EGR 2201 Unit 2
Basic Laws



Read Alexander & Sadiku, Chapter 2.
Homework #2 and Lab #2 due next
week.
Quiz next week.
Ohm’s Law


Ohm’s law says that the
voltage v across a resistor is
equal to the current i through
the resistor times the resistor’s
resistance R. In symbols:
v=iR
The voltage’s polarity and current’s
direction must obey the passive sign
convention, as shown in the diagram
at right. Otherwise, you need a
negative sign in this equation:
v = i  R.
Ohm’s Law Rearranged


The equation v = i  R is useful if we
know i and R, and we’re trying to find
v.
Often we’ll need to rearrange the
equation to one of the following
forms:
i=vR
or
R=vi
Ohm’s Law Game

Given values for two of the three
quantities in Ohm’s law, you must be
able to find the third quantity.

To practice, play my Ohm’s Law game.
Short Circuit



An element with R=0 (or with an
extremely small resistance) is called a
short circuit.
Since a short circuit’s resistance is
zero, Ohm’s law tells us that the
voltage across a short circuit must also
be zero:
v=iR=i0=0
But we can’t use Ohm’s law to
compute a short circuit’s current:
i = v  R = 0  0 = ???
Open Circuit



An element with R= (or with an
extremely large resistance) is called an
open circuit.
Since an open circuit’s resistance is
infinite, Ohm’s law tells us that the
current through an open circuit must
be zero:
i=vR=v=0
But we can’t use Ohm’s law to
compute an open circuit’s voltage:
v = i  R = 0   = ???
Power Dissipated by a Resistor


When current flows through a resistor,
electric energy is converted to heat, at
a rate given by the power law:
p=vi
Once the energy has been given off as
heat, we can’t easily reverse this
process and convert the heat back to
electric energy. We therefore say that
resistors dissipate energy.

In contrast, we’ll see later that capacitors
and inductors store energy, which can
easily be recovered.
Other Power Formulas for
Resistors


By combining the power law (p = v  i)
with Ohm’s law (v = i  R or i = v  R),
we can easily derive two other useful
formulas for the power dissipated by a
resistor:
p=i2R
p=v2R
Of course, each of these equations can
in turn be rearranged, resulting in a
number of useful equations that are
summarized in the “power wheel”….
The “Power Wheel”


This is a useful aid for
people who aren’t
comfortable with basic
algebra, but you
shouldn’t need it.
The important point is
that if you know any
two of these four
quantities (P, V, I, and
R), you can compute
the other two, as long
as you remember
p=vi and v=iR.
Power Calculation Games

Given values for two of the following
four quantities—voltage, current,
resistance, power—you must be able
to find the other two quantities.

To practice, play these games:
 Ohm’s Law
 Power Law
 Power-Current-Resistance
 Power-Voltage-Resistance
Conductance




It’s sometimes useful to work with
the reciprocal of resistance, which
we call conductance.
The symbol for conductance is G:
G=1R
Its unit of measure is the siemens
(S).
Example: If a resistor’s resistance is
20 , its conductance is 50 mS.
Review: Some Quantities and Their
Units
Quantity
Symbol
SI Unit
Symbol for
the Unit
Current
I or i
ampere
A
Voltage
V or v
volt
V
Resistance
R
ohm

Charge
Q or q
coulomb
C
Time
t
second
s
Energy
W or w
joule
J
Power
P or p
watt
W
Conductance
G
siemens
S
Ohm’s Law and the Power
Formulas Using Conductance


As the book discusses, Ohm’s law
and our power formulas can be
rewritten using conductance G
instead of resistance R.
Example: Instead of writing
v=iR
we can write

v=iG
But I advise you to ignore this, and
always use R instead of G.
Circuit Topology: Branches


When describing a circuit’s layout (or
“topology”), it’s often useful to identify the
circuit’s branches, nodes, and loops.
A branch represents a single circuit element
such as a voltage source or a resistor.
 Example: This circuit (from Figure 2.10)
has five branches.
Dots or No Dots?



In schematic diagrams,
our textbook sometimes
draws dots at the points
where two or more
branches meet, as in
this diagram.
But most of the time it
omits these dots, as in
this diagram.
There’s no difference in meaning.
Circuit Topology: Nodes

A node is the point of connection between
two or more branches.
 Example: This circuit has three nodes,
labeled a, b, and c.
Circuit Topology: Loops


A loop is any closed path in a circuit.
 Example: This circuit has six loops.
We won’t need to worry about the book’s
distinction between loops and independent
loops.
Elements in Series

Two elements are connected in series if
they are connected to each other at exactly
one node and there are no other elements
connected to that node.
 Example: In this circuit, the voltage
source and the 5- resistor are connected
in series.
Current Through SeriesConnected Elements

If two elements are connected in series, they
must carry the same current.


Example: In this circuit, the current through the
voltage source must equal the current through the
5- resistor.
But usually their voltages are different.

Example: In the circuit above, we wouldn’t expect
the voltage across the 5- resistor to be 10 V.
Elements in Parallel

Two or more elements are connected in
parallel if they are connected to the same
two nodes.
 Example: In this circuit, the 2- resistor,
the 3- resistor, and the current source
are connected in parallel.
Voltage Across Parallel-Connected
Elements

If two elements are connected in parallel,
they must have the same voltage across
them.


Example: In this circuit, the voltage across the 2-
resistor, the 3- resistor, and the current source
must be the same.
But usually their currents are different.

Example: In the circuit above, we wouldn’t expect
the current through the 3- resistor to be 2 A.
Some Connections are Neither
Series Nor Parallel

Sometimes elements are connected to each
other, but they are neither connected in series
nor connected in parallel.



Example: In this circuit, the 5- resistor and the 2-
resistor are connected, but they’re not connected in
series or in parallel.
In such a case, we wouldn’t expect the two elements
to have the same current or the same voltage.
This type of connection doesn’t have a special name.
Series Circuits


Two simple circuit layouts are series circuits
and parallel circuits.
In a series circuit, each connection between
elements is a series connection.


Therefore, current is the same for every element.
(But usually voltage is different for every element.)
A series circuit is a “one-loop” circuit.
A series circuit:
Same circuit on
the breadboard:
Another series circuit
+ v1 
is
i1
i2
i3
Analyzing a Series Circuit

Find the total resistance by adding the
series-connected resistors:
RT = R1 + R2 + R3 + ...
2. Use Ohm’s law to find the current
produced by the voltage source:
is = vs  RT
3. Recognize that this same current passes
through each resistor:
is = i1 = i2 = i3 = ...
4. Use Ohm’s law to find the voltage across
each resistor:
v1 = i1  R1 and v2 = i2  R2 and ...
1.
+ v2 
v3 +
Parallel Circuits

In a parallel circuit, every element is in
parallel with every other element.

Therefore, voltage is the same for every element.
(But usually current is different for every element.)
A parallel circuit:
Another parallel circuit
Same circuit on
the breadboard:
Analyzing a Parallel Circuit
+
v1

i1
+
v2

i2
+
v3

i3
Recognize that the voltage across each
resistor is equal to the source voltage:
vs = v1 = v2 = v3 = ...
2. Use Ohm’s law to find the current through
each resistor:
i1 = v1  R1 and i2 = v2  R2 and ...
1.
More Complicated Circuits


We’ve seen that series circuits and parallel
circuits are easy to analyze, using little more
than Ohm’s law.
But most circuits don’t fall into either of these
categories, and are harder to analyze.
Examples that are neither series circuits nor parallel circuits

Some authors call these series-parallel
circuits. Others call them complex circuits.
Kirchhoff’s Current Law (KCL)

Kirchhoff’s current law: The algebraic
sum of currents entering a node is zero.
 Example: In this figure,
i1  i2 + i3 + i4  i5 = 0

Alternative Form of KCL: The
sum of the currents entering
a node is equal to the sum
of the currents leaving the node.
 Example: In the figure,
i1 + i3 + i4 = i2 + i5
KCL Applied to a Closed Boundary

You can also apply
KCL to an entire
portion of a circuit
surrounded by an
imaginary boundary.
The sum of the
currents entering the
boundary is equal to
the sum of the
currents leaving the
boundary.

Example: In this figure, i1 + i5 = i2 + i7 + i8
Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s voltage law: Around any loop
in a circuit, the algebraic sum of the
voltages is zero.
Example: In this figure,
v1 + v2 + v3  v4 + v5 = 0

Alternative Form of KVL: Around any loop,
the sum of the voltage drops is equal to the
sum of the voltage rises.

Example: In the figure, v2 + v3 + v5 = v1 + v4
Equivalent Resistance


In analyzing circuits we will often
combine several resistors together to
find their equivalent resistance.
Basic idea: What single resistor would
present the same resistance to a source as
the combination of resistors that the source
is actually connected to?
Resistors in Series

The equivalent resistance of any
number of resistors connected in
series is the sum of the
individual resistances:
Req = R1 + R2 + ... + RN

We’ve already used this earlier in Step 1 of
our analysis of series circuits.
Parallel Resistors

The equivalent resistance of two
parallel resistors is equal to the
product of their resistances divided
by their sum:
𝑅𝑒𝑞

𝑅1 𝑅2
=
𝑅1 + 𝑅2
Note that the equivalent resistance is
always less than each of the original
resistances.
More Than Two
Resistors in Parallel



For more than two resistors in parallel, you
cannot simply extend the product-over-sum
rule like this:
𝑅1 𝑅2 𝑅3
𝑅𝑒𝑞 =
𝑅1 +𝑅2 +𝑅3
Instead, either use the product-over-sum rule
repeatedly (with two values at a time), or…
…use the so-called reciprocal formula:
𝑅𝑒𝑞
1
=
1
1
1
1
+
+
+ ⋯+
𝑅1 𝑅2 𝑅3
𝑅𝑁
Parallel Resistors: Two Shortcut
Rules for Special Cases

Special Case #1: For N parallel
resistors, each having resistance R,
𝑅𝑒𝑞 =

𝑅
𝑁
Special Case #2: When one
resistance is much less than another
one connected in parallel with it, the
equivalent resistance is very nearly
equal to the smaller one:
If R1 << R2, then Req  R1
Series-Parallel Combinations of
Resistors


In many cases, you can find the equivalent
resistance of combined resistors by
repeatedly applying the previous rules for
resistors in series and resistors in parallel.
Hint: Always start farthest from the source
or (in a case like the one above from Figure
2.36) farthest from the open terminals.
Voltage Division

For resistors in series, the total voltage
across them is divided among the
resistors in direct proportion to their
resistances.


Example: In the
circuit shown
(Figure 2.29), if
R1 is twice as big
as R2, then v1 will
be twice as big
as v2.
See next slide for a formula that captures this…
The Voltage-Divider Rule

The voltage-divider rule:
For N resistors in series, if the total
voltage across the resistors is v, then
the voltage across the nth resistor is
given by:
𝑣𝑛 =

𝑅𝑛
𝑣
𝑅1 +𝑅2 +⋯+𝑅𝑁
Example: In the circuit shown,
𝑣1 =
𝑅1
𝑣
𝑅1 +𝑅2
and
𝑣2 =
𝑅2
𝑣
𝑅1 +𝑅2
The Voltage-Divider Rule in More
Complex Circuits


The voltage-divider rule is easiest to
apply in series circuits, but it also
holds for series resistors in more
complex circuits.
Example: In the
circuit shown,
suppose we
know the value of
the voltage v. Then
we can say that
𝑣1 =
50
𝑣
50+30
and
𝑣2 =
30
𝑣
50+30
Current Division

For resistors in parallel, the total
current through them is shared by the
resistors in inverse proportion to their
resistances.


Example: In the
circuit shown
(Figure 2.31), if
R1 is twice as big
as R2, then i1 will
be one-half as big
as i2.
See next slide for a formula that captures this…
The Current-Divider Rule

The current-divider rule:
For two resistors in parallel, if the
total current through the resistors
is i, then the current through each
resistor is given by:
𝑖1 =
𝑅2
𝑖
𝑅1 +𝑅2
and
𝑖2 =
𝑅1
𝑖
𝑅1 +𝑅2
The Current-Divider Rule in More
Complex Circuits


The current-divider rule is easiest to
apply in parallel circuits, but it also
holds for parallel resistors in more
complex circuits.
Example: In the
circuit shown,
suppose we
know the value of
the current i. Then
we can say that
𝑖1 =
20
𝑖
60+20
and
𝑖2 =
60
𝑖
60+20
Review of Short Circuits



Recall that an element with R=0 (or
with an extremely small resistance) is
called a short circuit.
Recall also that, since the resistance of
a short circuit is zero, the voltage
across it must always be zero.
Sometimes short
circuits are introduced
intentionally into a
circuit, but often they
result from a circuit
failure.
New Observations about Short
Circuits



The equivalent resistance of a short circuit in
parallel with anything else is zero.
An element or portion of a circuit is shortcircuited, or “shorted out,” when there is a
short circuit in parallel with it.
No current flows in a short-circuited element;
instead all current is diverted through the
short circuit itself.
 Example: In this
circuit, no current
will flow through
R2 or R3.
Potentiometers & Rheostats


A potentiometer is an adjustable
voltage divider that is widely used
in a variety of electronic circuit
applications. It is a three-terminal
device.
A rheostat is an adjustable
resistance. It has only two
terminals.