Fundamental Electrical
Concepts
Electric Charge (Q)
• Characteristic of subatomic particles that
determines their electromagnetic interactions
• An electron has a -1.602∙10
-19
Coulomb charge
• The rate of flow of charged particles is called
current
Current
• Current = (Number of electrons that pass in
one second) ∙ (charge/electron)
-1 ampere = (6.242∙10
18
e/sec) ∙(-1.602 10
-19
Coulomb/e)
Notice that an ampere = Coulomb/second
• The negative sign indicates that the current inside is
actually flowing in the opposite direction of the
electron flow
Electrons
Current
Current
• i = dq/dt – the derivitive or slope of the charge
when plotted against time in seconds
• Q = ∫ i ∙ dt – the integral or area under the
current when plotted against time in seconds
4
3
2
1
Current
amps
5 sec
Q delivered in 0-5 sec= 12.5 Coulombs
AC and DC Current
•DC Current has a constant value
•AC Current has a value that changes sinusoidally
Notice that AC current
changes in value and
direction
No net charge is
transferred
Why Does Current Flow?
• A voltage source provides the energy (or
work) required to produce a current
Volts = joules/Coulomb = dW/dQ
• A source takes charged particles (usually
electrons) and raises their potential so they
flow out of one terminal into and through a
transducer (light bulb or motor) on their way
back to the source’s other terminal
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7
Terms to Remember
• The Source can be any source of electrical energy. In practice, there are
three general possibilities: it can be a battery, an electrical generator, or
some sort of electronic power supply.
• The Load is any device or circuit powered by electricity. It can be as simple as
a light bulb or as complex as a modern high-speed computer.
• (Path) a wire or pathway which will allow electron to flow throughout a
circuit.
• Electricity can be described as the flow of charged particles. If the
particles accumulate on an object, we term this static electricity.
• (Direct Current) An electrical current that travels in one direction and used
within the computer's electronic circuits.
• (Alternating Current) The common form of electricity from power plant to
home/office. Its direction is reversed 60 times per second.
• Circuit is a conducting path for electrons.
Voltage
• Voltage is a measure of the potential energy
that causes a current to flow through a
transducer in a circuit
• Voltage is always measured as a difference
with respect to an arbitrary common point
called ground
• Voltage is also known as electromotive force
or EMF outside engineering
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9
What is Ohm’s Law?
Ohm's Law states that, at constant temperature, the electric
current flowing in a conducting material is directly proportional
to the applied voltage, and inversely proportional to the
Resistance.
Why is Ohms Law important?
Ohm’s Law is the relationship between power, voltage, current
and resistance. These are the very basic electrical units we
work with. The principles apply to alternating current (ac),
direct current (dc), or radio frequency (rf) .
Ohm’s Law
I = V / R
Georg Simon Ohm (1787-1854)
I = Current (Amperes) (amps)
V = Voltage (Volts)
R = Resistance (ohms)
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11
Characteristics of Ohm’s Law
Voltage: Difference of potential, electromotive force, ability to do
work.
Unit of measure Volt Symbol V (Current: Flow of electrons Unit
of measure Ampere Symbol I
Resistance: Opposition to current flow Unit of measure Ohm
often seen as the Greek letter Omega Symbol R
A Circuit
• Current flows from the higher voltage terminal of the source
into the higher voltage terminal of the transducer before
returning to the source
+
Source
Voltage
-
I
+ Transducer -
Voltage
The source expends
energy & the transducer
converts it into
something useful
I
circuit diagram
cell switch lamp wires
Scientists usually draw electric circuits using symbols;
Simple Circuits
• Series circuit
– All in a row
– 1 path for electricity
– 1 light goes out and the
circuit is broken
• Parallel circuit
– Many paths for electricity
– 1 light goes out and the
others stay on
Series and Parallel Circuits
• Series Circuits
– only one end of each component is connected
– e.g. Christmas tree lights
• Parallel Circuits
– both ends of a component are connected
– e.g. household lighting
Passive Devices
• A passive transducer device functions only
when energized by a source in a circuit
Passive devices can be modeled by a resistance
• Passive devices always draw current so that
the highest voltage is present on the terminal
where the current enters the passive device
+ V > 0 -
I > 0
Notice that the voltage is
measured across the device
Current is measured
through the device
Active Devices
• Sources expend energy and are considered
active devices
• Their current normally flows out of their
highest voltage terminal
• Sometimes, when there are multiple sources
in a circuit, one overpowers another, forcing
the other to behave in a passive manner
Power
• The rate at which energy is transferred from
an active source or used by a passive device
• P in watts = dW/dt = joules/second
• P= V∙I = dW/dQ ∙ dQ/dt = volts ∙ amps = watts
• W = ∫ P ∙ dt – so the energy (work in joules) is
equal to the area under the power in watts
plotted against time in seconds
Conservation of Power
• Power is conserved in a circuit - ∑ P = 0
• We associate a positive number for power as
power absorbed or used by a passive device
• A negative power is associated with an active
device delivering power
I
+
V
-
If I=1 amp
V=5 volts
Then passive
P=+5 watts
(absorbed)
If I= -1 amp
V=5 volts
Then active
P= -5 watts
(delivered)
If I= -1 amp
V= -5 volts
Then passive
P=+5 watts
(absorbed)
Example
• A battery is 11 volts and as it is charged, it
increases to 12 volts, by a current that starts
at 2 amps and slowly drops to 0 amps in 10
hours (36000 seconds)
• The power is found by multiplying the current
and voltage together at each instant in time
• In this case, the battery (a source) is acting like
a passive device (absorbing energy)
Voltage, Current & Power
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22
In this second example, we will calculate the amount of resistance (R) in a
circuit, given values of voltage (E) and current (I):
What is the amount of resistance (R) offered by the lamp?
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23
In the last example, we will calculate the amount of voltage supplied by a
battery, given values of current (I) and resistance (R):
What is the amount of voltage provided by the battery?
Resistance
• Property of a device that indicates how freely it will
allow current to flow given a specific voltage
applied. If low in value, current flows more freely.
• Measured in ohms (Ω = V/A or KΩ = V/mA)
I
• Obeys the passive sign + -
convention V
if I > 0 then V > 0
R = or V = I∙R This is Ohm’s Law!
Voltage
Current
Resistors
• Resistors are devices that are used in a variety
of circuits to make the currents and voltages
what you desire
• They are color-coded and come in various
tolerances, ± 1%, , ± 5% and , ± 10% are most
common
Resistor Color Code
http://www.elexp.com/t_resist.htm
56 * 10 kΩ = 560 kΩ
237 * 1 Ω = 237 Ω
Conductance (G)
• Conductance is the reciprocal of resistance
• Its unit is the siemen = amps/volts = 1/Ω
• G = I / V = 1 / R
• An equivalent measure of how freely a current
is allowed to flow in a device
Resistivity (ρ)
• Property of a material that indicates how
much it will oppose current flow
• R= (ρ ∙ length) / (cross sectional area)
• Units are ohms ∙ meters (Ω ∙ m)
• As the wire gets bigger, so does the cross
section making the resistance smaller
• As the length gets longer, the resistance goes
up proportionately
Conductors
• Materials with electrons that are loosely
bound to the nucleus and move easily
(usually one electron in the outer shell)
• Their low resistance goes up as the
material is heated, due to the vibration of
the atoms interfering with the movement
of the electrons
• The best conductors are superconductors at
temperatures near 0
o
Kelvin
Semiconductors
• Materials with electrons that are bound
more tightly than conductors (usually 4
electrons in the outer shell)
• Impurities are added in controlled amounts
to adjust the resistivity down
• Semiconductors become better conductors
at higher temperatures because the added
energy frees up more electrons (even
though the electron flow is impeded by the
increased atomic vibration)
Insulators
• Materials that have all 8 electrons in the outer
shell tightly bound to the nucleus
• It takes high temperatures or very high electric
fields to break the atomic bonds to free up
electrons to conduct a current
• Very high resistivities and resistances
Resistivity of Materials
• Glass
• Gold
• Silver
• Silicon
• Aluminum
• Copper
Put a number 1 beside
the material that
is the best conductor,
a 2 next to the
material next most
conductive, etc.
Resistivity of Materials
1 Silver - A conductor - ρ=1.64 ∙10
-8
ohm-m
2 Copper - A conductor - ρ= 1.72∙10
-8
ohm-m
3 Gold – A conductor - ρ=2.45∙10
-8
ohm-m
4 Aluminum - A conductor - ρ= 2.8∙10
-8
ohm-m
5 Silicon - A semiconductor - ρ=6.4 ∙10
2
ohm-m
6 Glass – An insulator – ρ=10
12
ohm-m
Example
• A round wire has a radius of 1 mm = 10
-3
m
• The wire is 10 meters long
• The wire is made of copper
• R= (1.72 ∙ 10
-8
Ω∙m ∙ 10 m) /( π ∙(10
-3
m)
2
)
• R = 0.0547 Ω
• This is negligible in most circuits where
resistances are often thousands of ohms
Power used by Resistors
• We saw that P = I ∙ V
• If by Ohm’s law, V = I ∙ R, then P = I
2
∙ R
• Or since I = V /R, then P = V
2
/ R
• Since resistance is always positive for passive
devices, then power is always positive
(meaning that power is always absorbed or
used)
Kirchoff’s Laws
Circuit Definitions
• Node – any point where 2 or more circuit
elements are connected together
– Wires usually have negligible resistance
– Each node has one voltage (w.r.t. ground)
• Branch – a circuit element between two
nodes
• Loop – a collection of branches that form a
closed path returning to the same node
without going through any other nodes or
branches twice
Example
• How many nodes, branches & loops?
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
Example
• Three nodes
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
Example
• 5 Branches
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
Example
• Three Loops, if starting at node A
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A B
C
Kirchoff’s Voltage Law (KVL)
• The algebraic sum of voltages around
each loop is zero
– Beginning with one node, add voltages across
each branch in the loop (if you encounter a +
sign first) and subtract voltages (if you
encounter a – sign first)
• Σ voltage drops - Σ voltage rises = 0
• Or Σ voltage drops = Σ voltage rises
Example
• Kirchoff’s Voltage Law around 1
st
Loop
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A B
C
I
2
I
1
+
I
2
R
2
-
+ I
1
R
1
-
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1
st
Loop
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A B
C
I
2
I
1
+
I
2
R
2
-
+ I
1
R
1
-
Starting at node A, add the 1
st
voltage drop: + I
1
R
1
Example
• Kirchoff’s Voltage Law around 1
st
Loop
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A B
C
I
2
I
1
+
I
2
R
2
-
+ I
1
R
1
-
Add the voltage drop from B to C through R
2
: + I
1
R
1
+ I
2
R
2
Example
• Kirchoff’s Voltage Law around 1
st
Loop
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A B
C
I
2
I
1
+
I
2
R
2
-
+ I
1
R
1
-
Subtract the voltage rise from C to A through Vs: + I
1
R
1
+ I
2
R
2
– Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Circuit Analysis
• When given a circuit with sources and
resistors having fixed values, you can use
Kirchoff’s two laws and Ohm’s law to
determine all branch voltages and currents
+
12 v
-
I
7Ω
3Ω
A
B
C
+ V
AB
-
+
V
BC
-
Circuit Analysis
• By Ohm’s law: V
AB
= I·7Ω and V
BC
= I·3Ω
• By KVL: V
AB
+ V
BC
– 12 v = 0
• Substituting: I·7Ω + I·3Ω -12 v = 0
• Solving: I = 1.2 A
+
12 v
-
I
7Ω
3Ω
A
B
C
+ V
AB
-
+
V
BC
-
Circuit Analysis
• Since V
AB
= I·7Ω and V
BC
= I·3Ω
• And I = 1.2 A
• So V
AB
= 8.4 v and V
BC
= 3.6 v
+
12 v
-
I
7Ω
3Ω
A
B
C
+ V
AB
-
+
V
BC
-
Series Resistors
• KVL: +I·10Ω – 12 v = 0, So I = 1.2 A
• From the viewpoint of the source, the 7
and 3 ohm resistors in series are
equivalent to the 10 ohms
+
12 v
-
I
10Ω
+
I·10Ω
-
Series Resistors
• To the rest of the circuit, series resistors
can be replaced by an equivalent
resistance equal to the sum of all resistors
. . .
Σ Rseries
Series resistors (same current through all)
I
I
Kirchoff’s Current Law (KCL)
• The algebraic sum of currents entering a
node is zero
– Add each branch current entering the node
and subtract each branch current leaving the
node
• Σ currents in - Σ currents out = 0
• Or Σ currents in = Σ currents out
Example
• Kirchoff’s Current Law at B
+
-
Vs
Is
R
1
R
2
R
3
+
Vo
-
A
B
C
I
2
I
1
Assign current variables and directions
Add currents in, subtract currents out: I
1
– I
2
– I
3
+ Is = 0
I
3
Circuit Analysis
10 A
8Ω 4Ω
A
B
+
V
AB
-
By KVL: - I
1
∙ 8Ω + I
2
∙ 4Ω = 0
Solving: I
2
= 2 ∙ I
1
By KCL: 10A = I
1
+ I
2
Substituting: 10A = I
1
+ 2 ∙ I
1
= 3 ∙ I
1
So I
1
= 3.33 A and I
2
= 6.67 A
And V
AB
= 26.33 volts
I
1
I
2
+ +
- -
Circuit Analysis
10 A
2.667Ω
A
B
+
V
AB
-
By Ohm’s Law: V
AB
= 10 A ∙ 2.667 Ω
So V
AB
= 26.67 volts
Replacing two parallel resistors (8 and 4 Ω)
by one equivalent one produces the same
result from the viewpoint of the rest of the
circuit.
Parallel Resistors
• The equivalent resistance for any number
of resistors in parallel (i.e. they have the
same voltage across each resistor):
1
Req =
1/R
1
+ 1/R
2
+ ∙∙∙ + 1/R
N
• For two parallel resistors:
Req = R
1
∙R
2
/(R
1
+R
2
)
Example Circuit
Solve for the currents through each resistor
And the voltages across each resistor
Example Circuit
Using Ohm’s law, add polarities and
expressions for each resistor voltage
+ I
1
∙10Ω -
+
I
2
∙8Ω
-
+ I
3
∙6Ω -
+
I
3
∙4Ω
-
Example Circuit
Write 1
st
Kirchoff’s voltage law equation
-50 v + I
1
∙10Ω + I
2
∙8Ω = 0
+ I
1
∙10Ω -
+
I
2
∙8Ω
-
+ I
3
∙6Ω -
+
I
3
∙4Ω
-
Example Circuit
Write 2
nd
Kirchoff’s voltage law equation
-I
2
∙8Ω + I
3
∙6Ω + I
3
∙4Ω = 0
or I
2
= I
3
∙(6+4)/8 = 1.25 ∙ I
3
+ I
1
∙10Ω -
+
I
2
∙8Ω
-
+ I
3
∙6Ω -
+
I
3
∙4Ω
-
Example Circuit
Write Kirchoff’s current law equation at A
+I
1
– I
2
- I
3
= 0
A
Example Circuit
• We now have 3 equations in 3 unknowns,
so we can solve for the currents through
each resistor, that are used to find the
voltage across each resistor
• Since I
1
- I
2
- I
3
= 0, I
1
= I
2
+ I
3
• Substituting into the 1st KVL equation
-50 v + (I
2
+ I
3
)∙10Ω + I
2
∙8Ω = 0
or I
2
∙18 Ω + I
3
∙ 10 Ω = 50 volts
Example Circuit
• But from the 2
nd
KVL equation, I
2
= 1.25∙I
3
• Substituting into 1
st
KVL equation:
(1.25 ∙ I
3
)∙18 Ω + I
3
∙ 10 Ω = 50 volts
Or: I
3
∙ 22.5 Ω + I
3
∙ 10 Ω = 50 volts
Or: I
3
∙ 32.5 Ω = 50 volts
Or: I
3
= 50 volts/32.5 Ω
Or: I
3
= 1.538 amps
Example Circuit
• Since I
3
= 1.538 amps
I
2
= 1.25∙I
3
= 1.923 amps
• Since I
1
= I
2
+ I
3,
I
1
= 3.461 amps
• The voltages across the resistors:
I
1
∙10Ω = 34.61 volts
I
2
∙8Ω = 15.38 volts
I
3
∙6Ω = 9.23 volts
I
3
∙4Ω = 6.15 volts
Example Circuit
Solve for the currents through each resistor
And the voltages across each resistor using
Series and parallel simplification.
Example Circuit
The 6 and 4 ohm resistors are in series, so
are combined into 6+4 = 10Ω
Example Circuit
The 8 and 10 ohm resistors are in parallel, so
are combined into 8∙10/(8+10) =14.4 Ω
Example Circuit
The 10 and 4.4 ohm resistors are in series, so
are combined into 10+4 = 14.4Ω
Example Circuit
Writing KVL, I
1
∙14.4Ω – 50 v = 0
Or I
1
= 50 v / 14.4Ω = 3.46 A
+
I
1
∙14.4Ω
-
Example Circuit
If I
1
= 3.46 A, then I
1
∙10 Ω = 34.6 v
So the voltage across the 8 Ω = 15.4 v
+34.6 v -
+
15.4 v
-
Example Circuit
If I
2
∙8 Ω = 15.4 v, then I
2
= 15.4/8 = 1.93 A
By KCL, I
1
-I
2
-I
3
=0, so I
3
= I
1
–I
2
= 1.53 A
+ 34.6 v -
+
15.4 v
-
Circuits with Dependent
Sources
Circuit with Dependent Sources
V
1
= 60 volts because the 20Ω resistor is in
parallel; by Ohm’s law, V
1
= I
2
·20Ω;
so I
2
= V
1
/20Ω = 60v/20Ω = 3 A
Circuit with Dependent Sources
If I
2
= 3 A, then the 5Ω·I
2
dependent source
is 15 volts and if V
1
= 60 v., then the V
1
/4Ω
dependent source is 15 A
Circuit with Dependent Sources
Writing Kirchoff’s Voltage law around the
outside loop, -60 v + 5Ω·I
2
+ 5Ω·I
3
= 0
where I
2
=3 A, so I
3
= (60–15)v / 5Ω = 9 A
Circuit with Dependent Sources
Writing Kirchoff’s Current law at B
I
4
+ I
3
+ V
1
/4 = 0 (all leaving node B)
Since V
1
/4Ω =15 A and I
3
= 9 A, I
4
= -24 A
Circuit with Dependent Sources
Writing Kirchoff’s Current law at A
I
4
+ I
1
– I
2
= 0
Since I
2
= 3 A and I
4
= -24 A, I
1
= 27 A
Circuit with Dependent Sources
60 v source generating, P=-27A·60v=-1620 watts
5·I
2
source absorbing, P=24A·15v=360 watts
V
1
/4 source absorbing, P=15A·45v= 675watts
I
1
=27A
I
4
= -24A
I
1
/4Ω=15A
V
2
= 45v
Circuit with Dependent Sources
20 Ω resistor absorbing, P=3A·60v=180 watts
5 Ω resistor absorbing, P=9A·45v= 405 watts
-1620 w +360 w + 675 w + 180 w + 405 w = 0
I
2
=3A
I
3
=9A
V
1
=60v
V
2
=45v
VA = 28 volts
Find I
1
2 Amps by Ohm’s law (I
2
∙14=28)
Find I
2
2 Amps by KCL (4-I
1
-I
2
=0)
Find V
B
24 volts by KVL (-I
1
∙14+I
2
∙2+V
B
=0)
Find I
3
6 Amps by Ohm’s law (I
3
∙4=24)
Find I
4
4 Amps by KCL (I
2
+I
4
-I
3
=0)
Sources in Series
+
-
V2
V1
+
-
+
-
V1+V2
Voltage sources
In series add
algebraically
Sources in Series
-
+
V2
V1
+
-
+
-
V1-V2
Start at the top
terminal and add.
If hit a + (+V1)
If you hit a – (-V2)
Sources in Series
-
+
V2
8·Vx
+
-
+
-
8·Vx-V2
If one source is
dependent, then
so is the equivalent
Sources in Series
Is
Is
Current sources in
series must be the
same value and
direction
Is
Sources in Parallel
I1 I2
Current sources in parallel add algebraically
I1+I2
Sources in Parallel
I1 I2
Current sources in parallel add algebraically
-I1+I2
Sources in Parallel
I1 5·Ix
If any source is dependent, then the
combination is also dependent
-I1+5·Ix
Sources in Parallel
Vs Vs
Voltage sources in parallel must be the same
value and same direction
Vs
+
-
+
-
+
-
Source Transformation
Source Transformation
• Practical voltage sources are current limited and
we can model them by adding a resistor in
series
• We want to create an equivalent using a current
source and parallel resistance for any R
L
+
-
Vs
R
S
R
L
+
V
L
-
I
L
Practical Source
Source Transformation
• V
L
and I
L
must be the same in both circuits
for any R
L
Ip
R
p
R
L
+
V
L
-
I
L
Practical Source
Source Transformation
• V
L
and I
L
must be the same in both circuits
for R
L
= 0, or a short circuit
• Ip = I
L
and V
L
= 0
Ip
R
p
R
L
=0
+
V
L
-
I
L
Practical Source
Source Transformation
• Now look at the voltage source in series
with the resistor with a short circuit
• I
L
= Vs/Rs and V
L
= 0
• So Ip = Vs/Rs
+
-
Vs
R
S
R
L
=0
+
V
L
-
I
L
Practical Source
Source Transformation
• V
L
and I
L
must also be the same in both
circuits for R
L
= ∞, or an open circuit
• I
L
= 0 and V
L
= Ip·Rp
Ip
R
p
R
L
=∞
+
V
L
-
I
L
Practical Source
Source Transformation
• Now look at the voltage source in series
with the resistor with an open circuit
• I
L
= 0 and V
L
= Vs, so Vs = Ip·Rp
• If Ip = Vs/Rs, then Rp = Rs
+
-
Vs
R
S
R
L
=∞
+
V
L
-
I
L
Practical Source
• We can transform the voltage source
• Why? Gets all components in parallel
Example
+
-
20v
4A
10Ω
15Ω
6Ω
+
Vo
-
I
1
• We can combine sources and resistors
• Ieq = 2A+4A = 6A, Req = 3Ω
Example
20v/10Ω
=2A
4A
10Ω
15Ω
6Ω
+
Vo
-
• Vo = 6A· 3Ω = 18 v
Example
6A 3Ω
+
Vo
-
• Going back to the original circuit, Vo=18 v
• KCL: I
1
+ 4A - 1.2A - 3A=0, so I
1
=0.2A
Example
+
-
20v
4A
10Ω
15Ω
6Ω
+
Vo = 18 v
-
18/15
=1.2A
18/6
=3A
I
1
• We can transform the current source after
first combining parallel resistances
• Why? Gets all components in series
Example
+
-
20v
4A
10Ω
15Ω
6Ω
+
Vo
-
• We can transform the current source after
first combining parallel resistances
• Req=6·15/(6+15)=30/7 Ω
Example
+
-
20v
4A
10Ω
30/7Ω
+
Vo
-
• We can now add the series voltage
sources and resistances
• Rtotal=100/7 Ω and Vtotal=20/7 volts
Example
+
-
20v
120/7v
10Ω
30/7Ω
+
-
I
1
• We can easily solve using KVL for I
1
• I
1
= 20/7 ÷ 100/7 = 0.2 A
Example
+
-
20/7 v
100/7 Ω
I
1
Voltage & Current Division
• We could have a circuit with multiple
resistors in series where we want to be
able to find the voltage across any resistor
• Clearly Req = Σ Ri, and I = Vs/Req
•
So Vi = I·Ri = Vs ·(Ri/Req)
Voltage Division
+
-
Vs
R1
Ri
I
. . . . . .
Rn
+ Vi -
• You have a 12 volt source, but some
devices in your circuit need voltages of 3
and 9 volts to run properly
• You can design a voltage divider circuit to
produce the necessary voltages
Voltage Division Application
+
-
Vs
R1
Ri
I
. . . . . .
Rn
+ Vi -
• To get 3, 6 and 3 across the three
resistors, any R, 2·R and R could be used
• 9 volts is available at A, 3 volts at B
Voltage Division Application
+
-
12 v
R
2·R
R
+ 6v -
+
9v
-
+ 3v - +
3v
-
A
B
Wheatstone Bridge
• This circuit is often used to measure
resistance or convert resistance into a
voltage.
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
Wheatstone Bridge
• Using the voltage divider at A,
• V
AD
= 100 v ∙ R/(100+R)Ω
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
D
Wheatstone Bridge
• Find the Voltage at B, using the voltage
divider theorem
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
Wheatstone Bridge
• V
BD
= 100 v ∙ 500Ω/(300+500)Ω = 62.5 v
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
D
Wheatstone Bridge
• Let’s find the relationship between V
AB
& R
V
AB
= V
AD
– V
BD
= 100∙R/(100+R) – 62.5
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
D
Wheatstone Bridge
• The Wheatstone bridge is considered
balanced when V
AB
=0 v.
• Find R
• R=167Ω
+
100v
-
100Ω
R
500Ω
300Ω
+ V
AB
-
A
B
Wheatstone Bridge
• What if we want to find the current through any
parallel resistor?
• Req = 1 / Σ(1/Ri) and V = Is·Req
• So Ii = V / Ri = Is·(Req/Ri)
Current Division
Is
R1
Ri
Rn
+
V
-
Ii
. . . . . .
Wheatstone Bridge
• The 10 A current source divides between
the two branches of the bridge circuit
10 A
100Ω
100Ω
500Ω
300Ω
+ V
AB
-
D
I
2
I
1
Wheatstone Bridge
• First, simplify by combining the series
resistances in each branch of the bridge
10 A
100Ω
100Ω
500Ω
300Ω
+ V
AB
-
D
I
2
I
1
Wheatstone Bridge
• First, simplify by combining the series
resistances in each branch of the bridge
10 A
100+100Ω
300+500Ω
I
2
I
1
Wheatstone Bridge
• Find the parallel equivalent resistance
• Req = 200∙800/(200+800) Ω = 160 Ω
10 A
100+100Ω
300+500Ω
I
2
I
1
Wheatstone Bridge
• I
1
= 10 A ∙(160/200) = 8 A
• I
2
= 10 A ∙(160/800) = 2 A
10 A
100+100Ω
300+500Ω
I
2
I
1
Voltage vs Current Division
• Voltage Division
• Vi = Vs · Ri/Req
• Req = Σ Ri
• Ri/Req < 1
• Vi < Vs
• Series resistors only
• Current Division
• Ii = Is · Req/Ri
• Req = 1÷ Σ (1/Ri)
• Req/Ri < 1
• Ii < Is
• Parallel resistors only
Application
• A practical source of 12 volts has a 1Ω
internal resistance. Design a voltage
divider that will be used to power up to ten
6 volt light bulbs in parallel.
1Ω
R
1
+
-
12 v
R
2
100Ω
100Ω
100Ω
. . . .
10 lights
Application
• The voltage across each bulb must be
within the range of 5.5 to 6 volts in order
for the bulbs to be bright enough. Find R
1
and R
2
.
1Ω
R
1
+
-
12 v
R
2
100Ω
100Ω
100Ω
. . . .
10 lights
Application
• With no bulbs, the maximum voltage is
obtained:
12v·R
2
1Ω+R
1
+R
2
1Ω
R
1
+
-
12 v
R
2
+
Vbulb
-
Vbulb = 6.0 v =
Application
• The parallel combination of ten 100Ω
bulbs is 10Ω, the parallel combination with
R2 is 10·R
2
/(10+R
2
)
10·R
2
/(10+R
2
)
1Ω
R
1
+
-
12 v
R
2
100Ω
100Ω
100Ω
. . . .
10 lights
+
Vbulb
-
Ω
Application
• Using the voltage divider and the minimum
voltage allowed:
12v·10·R
2
/(10+R
2
)
1Ω+R
1
+10·R
2
/(10+R
2
)
1Ω
R
1
+
-
12 v
R
2
100Ω
100Ω
100Ω
. . . .
10 lights
+
Vbulb
-
Vbulb = 5.5 v =
Solution
• Solving the two equations simultaneously,
R
1
= .82Ω and R
2
= 1.82Ω
• With 10 bulbs, R
2
in parallel with ten 100Ω
resistors would be 1.54Ω
• Using the voltage divider, with 10 bulbs
Vbulb = 12∙1.54Ω÷(1+.82Ω+1.54Ω) = 5.5 v
• Using the voltage divider, with no bulbs
Vbulb = 12∙1.82Ω÷(1+.82Ω+1.82Ω) = 6.0 v
Capacitors and
RC circuits
Capacitors
• Composed of two conductive plates separated
by an insulator (or dielectric).
– Commonly illustrated as two parallel metal plates
separated by a distance, d.
C = ε A/d
where ε = ε
r
ε
o
ε
r
is the relative dielectric constant
ε
o
is the vacuum permittivity
Effect of Dimensions
• Capacitance increases with
– increasing surface area of the plates,
– decreasing spacing between plates, and
– increasing the relative dielectric constant of the
insulator between the two plates.
Types of Capacitors
• Fixed Capacitors
– Nonpolarized
• May be connected into circuit with either terminal of capacitor
connected to the high voltage side of the circuit.
– Insulator: Paper, Mica, Ceramic, Polymer
– Electrolytic
• The negative terminal must always be at a lower voltage than
the positive terminal
– Plates or Electrodes: Aluminum, Tantalum
Nonpolarized
• Difficult to make nonpolarized capacitors that
store a large amount of charge or operate at
high voltages.
– Tolerance on capacitance values is very large
• +50%/-25% is not unusual
PSpice
Symbol
Electrolytic
Symbols Fabrication
Variable Capacitors
• Cross-sectional area is changed as one set of
plates are rotated with respect to the other.
http://www.tpub.com/neets/book2/3f.htm
PSpice
Symbol
Electrical Properties of a Capacitor
• Acts like an open circuit at steady state when connected
to a d.c. voltage or current source.
• Voltage on a capacitor must be continuous
– There are no abrupt changes to the voltage, but there may be
discontinuities in the current.
• An ideal capacitor does not dissipate energy, it takes
power when storing energy and returns it when
discharging.
Properties of a Real Capacitor
• A real capacitor does dissipate energy due
leakage of charge through its insulator.
– This is modeled by putting a resistor in
parallel with an ideal capacitor.
Energy Storage
• Charge is stored on the plates of the capacitor.
Equation:
Q = CV
Units:
Farad = Coulomb/Voltage
Farad is abbreviated as F
Sign Conventions
• The sign convention used with a
capacitor is the same as for a power
dissipating device.
• When current flows into the positive side
of the voltage across the capacitor, it is
positive and the capacitor is dissipating
power.
• When the capacitor releases energy back
into the circuit, the sign of the current will
be negative.
Charging a Capacitor
At first, it is easy to store charge in the
capacitor.
As more charge is stored on the plates of the
capacitor, it becomes increasingly difficult to
place additional charge on the plates.
Coulombic repulsion from the charge already on
the plates creates an opposing force to limit the
addition of more charge on the plates.
Voltage across a capacitor increases rapidly as charge
is moved onto the plates when the initial amount of
charge on the capacitor is small.
Voltage across the capacitor increases more slowly as
it becomes difficult to add extra charge to the plates.
Adding Charge to Capacitor
• The ability to add charge to a capacitor depends
on:
– the amount of charge already on the plates of the
capacitor
and
– the force (voltage) driving the charge towards the
plates (i.e., current)
Discharging a Capacitor
At first, it is easy to remove charge in the capacitor.
Coulombic repulsion from charge already on the plates creates a
force that pushes some of the charge out of the capacitor once
the force (voltage) that placed the charge in the capacitor is
removed (or decreased).
As more charge is removed from the plates of the capacitor, it
becomes increasingly difficult to get rid of the small amount of
charge remaining on the plates.
Coulombic repulsion decreases as charge spreads out on the
plates. As the amount of charge decreases, the force needed to
drive the charge off of the plates decreases.
Voltage across a capacitor decreases rapidly as charge is removed
from the plates when the initial amount of charge on the capacitor is
small.
Voltage across the capacitor decreases more slowly as it becomes
difficult to force the remaining charge out of the capacitor.
RC Circuits
RC circuits contain both a resistor R and a capacitor C (duh).
Until now we have assumed that charge is
instantly placed on a capacitor by an emf.
The approximation resulting from this
assumption is reasonable, provided the
resistance between the emf and the capacitor
being charged/discharged is small.
If the resistance between the emf and the
capacitor is finite, then the charge on the
capacitor does not change instantaneously.
Q
t
Q
t
Switch open, no current flows.
Charging a Capacitor
ε
R
switch
C
t<0
Close switch, current flows.
t>0
I
Apply Kirchoff’s loop rule*
(green loop) at the instant
charge on C is q.
*Convention for capacitors is “like” batteries: negative if going across from + to -.
ε
q
- -IR =0
C
This equation is
deceptively
complex because
I depends on q
and both depend
on time.
When t=0, q=0 and I
0
=ε/R.
Limiting Cases
ε
R
switch
C
When t is “large,” the capacitor
is fully charged, the current
“shuts off,” and Q=Cε.
I
ε
q
- -IR =0
C
ε
q
- -IR =0
C
Math:
ε q
I= -
R RC
ε εεdq q C q C -q
= - = - =
dt R RC RC RC RC
ε
dq dt
=
C -q RC
ε
dq dt
= -
q-C RC
More math:
0
ε
∫∫
qt
0
dq dt
= -
q-C RC
( )
0
ε
∫
t
q
0
1
ln q - C = - dt
RC
ε
ε



q-C t
ln = -
-C RC
ε
ε
t
-
RC
q-C
= e
-C
εε
t
-
RC
q-C =-C e
Still more math:
εε
t
-
RC
q=C -C e
ε



t
-
RC
q=C 1-e
( )



t
-
RC
q t =Q 1-e
( )
ε εε
t tt
- --
RC RC RC
dq C C
It= = e= e=e
dt RC RC R
RC is the “time constant” of the circuit; it tells us “how fast”
the capacitor charges and discharges.
ε
q
- -IR =0
C
Why not just
solve this for
q and I?
Charging a capacitor; summary:
( )



t
-
RC
final
q t =Q 1-e
( )
ε
t
-
RC
It= e
R
Charging Capacitor
0
0.002
0.004
0.006
0.008
0.01
0 0.2 0.4 0.6 0.8 1
t (s)
q (C)
Charging Capacitor
0
0.01
0.02
0.03
0.04
0.05
0 0.2 0.4 0.6 0.8 1
t (s)
I (A)
Sample plots with ε=10 V, R=200 Ω, and C=1000 µF.
RC=0.2 s
recall that this is I
0
,
also called I
max
In a time t=RC, the capacitor charges to Q(1-e
-1
) or 63% of its
capacity…
Charging Capacitor
0
0.002
0.004
0.006
0.008
0.01
0 0.2 0.4 0.6 0.8 1
t (s)
q (C)
Charging Capacitor
0
0.01
0.02
0.03
0.04
0.05
0 0.2 0.4 0.6 0.8 1
t (s)
I (A)
RC=0.2 s
…and the current drops to I
max
(e
-1
) or 37% of its maximum.
τ=RC is called the time constant of the RC circuit
Capacitor charged, switch
open, no current flows.
Discharging a Capacitor
R
switch
C
t<0
Close switch, current flows.
t>0
I
Apply Kirchoff’s loop rule*
(green loop) at the instant
charge on C is q.
*Convention for capacitors is “like” batteries: positive if going across from - to +.
q
-IR =0
C
+Q
-Q -q
+q
q
-IR =0
C
Math:
q
IR =
C
−
dq
I=
dt
dq dt
= -
q RC
dq q
-R =
dt C
negative because
charge decreases
More math:
∫∫ ∫
qt t
Q0 0
dq dt 1
= - = - dt
q RC RC
( )
∫
t
q
Q
0
1
ln q = - dt
RC



qt
ln = -
Q RC
t
-
RC
q(t)=Q e
tt
--
RC RC
0
dq Q
I(t)=- = e =I e
dt RC
same equation
as for charging
Disharging a capacitor; summary:
( )
0
t
-
RC
I t =I e
Sample plots with ε=10 V, R=200 Ω, and C=1000 µF.
RC=0.2 s
t
-
RC
0
q(t)=Q e
Discharging Capacitor
0
0.002
0.004
0.006
0.008
0.01
0 0.2 0.4 0.6 0.8 1
t (s)
q (C)
Discharging Capacitor
0
0.01
0.02
0.03
0.04
0.05
0 0.2 0.4 0.6 0.8 1
t (s)
I (A)
Discharging Capacitor
0
0.01
0.02
0.03
0.04
0.05
0 0.2 0.4 0.6 0.8 1
t (s)
I (A)
Discharging Capacitor
0
0.002
0.004
0.006
0.008
0.01
0 0.2 0.4 0.6 0.8 1
t (s)
q (C)
In a time t=RC, the capacitor discharges to Qe
-1
or 37% of its
capacity…
RC=0.2 s
…and the current drops to I
max
(e
-1
) or 37% of its maximum.
Notes
( )
ε
t
-
RC
It= e
R
This is for charging a capacitor.
ε/R = I
0
= I
max
is the initial current,
and depends on the charging emf
and the resistor.
( )
0
t
-
RC
I t =I e
This is for discharging a capacitor.
I
0
= Q/RC, and depends on how
much charge Q the capacitor
started with.
I
0
for charging is equal to I
0
for discharging only if the
discharging capacitor was fully charged.
SUMMARY
( )



t
-
RC
final
q t =Q 1-e
t
-
RC
0
q(t)=Q e
V =IR
Q(t)=CV(t)
This is always true for a capacitor.
Q
final
= CV, where V is the potential
difference of the charging emf.
Q
0
is the charge on the capacitor
at the start of discharge. Q
0
= Cε
only if you let the capacitor charge
for a “long time.”
Ohm’s law applies to resistors, not
capacitors. Sometimes you can get
away with using this. Better to
take dq/dt.
Capacitors in Parallel
C
eq
for Capacitors in Parallel
i
4321eq
4321
4433
2211
4321
C
CCCC
dt
dv
Ci
dt
dv
C
dt
dv
C
dt
dv
C
dt
dv
Ci
dt
dv
Ci
dt
dv
Ci
dt
dv
Ci
dt
dv
Ci
iiiii
eqin
in
in
+++=
=
+++=
==
==
+++=
Capacitors in Series
C
eq
for Capacitors in Series
i
( ) ( ) ( ) ( )
[ ]
1
4321eq
t
t
t
t
4
t
t
3
t
t
2
t
t
1
t
t
4
4
t
t
3
3
t
t
2
2
t
t
1
1
4321
1111C
idt
1
idt
1
idt
1
idt
1
idt
1
idt
1
idt
1
idt
1
idt
1
1
o
1
o
1
o
1
o
1
o
1
o
1
o
1
o
1
o
−
+++=
=
+++=
==
==
+++=
∫
∫∫∫∫
∫∫
∫∫
CCCC
C
v
CCCC
v
C
v
C
v
C
v
C
v
vvvvv
eq
in
in
in
General Equations for C
eq
Parallel Combination Series Combination
• If P capacitors are in
parallel, then
• If S capacitors are in
series, then:
1
1
1
−
=
=
∑
S
s
s
eq
C
C
∑
=
=
P
p
Peq
CC
1
Summary
• Capacitors are energy storage devices.
• An ideal capacitor act like an open circuit at steady state
when a DC voltage or current has been applied.
• The voltage across a capacitor must be a continuous function;
the current flowing through a capacitor can be discontinuous.
• The equations for equivalent capacitance for
capacitors in parallel capacitors in
series
1
1
1
−
=
=
∑
S
s
s
eq
C
C
∑
=
=
P
p
Peq
CC
1
∫
==
1
1
t
t
CC
C
C
o
dti
C
v
dt
dv
Ci
Measuring Instruments
You know how to calculate the
current in this circuit:
Measuring Instruments: Ammeter
V
R
.
V
I =
R
If you don’t know V or R, you can
measure I with an ammeter.
.
V
I =
R+r
To minimize error the ammeter resistance r should very small.
Any ammeter has a resistance r. Any ammeter has a resistance r. The current you measure is
r
Example: an ammeter of resistance 10 mΩ is used to measure
the current through a 10 Ω resistor in series with a 3 V battery
that has an internal resistance of 0.5 Ω. What is the percent
error caused by the nonzero resistance of the ammeter?
V=3 V
R=10 Ω
r=0.5 Ω
Actual current:
V
I =
R+r
3
I =
10+0.5
I = 0.286 A = 286 mA
You might see the symbol ε
used instead of V.
V=3 V
R=10 Ω
r=0.5 Ω
Current with ammeter:
A
V
I =
R+r+R
3
I =
10+0.5+0.01
I = 0.285 A = 285 mA
R
A
×
0.286-0.285
% Error = 100
0.286
% Error = 0.3 %
A Galvanometer
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/galvan.html#c1
When a current is passed through a coil connected to a needle,
the coil experiences a torque and deflects.
An ammeter (and a voltmeter) is based on a galvanometer.
For now, all you need to know is that the
deflection of the galvanometer needle is
proportional to the current in the coil (red).
A typical galvanometer has a resistance of a few tens of ohms.
A galvanometer-based ammeter uses a galvanometer and a
shunt, connected in parallel:
A
I
G
R
G
R
SHUNT
I
G
I
SHUNT
I
⇒
Everything inside the blue box is the ammeter.
The resistance of the ammeter is
= +
A G SHUNT
11 1
RRR
=
+
G SHUNT
A
G SHUNT
R R
R
RR
G
R
G
R
SHUNT
I
G
I
SHUNT
I
Homework hint: “the galvanometer reads 1A full scale”
means a current of I
G
=1A produces a full-scale deflection of
the galvanometer needle. The needle deflection is
proportional to the current I
G
.
If you want the ammeter shown to read 5A full scale, then
the selected R
SHUNT
must result in I
G
=1A when I=5A. In that
case, what are I
SHUNT
and V
AB
(=V
SHUNT
)?
A
B
A galvanometer-based ammeter uses a galvanometer and a
shunt, connected in parallel:
G
R
G
R
S
I
G
I
S
I
= +
A GS
111
RRR
Example: what shunt resistance is required for an ammeter to
have a resistance of 10 mΩ, if the galvanometer resistance is
60 Ω?
= −
S AG
111
RRR
( ) ( )
= = = Ω
GA
S
GA
60 .01
R R
R 0.010
R -R 60-.01
The shunt resistance is chosen so that I
G
does not exceed the
maximum current for the galvanometer and so that the
effective resistance of the ammeter is very small.
(actually 0.010002 Ω)
To achieve such a small resistance, the shunt is probably a
large-diameter wire or solid piece of metal.
( ) ( )
= = = Ω
GA
S
GA
60 .01
R R
R 0.010
R -R 60-.01
Web links: ammeter design, ammeter impact on circuit,
clamp-on ammeter (based on principles we will soon be
studying).
Measuring Instruments: Voltmeter
You can measure a voltage by placing a galvanometer in
parallel with the circuit component across which you wish to
measure the potential difference.
V=3 V
R=10 Ω
r=0.5 Ω
G
R
G
a
b
V
ab
=?
Example: a galvanometer of resistance 60 Ω is used to
measure the voltage drop across a 10 kΩ resistor in series with
a 6 V battery and a 5 kΩ resistor (neglect the internal
resistance of the battery). What is the percent error caused by
the nonzero resistance of the galvanometer?
First calculate the actual voltage drop.
V=6 V
R
1
=10 kΩ
R
2
=5 kΩ
a
b
= ×Ω
3
eq 1 2
R R +R =15 10
= = = ×
×Ω
-3
3
eq
V 6 V
I 0.4 10 A
R 15 10
( )( )
= × × Ω=
-3 3
ab
V =IR 0.4 10 10 10 4 V
The measurement is made with the galvanometer.
V=6 V
R
1
=10 kΩ
R
2
=5 kΩ
G
R
G
=60 Ω
a
b
60 Ω and 10 kΩ resistors in parallel
are equivalent to an 59.6 Ω resistor.
The total equivalent resistance is
5059.6 Ω, so 1.19x10
-3
A of current
flows from the battery.
I=1.19 mA
The voltage drop from a to b is then
measured to be
6-(1.19x10
-3
)(5000)=0.07 V.
The percent error is.
×
4-.07
% Error = 100=98%
4
To reduce the percent error, the device being used as a
voltmeter must have a very large resistance, so a voltmeter
can be made from galvanometer in series with a large
resistance.
V
G
R
G
R
Ser
⇒
Everything inside the blue box is the voltmeter.
a
b
V
ab
a
b
V
ab
Homework hints: “the galvanometer reads 1A full scale” would mean a current of I
G
=1A would produce
a full-scale deflection of the galvanometer needle.
If you want the voltmeter shown to read 10V full scale, then the selected R
Ser
must result in I
G
=1A
when V
ab
=10V.
Example: a voltmeter of resistance 100 kΩ is used to measure
the voltage drop across a 10 kΩ resistor in series with a 6 V
battery and a 5 kΩ resistor (neglect the internal resistance of
the battery). What is the percent error caused by the nonzero
resistance of the voltmeter?
We already calculated the actual
voltage drop (3 slides back).
V=6 V
R
1
=10 kΩ
R
2
=5 kΩ
a
b
( )( )
= × × Ω=
-3 3
ab
V =IR 0.4 10 10 10 4 V
The measurement is now made with the voltmeter.
V=6 V
R
1
=10 kΩ
R
2
=5 kΩ
G
R
G
=100 kΩ
a
b
100 kΩ and 10 kΩ resistors in
parallel are equivalent to an 9090 Ω
resistor. The total equivalent
resistance is 14090 Ω, so 4.26x10
-3
A of current flows from the battery.
I=4.26 mA
The voltage drop from a to b is then
measured to be
6-(4.26x10
-3
)(5000)=3.9 V.
The percent error is.
×
4-3.9
% Error = 100=2.5%
4
Not great, but much better. Larger R
ser
is needed for high accuracy.
An ohmmeter measures resistance. An ohmmeter is made
from a galvanometer, a series resistance, and a battery.
G
R
G
R
Ser
R=?
The ohmmeter is connected in parallel with the unknown
resistance with external power off. The ohmmeter battery
causes current to flow, and Ohm’s law is used to determine
the unknown resistance.
Ω
Measuring Instruments: Ohmmeter
Everything inside the blue
box is the ohmmeter.
To measure a really small resistance, an ohmmeter won’t
work.
Solution: four-point probe.
V
A
reference: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/movcoil.html#c4
Measure current and voltage separately, apply Ohm’s law.