flames, even explode, but some resistors are designed
to be flame proof, and some are described as fusible resistors”,17 they blow gently.
“For higher power, wire-wound resistors are used, and come in rated powers to thousands of watts.”

Noise in a Resistor

I added: “Remember the symbol for resistor, you might expect electrons would make a lot of noise bouncing all
around those lines.”
Then Shristi said “Silly silly silly!”
“But .... we have just learned that resistors do get hot trying to keep those poor electrons cool when flowing
through them. And they do generate noise that shows up as hiss in an audio circuit. It is like the flow of water
through pipes; you can hear it. For most applications, noise from a resistor does not matter. But some cases such
as audiophile preamplifiers and photodetectors require low-noise design.”
“It is really complicated so I will just touch on it to make you aware it exists. Do your own research if you need
to know; see the footnotes. The most relevant types of noise for audio applications are thermal noise and current
noise. Thermal noise18 (also known as Johnson19 or Nyquist20 noise) is generated by the thermal agitation of
electrons inside a conductor which happens regardless of any applied voltage; it doesn't matter what type of
resistor, they all produce the same amount of thermal noise.

Thermal noise voltage en = √4kTRB.

k = Boltzmann's constant 1.38x10-23, T=temperature in Kelvin, R=resistance,
B=bandwidth in Hz over which the noise is measured.

8
“On the other hand, current noise results from the flow of current in the resistor, and is directly related to the
type of resistive material. It is also called flicker noise (like a flickering candle),21 pink noise, and 1/f noise.22 It
is inversely proportional to the frequency and continues to increase as frequency decreases.
“Current noise can be controlled by selection of the resistor technology.23 24. Current noise is directly
proportional to the current flowing through the device, and is expressed in “microvolts-per-volt” of applied DC
voltage, transmitted in a single frequency decade (a decade is a factor of 10, e.g., 10 – 100 Hz), providing a
resistor noise-quality index. Thermal noise can also be reduced by using cryogenic cooling.
“There are other types of noise including shot noise,25 and popcorn (or burst) noise in semiconductors.26

“Download a noise calculator spreadsheet

created by Bruce Trump.”27

Resistor Types and Properties

“Different Types of resistors and their properties are listed in the following table in the order of likely use. To
look at data sheets and prices, see a distributor's website, e.g. DigiKey.com, under the category of resistors. You
can also get custom non-standard values when needed, for a price.28 Practical values of resistors go as low as 0.1
milliohms, and as high as 500 MΩ. When you use a 0.1 mΩ resistor, its inductance actually dominates over its
resistance.

“Don't forget that lower-case m stands for milli, and upper case M stands for mega. For example, 1 mΩ is 0.001
ohms; 1 MΩ is 1,000,000 ohms.

Shristi asked “How do you get color codes on those small chip resistors?”

I replied “For typical quarter watt resistors, the value of resistance is usually printed on the body of the resistor
in color code; for small surface mount resistors, the value is laser etched on the body. You might need a
magnifying glass to read them. They take the form of three digits; the first two specify the value, and the third
specifies the number of zeroes that follow. For example, 523 means 52,000 or 52k; 106 means 10,000,000 or
10M.

“For precision resistors like 1% metal film, they have four digits; the first three specify the value, and the fourth
specifies the number of zeroes that follow. For example, 4993 means 499,000 or 499k; 1005 means 10,000,000
or 10M.”

Shristi continued “So, what if you see 100? is that 100 ohms, or 10 with no following zeroes?”

“If it says 100Ω with the ohm symbol that is 100 ohms. But sometimes you can't tell. Pull out a meter and
measure it, or get the data sheet.

“Some components are too small to put any marking. You have to be careful with them; don't take them out of
their package until you are ready to use them.”

9
 Resistor Table
Type Fabrication Toler- TCR Note Other Features Current Rough
ance 2 Noise Approx
(note 1) (notes 3, Cost $
4)
Carbon Carbon film is 5% 500 General Purpose, most common, Low 0.01 leaded
Film29 deposited on ceramic typical ppm/°C inexpensive. Has 4 color stripes (2 digits 5%
substrate plus multiplier plus tolerance).
Metal Film30 Thin metal layer is From 5 ppm/°C Precision resistor, good tolerance, Low, -32 0.03 (1%
(thin film) evaporated or sputtered 0.05% stability, TCR, low noise, low inductance dB to -16 chip)
(vacuum deposition) on up to and capacitance (unless made with dB
on a ceramic substrate 2% cylinder shape and spiral cut), and high
linearity. Best operated at 20 – 80%
specified power rating (50% is best), but
less than 20% in a humid environment
reduces reliability. Has 5 color stripes (3
digits plus multiplier plus tolerance).
Thick Film31 Firing a special paste of 0.5% 50 ppm/°C More resistant to moisture than thin film Large, -18 0.001 (5%
glass and metal oxides since they are glass like. Lowest cost dB to +20 chip), 0.17
using screen and stencil resistor on the market, used in many dB (5% leaded)
printing, onto a applications if performance requirements - 6 (1%
substrate, 1,000 times are low. leaded)
thicker than thin film.
Power Wire- Wire wound on a 0.1% 10 ppm/°C Power rating from maybe 1/8 watt to Low 0.65 (10 watt
wound ceramic substrate hundreds or thousands of watts. Due to 5%)
coiled-wire construction, inductance and
capacitance produce poor high frequency
characteristics.
Metal Oxide Oxide of metal is From 50 ppm/°C Non-inductive power resistor. Can Low 3 (0.5%);
Power deposited on ceramic 0.5% withstand higher temperatures and better 0.25 (5%)
(film) substrate short term surges than other film types.
High voltage versions available.
Carbon Made from a slug of 5% Not Non-Inductive. Can withstand larger Large, 0.56
Composition carbon, with plastic specified; short-term pulses and higher voltages than -12 dB to
32
coating they are film resistors and are virtually impervious +6 dB
too to ESD (Electro-static discharge).
unstable to
provide a
rating.

Precision Wire wound on ceramic From 1 ppm/°C Better ESD stability, lower TCR, and Very low 6 (.01%)
Wire-wound substrate 0.005% lower noise than film types. Due to or none,
coiled-wire construction, inductance and -38 dB
capacitance produce poor high frequency typical
characteristics.
Bulk Metal Pattern etched in metal 0.001% 0.2 ppm/°C Non-inductive, Non-Capacitive Very low 35;
Foil33 or metal alloy. or none; 25 for
-42 dB 0.005%
1) Tolerance of resistance is the difference between the labeled value and its actual value.
2) TCR = Temperature Coefficient of Resistance – how much the resistance changes with temperature. ppm =
parts per million; 10,000 ppm = 1%. 1 ppm = 0.0001%.
3) Noise decreases with increasing mechanical size or volume of the same type of resistor.34
4) A current noise index is expressed in decibels, and the equation converting μV/V to dB is:
dB = 20 x log (noise voltage in μV / DC voltage in V); where μV is microvolts.
For example, 0 dB equates to 1.0 μV/V, and 15 dB equates to 5.6 μV/V. dB is explained later.
10
Potentiometers35
36

A potentiometer
(“pot”) is just a
manually adjustable
variable resistor with
three terminals. Two
terminals connect to a
resistor, and a third
terminal connects to a
sliding contact called a
Potentiometer wiper which moves over
schematic the resistor.
Power wire-wound potentiometer symbol
37 38

Potentiometers used as trimmers Slide potentiometer

Potentiometer
5, 10, or 20 turn trimmers

“Volume, bass, and treble controls on an amplifier are implemented with potentiometers. A rheostat is a two-
terminal variable resistor, but this term is becoming obsolete. Potentiometers come in linear taper and
logarithmic (or audio) taper. Single turn and multi-turn precision (5, 10, or 20 turn) are also available.”

“What is taper?” asked Shristi.

I replied “Taper is the relationship between the slider's position, and the resistance. In linear taper, if you turn
the knob one fourth of the total, the resistance would change one fourth of the total. In log taper, the resistance
would change logarithmically.”39

“Why would you want to use log taper?” she asked.

I replied “Log taper is used for volume controls because the ear has a logarithmic response. With log taper the
volume seems to increase as the control increases. With linear taper, it seems to reach maximum at about ¼ the
way open. Different log tapers are available; consult the data sheet.”

11
 Materials and Properties of Potentiometers

Type Composition Properties

Carbon Ink composed of carbon and Very common, low cost, reasonable noise and
composition filler material, on a phenolic wear.
substrate.
Cermet Mixture of non-conducting High degree of stability over a wide temperature
ceramic material and range, low temperature coefficient (150 ppm/°C),
conducting metals fixed to a high temperature capable, lower noise level than
non-conducting base. carbon composition, and expensive. Limited
number of cycles restrict it to trimmer
applications where it is not adjusted often.
Wirewound Wire wound on a ceramic High power, precision, long lasting, limited
core resolution due to discrete nature of the wires.
Conductive Conductive plastic based ink High resolution, long life, low noise, limited
plastic film deposited onto a plastic power, expensive, used in high-end audio
substrate. equipment. Not as temperature stable as cermet.

A potentiometer can get dirty and

cause noise when the knob is turned;
40 clean with Caig DeoxIT chemical spray
through the opening. A totally sealed
pot with a metal frame can be cleaned
by drilling a small hole in the frame.

Resistor Networks

“Before we look at resistors, let's look at batteries since resistors cannot be looked at in total isolation:”

Batteries in Series and Parallel

“The long line of the battery is the positive side. When batteries are connected in series, the values add
together”:

Voltages add Same voltage

Batteries in Series Batteries in Parallel

“With three batteries in parallel, each battery only supplies one third of the total load current. Generally, only
batteries of the same voltage, type, brand, and age should be put in parallel. Otherwise, a weak battery can
discharge a strong one. Unless you have a good reason, don't parallel batteries.”

12
 Batteries have a small internal
resistance , and it is not linear. If
you applied an AC voltage coupled
to a battery by a capacitor, the
battery would act like almost a
short circuit to the AC signal and
allow it to pass through .

“When I was a kid, a friend gave me about 50 6-volt lantern batteries, discards from the railroad company but
they still had juice in them. I connected them in series, and I enjoyed shorting the 300 volt string and drawing a
¼ inch arc. That is, until I inadvertently touched them myself. I didn't realize the danger, and boy was that shock
a real surprise, I didn't know what I was doing. I didn't associate danger with a harmless lantern battery.”

Resistors in Series and Parallel

“Resistors can be combined in series, parallel, and in loops, and we are going to learn how to compute their
combined resistances, voltages, and currents.”

Resistances add:
Rtotal = R1 + R2 + R3

“In parallel they look like this:”

1 = 1 + 1 + 1
Rtotal R1 R2 R3

9KΩ 1KΩ 900Ω

Here is a simpler formula for two resistors in
parallel. Use this formula multiple times for
more than two:

Rtotal = R1 x R2 Memorize This

R1 + R 2

“Shristi, would you like to apply the formula to the above three-resistor schematic?”
She wrote on her notepad.
“Okay, here is the formula.” she wrote:
Rtotal = R1R2/( R1+R2);
She continued “For the first two resistors we get”:
= 9kΩ x 1kΩ / (9kΩ+1kΩ) = 9MΩ/10kΩ = 0.9009kΩ, round to 900Ω
“Combine this result with the third resistor and we get”:
900Ω x 900Ω / (900Ω + 900Ω) = 450Ω

13
“Right;” I continued:
“Remember that capital M means mega, or 1,000,000. I see you placed a zero before the decimal point” I said.
“It's a good idea so it doesn't get lost, it's so small. There is a notation you may see on schematics, that helps
from losing the decimal point, especially when making copies of copies:
1Ω = 1R or 1R0
1.2Ω = 1R2
1kΩ = 1K or 1K0
1.2kΩ = 1k2
1.2MΩ = 1M2
“The k (x 1,000), M (x 1 million), or R (x1) stands for the decimal point, and also the multiplier. You can use
lower or upper case for the k, but use capital M so it is not confused with milli.
“The inverse of resistance R is called conductance G measured in mhos41 (ohms spelled backwards, with the
backwards symbol ℧), but officially named siemens; we don't use the term much.

Analyzing a Simple Battery and Resistor Circuit

10,000 (“10K ohms” or “10kΩ”)

Current flows
from + to -.
i = 1mA Constriction

(m = .001)
Pump
“1 milliamp”

i = 10V / 10kΩ = 1 mA

“You can calculate the current in the above circuit using Ohm's law”:

voltage V
current resistance i R

“To find current, cover current with your finger to get the formula: i = V/R
Here is a graph of voltage vs. current of a resistor”:

14
 “Notice that the line is straight. This means
that the resistor is a “linear” device. If you
double the voltage across the resistor, you
double the current through it.

“You can also look at the graph to see that

when V = 10V, i = 10mA. If you vary the
voltage from 0 to 10 volts, the graph will show
Milliamps

you the resulting current for each voltage. That

is about all there is to such a simple circuit.”

Volts

“Remember, the battery is like a water pump, and is the source of electrical pressure. The wire is like a water
pipe, and the resistor is like a constriction in the pipe. A variable resistor is like a water valve that lets you
change the flow rate.”

Then Shristi said “But it is not a perfect analogy; water will flow out the end of an open pipe, not out the end of
an open wire.”

I replied “This is true unless the voltage is thousands of volts. And wire usually has insulation to prevent
undesirable connection should wires touch; pipe doesn't have this problem.

“And here is another thing to be aware of. Electrons flow out from the minus terminal, and return to the plus
terminal. This is the true current flow, but by convention current flows in the opposite direction, from the plus
terminal, the one with the long line, to the minus terminal, the one with the short line.42 Thus, the arrows on
component symbols point in the opposite direction of actual electron flow.”

Shristi asked “Why was this done?”

“A long time ago” I replied “someone took a guess whether the positive or negative charge was the current
carrier; he guessed positive and he guessed wrong, so now we are stuck with it.”
She continued “Does it make a difference?”
“Not to us; follow the convention and all the arrows on the transistors and diodes will point in the direction of
conventional current flow.”

Analyzing a Simple Battery and Two Resistor Circuit

“When you measure a voltage, unless you are told to measure across a component, your measurement will
always be with respect to a reference point, sometimes called “common” or “ground” or “return” or “zero
volts”. All (low frequency) currents must return to the source, and that return path has resistance and needs to be
considered. It is also usually the negative terminal of the battery. In the following diagrams, the reference point
is the line at the bottom of the diagram.
Shristi asked “Why do you call the return path 'ground' ”?
“It is common practice” I replied “to tie the negative power supply to the chassis, and for safety reasons to
15
prevent electrical shock, the chassis is tied to earth ground through the ground wire of the power cord. That
ground wire actually connects to the earth at the power distribution pole with a stake driven into the ground. If
for any reason a dangerous voltage should touch or connect to the chassis, it will conduct to the ground instead
of conducting through you when you touch the chassis.”

“What about portable equipment with no power cord?” she asked.

“We still call the return path 'ground', it is a custom, and that can lead to confusion. It should properly be called
'return'.

I continued “Now suppose we connect a second resistor to the previous circuit. Shristi, can you determine the
voltage across each resistor?”

“9 kilo ohms” = 9 kΩ
= 9 thousand ohms

See how the voltage

from the battery
divides between the
resistors? Ohm's law
is applied throughout.

She replied “We use the defining equation for resistors, V = iR. We know the total resistance R = R1 + R2, so to
find V we first have to compute the current i.”

“Right” I said. “So what is i?”

“The current in the loop” she said “is also defined by V = i x R, but R is the sum of the resistors in the loop, 9
kΩ + 1 kΩ = 10kΩ. And V = 10 volts. So i = V/R = 10V/10kΩ = 1 mA.”

I asked “Now you know the current is 1 mA, so the voltage across the resistors is …?

She replied “VR1 = i x R1 = 1 mA x 9 kΩ = 9 volts. The milli and kilo cancel each other.
“And VR2 = i x R2 = 1 mA x 1 kΩ = 1 volt!”

“Right. You can see that even a simple circuit can be complicated.”

Solving Resistor Networks

43
“Now we will look at four different methods for determining the voltages,
currents, and resistances in networks44: voltage division, source transformation,
Thevenin equivalent, and Kirchhoff's loop analysis. But first we will touch on
Kirchhoff's voltage and current laws which are fundamental to everything.”

Kirchhoff's Circuit Laws45

“Kirchhoff's voltage law (KVL) states that the sum of the voltages in a loop must Gustav Robert Kirchhoff
sum to zero. In terms of the water analogy, the total back pressures must equal the 1824 – 1887
forward pressure from the pump. And Kirchhoff's current law (KCL) states that the
16
sum of the currents into a point must equal the currents leaving the point, which also applies to water networks.
With these two laws we can write equations to determine the voltages and currents in networks. These ideas are
simple and almost obvious but need to be stated, and they are shown in the following circuit with two loops:

Currents entering a point must

equal currents leaving the point.

The sum of voltages The sum of voltages

across R1 and R2 across R2 and R3
must equal the must equal the
battery voltage V1. battery voltage V2.

Voltage Division

“There is a simpler approach to solving the voltages across two resistors that does not require computing the
current, called voltage division. It is used extensively and should be memorized. I will redraw the earlier circuit
slightly to emphasize that we are looking at voltage division; it's the same circuit:”

Notice we don't compute the loop current. Vin is

given as 10V, which divides proportionally across
the resistors. We want to find Vout:
Vout = Vin x R2/(R1+R2)
= 10V x 1K/(9K + 1K)
= 10V x 1K/10K = 1V
Memorize This: Vout = Vin (R2/(R1+R2))

Voltage reference point: “common”, “ground”, or “return”.

“When you have two resistors in series, the voltages divide between them proportionally. The 9 kΩ resistor will
have 9 times the drop of the 1 kΩ resistor. The formula is:

Vout = Vin x R2 / (R1+R2)

= 10V x 1kΩ / (9kΩ+1kΩ) = 10V (1k/10k) = 1V.
“It is the same answer as before, but you don't have to compute the current. Now let's try something more
complicated.” I put up this slide:

“Let's say you are designing a circuit. The battery is 10V, and you need Vout = 3V; what must be the value of R1?
If ever you can't see the solution, just start writing. What do you see, Shristi?”

She wrote on her notepad: “Since I know the voltage across R2 is 3V, I will write the equation for R2 using
17
voltage division:

Vout = VR2 = Vin x R2 / (R1+R2) “Vin is 10V, VR2 is 3V, and R2 is 1 kΩ. Substituting:”
3V = 10V x 1K / (R1+1K)

“I solve for R1 using simple algebra”:

0.3V = 1k/(R1+1k); 0.3 x (R1+1k) = 1k; 0.3R1 + 0.3k = 1k; 0.3R1 = 0.7k
“And R1 = 0.7k/0.3 = 2.33k. It's a lot easier, and we didn't have to compute the current” she said.

Source Transformation

“Source Transformation46 is the method of transforming voltage sources into current sources, and vice versa, in
order to simplify complex circuits. It is an application of Thevenin's theorem47 and Norton's theorem48;
Thevenin's theorem states that any network consisting of voltage and current sources and resistances can be
replaced by a single voltage source and a resistor in series. Norton's theorem is similar; the network can be
replaced by a current source and a resistor in parallel. First we look at how to do the source transformations,
then we apply it to simplify a network. It's very easy to use, and very helpful.

“In the following figure, the voltage-source and resistor on the left is converted to a current-source and resistor
on the right by using Ohm's law, V = i x R. The resistance R is the same for both figures:

No connection allowed at point “C”.

“In the example, V = 10V and R=10K. The current-source for the right figure is computed by i = V/R =
10V/10K = 1mA.

“We have just converted from a voltage source into a current source! Wasn't that easy? You can go both
directions.”

“So what?” Shristi asked.

18
“Let's do an example. Consider this circuit; it consists of two interacting loops. What is Vout?

“To simplify this network, we convert the voltage-source V1 in series with R1, into a current-
source i in parallel with the same resistor, R1, setting i = V1 / R1:

i = V1 / R1 = 10V / 5 kΩ = 2 mA Notice that there are

now two resistors in
parallel? They are
easy to combine. This
makes source
transformation so
useful.

“We combine the two parallel resistors R1 and R2 into one resistor R5 using R║
=R1R2/(R1+R2); this gives us 0.833k. (║ = “parallel”)

“Then we convert current-source i and R5 back into a voltage-source V3 and R5; R5 keeps its
value, setting V3 = i x R5.

V3 = i x R5 = 2mA x 0.833 kΩ = 1.67V

“Now since everything is in series, it is a simple matter of adding the resistances together, and the source
voltages together:

Vtotal = 1.67V – 3V = -1.33V “V2 opposes V3, so we subtract.”

Rtotal = 0.833kΩ + 2kΩ + 4kΩ = 6.833kΩ

“To find the current in the loop, use i = Vtotal/Rtotal: = -1.33V / 6.833kΩ = -0.195mA”
19
“Current is counter-clockwise (opposite the arrow) since V2 > V3.”

“And to answer the question: The output voltage Vout = i x R4 = -0.195mA x 4 kΩ = -0.779V.”

Thevenin's Theorem

“Thevenin's theorem49 states that any network consisting of voltage and current sources and resistances can be
replaced by a single voltage source and a resistor in series. Surprisingly, Thevenin's Theorem provides a really
simple and easy method for finding the voltage at a point in a network.

“It is easy to remember because it sounds like Ohm's law (voltage = current x resistance):

Open circuit voltage = Short current current x Resistance.

“Let's do an example.

“If you want to know the voltage Vx, this is the “open circuit voltage” you are looking for, so called because it is
not yet short-circuited, the first of four steps:

Step 1: You short R2 which is the unknown voltage Vx, and compute the current going through the short. This is
just ishort = V1/R1 + V2/R3 = 1mA + 0.667mA = 1.667 mA. If there were any current sources, you would add them
in (there aren't any here).

Step 1: Short R2.

Compute the currents
in the short.

Step 2: You un-short the resistor just shorted, and you short out all voltage sources in the network so they are
like wire, and you open or disconnect all current sources:

Step 2: Short all

voltage sources, and
open any current
sources.

20
Step 3: Then you compute the resistance you see, “looking into” the point at Vx. This includes R2 unshorted.
Step 3:
In this example we see three resistors in parallel, R1, R2, and R3:
“Looking into”
R║ = 10kΩ ║ 20kΩ ║ 30kΩ = 5.46kΩ (║ means “parallel”). Vx, compute
Use the formula 1/R = 1/10kΩ + 1/20kΩ + 1/30kΩ = 1/5.46kΩ the resistance
R║ = 5.46kΩ you see.

Step 4: Vx = ishort-circuit x resistance =1.667mA x 5.46kΩ = 9.1V. Your answer.

“Notice that we converted the network into its Norton and Thevenin equivalents!”:

Step 4: Apply
the magic
formula: Voc = isc
x R║
Norton Thevenin

Kirchhoff”s Loop Equations

“The techniques I have given you so far are a lot easier to use but don't work in all cases, so I am now giving
you Kirchoff's loop analysis. You will find it in a lot of the literature.

“Remember that Kirchhoff's voltage law (KVL) states: The sum of voltages in a loop must equal zero. And
Kirchoff's current law (KCL) states: The currents entering a point must equal the currents leaving the point.”

Then Shristi asked “When I build a radio transmitter, currents flow into the antenna and travels far without
returning to the source in a loop. How do you explain that?”

“Kirchhoff's laws do not apply at higher frequencies, whether radio waves, or fast-rising pulses moving down a
wire. A curious point is that free space itself has a space impedance of about 377 ohms, the ratio of the electric
field to the magnetic field of the radio wave.50 The speed of light is 1.02 nanoseconds (billionths of a second)
per foot. A fast-rising pulse sent down a wire will travel at about 67% the speed of light. If you have several feet
of wire, it will take several nanoseconds before the source voltage hits its load resistance. Obviously, while a
pulse is traveling, the source voltage will not equal the load voltage. During that time, the wire itself acts like a
load to the source, and has a characteristic impedance equal to the ratio of voltage to current, determined by the
inductance and capacitance and the wire's geometry and materials, and not resistance which may be nearly
zero.51 Cables are specified by their impedance; 50 ohm and 75 ohm cables are very common. But I digress; that
is a topic saved for later.”

A pulse travels through wire at about 1ns (nanosecond) per foot. Memorize this.

“Let's go through an example. You have to pay careful attention to polarity. Follow these instructions:

“First, draw a loop showing the current in each loop. The direction is arbitrary, but I suggest all loops be drawn
clockwise as shown in this case. That way, any contribution from each adjacent loop will be subtracted from the
equation of the loop you are computing. If the loop direction is opposite of the actual direction of flow, the
computed current will come out negative.

21
“Second, pick any point on the loop to start; it doesn't matter where.

“Third, move in the direction of the loop, and add the voltages contributed by each component. When the loop
current enters a resistor, it produces a positive voltage, regardless of the polarity of the battery. When the loop
current enters a battery, the polarity is the sign of the battery when entering; the battery voltage is independent
of the loop current. Set the total to zero since the sum of all voltages in the loop must add to zero.

“Now study each term of the two loop equations to understand where each came from”:

Loop 1 equation -10V + (i1 x 5K) + (i1 x 1K) – (i2 x 1K) = 0

Contribution of Loop 2

The direction of i2
is opposite i1, so
i1 hits the “-” of i2's contribution is
the battery, so the subtracted.
voltage is negative.
Loop 1 starts here Loop 2 starts here

Loop 2 equation 3V + (i2 x 1K) – (i1 x 1K) + (i2 x 2K) = 0

Contribution of Loop 1

“The voltage across R2 depends on the contribution of each loop. You write one equation for each loop, which
contains the contribution of the other loops. Look at the loop equations; you will see that each one contains the
contribution of the other loop.
This is important, and it is the key and purpose of the loop
analysis. R2 is shared by both loops, and you must include the
contribution of each loop current. You have to keep the polarities
straight.

“You have two equation and two unknowns; go through the drudgery and algebraically solve the two equations
simultaneously; I won't do it here. Get your answer: i1 = 1.59mA; i2 = -0.47mA. Notice i2 is negative, which
means the actual current is opposite that shown by the loop.

“According to Kirchhoff's current law, since i1 enters a point and i2 leaves the point, the difference must exit via
R2 since there is no other place to go. So the current in R2 is i1 - i2 = 1.59mA - -0.47mA = 2.06mA, and the
voltage across R2 is 2.06mA x 1KΩ = 2.06V.

“To verify our answer, let's apply Thevenin's theorem to R2 (open circuit voltage = short-circuit current x
resistance). In this case we will do without the loops, and follow the natural direction of current determined by
the voltage sources:

22
 Compute the current through
R2 with R2 shorted
Look into R2 with batteries shorted;
Voc = isc x Rlooking into R2 compute the resistance you see.
= (V1/R1 + V2/R3) x (R1 ║ R2 ║ R3);
= (10V/5KΩ + 3V/2KΩ) x 0.588KΩ ; (1/R║ = 1/5KΩ +1/1KΩ +1/2KΩ = 1/0.588KΩ)
= (2mA + 1.5mA) x 0.588KΩ
= 3.5mA x 0.588KΩ
= 2.06V – the voltage across R2

“We get the same result from unrelated computations, but the method is much easier.”

Chapter 1 Summary

A battery is like a water pump providing electrical pressure. Wire is like a pipe, and a resistor is like a
constriction in the pipe to control the flow.

Resistors come in different types such as carbon film, metal film, thick film, wire wound, metal oxide, carbon
composition, and the most stable noise free and expensive bulk-metal foil. Each emphasizes different
properties, e.g. low inductance, low noise, high stability, high power, low cost.

Resistors are defined by Ohm's Law: voltage = current x resistance.

To remember this formula, cover the desired quantity with your
finger and perform the remaining operation.

Energy is like “jewels” in a bucket; the quantity is measured in joules. Power is the rate of pouring the
“jewels” out in watts, or joules per second.

Power in a resistor is defined by the following equations:

Power = Vi = i2R = V2/R
Resistors have a maximum power rating; conservative design suggests operation at about 50% of their rating,
and should generally be derated above 70°C ambient.

Resistors generate two main types of noise, thermal noise en = √4kTRB which occurs in any conductor
regardless of applied voltage, and current noise which is proportional to the current flowing through the resistor
and is controlled by choosing the proper resistor type.

Potentiometers are variable resistors. The main types are carbon composition which are common and low cost;
cermet for high stability and low noise, but expensive and only useful as trimmers; wirewound for high power
and precision; and conductive plastic for high end audio, but expensive. Potentiometers come in linear taper, or
logarithmic taper used for audio applications to provide a more natural feel to volume control.

When resistors are placed in series, their values add: Rtotal = R1 + R2 + R3.
Resistors placed in parallel are defined by:
1 = 1 + 1 + 1 For just two resistors: Rtotal = R1 x R2 / (R1 + R2)
Rtotal R1 R2 R3

When batteries are placed in series, their values add: Vtotal = V1 + V2 + V3. Batteries placed in parallel have the
23
same voltage and should be well matched.

Current, by convention, flows from plus to minus. In actuality, electrons flow from minus to plus. The arrows on
electronic components agree with convention, so follow convention.

Voltages in a circuit are usually measured with respect to a point, often called “ground” or “common” or
“return”.

Kirchhoff's voltage law (KVL) states that the sum of the voltages in a loop must sum to zero.
Kirchhoff's current law (KCL) states that the currents entering a point must equal the currents leaving the point.
Kirchhoff's laws do not apply at higher frequencies.

Four methods were given for solving resistor networks: Voltage division: Vout = Vin x R2/(R1+R2); source
transformation: a voltage source in series with a resistor can be replaced with a current source in parallel with
the resistor, and vice versa; Thevenin equivalent: Voc = isc x R║ (open circuit voltage = short current current x
resistance); and Kirchhoff's loop analysis.

A pulse travels down a wire at nearly the speed of light, about 1ns (nanosecond) per foot.

24
Chapter 1 Problems

“The problems are not optional! The only way to know if you have learned the material is to test yourself, and
grapple with the problems.”

For problems 1 to 4, find Vout for the above circuit using the following methods:

1) Voltage division (Vout = Vin x R2/(R1+R2) )

2) Source transformation (convert voltage sources into current sources and vice versa).

3) Thevenin method (Voc = isc x R).

4) Kirchhoff's Loop method (the voltages in a loop must sum to zero).

5) For the following circuit, use Kirchhoff's Loop method (KVL) to find Vout:

6) What is the value of R1 (which is the same as R2) so that the input resistance to the following circuit is 100
ohms?

25
Solutions to Problems

1) Using voltage division:

Vout = Vin x R2/(R1+R2) = 10V x 1k/(9k+1k) = 1V.

2) Using source transformation:

“We convert the voltage source in series with a resistor, into a current source in parallel with the same resistor”:

“Now that we have two resistors in parallel, we can combine them”:

R║ = R1R2/(R1+R2) = 9k x 1k/(9k+1k) = 0.9kΩ

And Vout = i x R║ = 1.11mA x 0.9kΩ =1.0V

3) Using the Thevenin method:

Voc = isc x R║

“We short R2 and measure the current through the short:

isc = 10V/R1 = 10V/9k = 10/9 mA

“Then we open the short-circuit, and measure the resistance looking into R2 (battery is shorted):
R║ = R1R2/(R1+R2) = 9k x 1k/(9k+1k) = 9/10 kΩ

“Finally, Voc = Vout = isc x R║ = 10/9 mA x 9/10 kΩ = 1V.

4) Using Kirchhoff's Loop method

“The KVL loop equation, starting from the minus terminal of the battery and moving clockwise, is”:

26
-10V + (i x 9K) + (i x 1K) = 0
-10V + i(9K + 1K) = 0 ; Simplifying and solving for i:
-10V + i(10K) = 0
i(10K) = 10V
i = 10V/10K = 1mA
And Vout = i x R2 = 1mA x 1kΩ = 1V.

5) Kirchhoff's Loop method with two loops

-10V + iloop-1 2K + iloop-1 2K - iloop-2 2K = 0 5V + iloop-2 4K + iloop-2 2K - iloop-1 2K = 0

-10V + iloop-1 4K - iloop-2 2K = 0 5V + iloop-2 6K - iloop-1 2K = 0

iloop-2 is in the opposite direction
of iloop-1, so they are subtracted.

Loop 1: Start (for example) at the negative terminal of V1, and move clockwise.
Loop 2: Start at the positive terminal of V2, and move clockwise.

The two loop equations are:

-10V + iloop-1 4K - iloop-2 2K = 0

5V + iloop-2 6K - iloop-1 2K = 0 ; Multiply the 2nd equation by 2 and rearrange:

-10V + iloop-1 4K + iloop-2 2K = 0

10V - iloop-1 4K + iloop-2 12K = 0; Adding the two equations together gives:

0 + 0 + iloop-2 14K = 0

“And, iloop-2 = 0 since this is the only value that will solve the equation; no current flows through R4, Vout = 0. It
does not matter what the value of R4 is. Consider that R1 and R2 divide V1=10V by half to 5V, which balances
out V2=5Vso no current flows.”

6) Find the value of R1 = R2 so that R-input = 100 ohms:

Write the defining equation:

27
Rin = 100 R2/(100 + R2) + R1

100 = 100 R2/(100 + R2) + R1

Rin R3 R3

100 = 100 R1/(100 + R1) + R1 Set R2=R1 since they are both equal.

100 – R1 = 100 R1/(100 + R1) Subtract R1 from both sides.

(100 – R1)(100+R1) = 100 R1 Multiply both sides by 100+R2

10,000 – R12 = 100 R1

R12 + 100 R1 – 10,000 = 0 You could use WolframAlpha.com.

R1 = (-100 +/- √50,000) /2 Solve the quadratic equation; only the + root makes sense.

R1 = 61.8 ohms

28
1 Attribution: By Nunikasi - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=12161146
2 Attribotion: By Afrank99 - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=456666
3 Stackpole RSM Series
4 Attribution: By Olli Niemitalo - {www.danotherm.dk}, CC0, https://commons.wikimedia.org/w/index.php?
curid=27187147
5 Attribution: http://www.te.com/usa-en/product-1879453-1.html
6 Attribution: http://www.danotherm.dk/page/product/59/10/adjustable-resistors
7 Attribution: By Nkendrick (talk) - I (Nkendrick (talk)) created this work entirely by myself., Public Domain,
https://en.wikipedia.org/w/index.php?curid=23880822
8 Attribution: By User Mike1024 on en.wikipedia - Photographed by User:Mike1024, Public Domain,
https://commons.wikimedia.org/w/index.php?curid=1607140
9 Attribution: www.vishay.com%2Fdocs%2F52032%2Fchpht.pdf
10 For a detailed reference, see: http://www.electricaltechnology.org/2015/01/resistor-types-resistors-fixed-variable-linear-
non-linear.html
11 Georg Simon Ohm: https://commons.wikimedia.org/wiki/File%3AGeorg_Simon_Ohm3.jpg; By The original uploader
was BerndGehrmann at German Wikipedia (Transferred from de.wikipedia to Commons.) [Public domain], via
Wikimedia Commons
12 By User:Dysprosia, traced by User:Stannered, BSD, https://commons.wikimedia.org/w/index.php?curid=3348470
13 No copyright: https://commons.wikimedia.org/wiki/File:4-Band_Resistor.svg
14 Resistor Color Codes: Color band calculator, and other calculators: https://www.digikey.com/en/resources/conversion-
calculators/conversion-calculator-resistor-color-code-4-band
15 Joule: https://en.wikipedia.org/wiki/Joule
16 James Watt attribution: Carl Frederik von Breda [Public domain], via Wikimedia Commons,
https://commons.wikimedia.org/wiki/File%3AWatt_James_von_Breda.jpg
17 Fusible resistors: Eg: Tyco Electronics FRN series.
18 Thermal (Johnson or Nyquist) noise: https://en.wikipedia.org/wiki/Johnson–Nyquist_noise
19 John Bertrand Johnson (1887–1970): https://en.wikipedia.org/wiki/John_Bertrand_Johnson
20 Harry Nyquist (1889–1976): https://en.wikipedia.org/wiki/Harry_Nyquist
21 Flicker noise: http://www.edn.com/electronics-blogs/the-signal/4408242/1-f-Noise-the-flickering-candle-;
https://en.wikipedia.org/wiki/Flicker_noise
22 Pink (1/f) noise: https://en.wikipedia.org/wiki/Pink_noise
23 Noise and resistor technologies: http://www.eetimes.com/document.asp?doc_id=1272283
24 Current noise: Resistor Current Noise Measurements, Frank Seifert (Frank.Seifert@aei.mpg.de)
https://dcc.ligo.org/public/0002/T0900200/001/current_noise.pdf. See the footnotes therein, and the charts showing
noise measurements of commercially available resistors.
25 Shot noise: https://en.wikipedia.org/wiki/Shot_noise Shot noise occurs when the number of current carrying electrons is
sufficiently small that independent random electron events become significant statistical fluctuations.
26 Popcorn noise in semiconductors: https://en.wikipedia.org/wiki/Burst_noise
27 Noise Voltage – Bruce Trump, spreadsheet file download,
http://e2e.ti.com/blogs_/archives/b/thesignal/archive/2013/03/03/1-f-noise-the-flickering-candle. See
http://www.edn.com/electronics-blogs/the-signal/4408242/1-f-Noise-the-flickering-candle-
28 Custom resistors: http://www.caddock.com/
29 Carbon Film Resistor e.g.: http://www.digikey.com/product-detail/en/stackpole-electronics-
inc/CF14JT1K00/CF14JT1K00CT-ND/1830350
30 Metal film resistor: http://www.resistorguide.com/metal-film-resistor/
31 Thin and Thick Film resistor: http://www.resistorguide.com/thin-and-thick-film/
32 Carbon Composition Resistor e.g.: http://www.digikey.com/product-detail/en/te-connectivity-passive-
product/CBT25J100R/A105998CT-ND/3477604
33 Bulk Metal Foil Resistor e.g.: http://www.digikey.com/product-detail/en/vishay-foil-resistors-division-of-vishay-
precision-group/Y1453100R000V9L/Y1453-100A-ND/2609893
34 Noise decreases with increasing physical size: Resistor Current Noise Measurements, Frank Seifert
(Frank.Seifert@aei.mpg.de) https://dcc.ligo.org/public/0002/T0900200/001/current_noise.pdf, p. 8.
35 Potentiometers: https://en.wikipedia.org/wiki/Potentiometer
36 Power potentiometer attribution: By Gdead at English Wikipedia (Transferred from en.wikipedia to Commons.) [Public
domain], via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Pot1.jpg
37 Trimmer potentiometers: Attribution: By Junkyardsparkle - Own work, CC0,
https://commons.wikimedia.org/w/index.php?curid=39291275
38 Potentiometer attribution: By Iainf (Self-photographed) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0
 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via
Wikimedia Commons.
39 Linear and Logarithmic taper: http://sound.whsites.net/pots-f4.gif
40 Cleaning spray: DeoxIT – available from Radio Shack – https://www.radioshack.com/products/deoxit-d5s-6-spray-
contact-cleaner-and-rejuvenator; for reviews, see http://antiqueradios.com/forums//viewtopic.php?f=9&t=194334.
41 mho: https://en.wikipedia.org/wiki/Siemens_(unit)#Mho
42 Conventional current flow vs. actual current flow: https://en.wikipedia.org/wiki/Electric_current#Conventions
43 Gustav Robert Kirchhoff: This media file is in the public domain in the United States. This applies to U.S. works where
the copyright has expired, often because its first publication occurred prior to January 1, 1923. See
https://commons.wikimedia.org/wiki/File:Gustav_Robert_Kirchhoff.jpg#filelinks
44 Network Analysis https://en.wikipedia.org/wiki/Network_analysis_(electrical_circuits)
45 Kirchhoff's Circuit Laws: https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws
46 Source Transformation https://en.wikipedia.org/wiki/Source_transformation
47 Thevenin's theorem https://en.wikipedia.org/wiki/Thévenin's_theorem
48 Norton's theorem https://en.wikipedia.org/wiki/Norton's_theorem
49 Thevenin's theorem: https://en.wikipedia.org/wiki/Thévenin's_theorem
50 Space impedance: https://en.wikipedia.org/wiki/Impedance_of_free_space; Wave impedance:
https://en.wikipedia.org/wiki/Wave_impedance
51 Characteristic impedance: https://en.wikipedia.org/wiki/Characteristic_impedance;
https://www.allaboutcircuits.com/textbook/alternating-current/chpt-14/50-ohm-cable/