PHY305F
Electronics Laboratory
Lecture Notes
Department of Physics
University of Toronto
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 1
PHY305F
Electronics Laboratory
Section 1
DC Circuit Basics: Passive and
Linear Components and Circuits
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 2
1
Basic Concepts
• Direct current (DC) circuit analysis
→ deals with constant currents and voltages
• Alternating current (AC) circuit analysis
→ deals with time-varying current and voltage signals whose time
average values are zero
• DC circuit components
→ constant voltage source
→ constant current source
→ resistor
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 3
Electric Current
• The fundamental electric quantity is charge, q, and the smallest
amount of charge that exists is that carried by an electron (-1.602
× 10-19 C). The SI unit of charge is the Coulomb (C).
• Electric current, i, is the time rate of change of positive charge
passing through a reference area (typically the cross section of a
conducting wire). The SI unit of current is the Ampere (A):
1 A = 1 C/s.
dq C
i=
dt
s
• The charges in motion that give rise to the current are negatively
charged electrons, but by convention, the positive direction of
current flow is that of positive charges. So electrons move in the
opposite direction to the current flow.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 4
2
Potential Difference (or Voltage)
• Work or energy is needed to move charge. The total work per
unit charge associated with the motion of charge between two
points is the voltage. The voltage is more correctly the potential
difference arising from an electric field E. The change in potential
dv across a distance in an electric field is
dv = −E • d r
• A positive charge will move from a higher to a lower potential.
The potential difference, or voltage, between two points is
V2
v = v 21 = v 2 − v1 = ∫ dv
V1
• The SI unit of voltage is the Volt (V): 1 V = 1 J/C.
• Note: current flowing in a conductor is due to a potential
difference between its ends. Electrons move from a point of less
positive potential to a point of more positive potential, and the
current flows in the opposite direction.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 5
Resistance and Ohm’s Law
• For most materials: v ∝ i
with
v = Ri .
This is Ohm’s Law.
• Here,
→ v = v 2 − v1 is the voltage across the object
→ i is the current through the object
→ R is a constant of proportionality called the resistance of the object
• Resistance is a function of material and shape of the object.
• The SI unit of resistance is the Ohm (Ω).
• The inverse of resistivity is conductivity.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 6
3
Resistors
i
i
+
l
l R=
A
R
v
1/R
–
A
v
Physical resistors
with resistance R.
Typical materials are
carbon, metal film.
Circuit symbol
i-v characteristic
(from Rizzoni Figure 2.20)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 7
i vs. v Curves
• i-v curves provide a graphical means of representing the terminal
characteristics of circuit elements
• An i-v curve defines the circuit element, in that if any prescribed
voltage (or current) is applied, then the resulting current (or
voltage) is directly obtainable from the i-v curve.
• Power can thus also be derived from an i-v curve.
Example: a tungsten filament light bulb
→
→
→
→
use a variable voltage source and measure current for each voltage
positive voltage gives positive current, negative v gives negative i
in both cases, power is positive as the bulb dissipates power
only two quadrants are used; in the other two power would be
negative implying that the bulb generated power
→ not linear because resistance is function of temperature (i.e. power)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 8
4
i-v Curve for a Tungsten Light Bulb
i (amps)
0.5
0.4
0.3
0.2
0.1
Current
meter
Variable
voltage
source
10 20 30 40 50 60 v (volts)
–60 –50 –40 –30 –20 –10 0
– 0.1
–0.2
i
+
–0.3
v
–0.4
–
–0.5
(from Rizzoni Figure 2.18)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 9
The Resistor Colour Code
b
4
b
b
3
2
b
1
Color bands
black
brown
red
orange
yellow
green
0
1
2
3
4
5
blue
violet
gray
white
silver
gold
Resistor value = ( b 1 b 2 )
10 b 3;
b 4 = % tolerance in actual value
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
6
7
8
9
10%
5%
(from Rizzoni Figure 2.22)
Section 1, Page 10
5
Electric Power
• Recall that power is work/time. The power generated or
dissipated by a circuit element can be represented as
Power =
• So, P =
Work
Work Charge
=
×
= Voltage × Current
Time Charge Time
dW d(qv )
=
= vi
dt
dt
• Power (like voltage) can be positive or negative. The passive
sign convention gives
→ P is positive if device absorbs energy
i.e., if current flows from a higher to a lower voltage (+ to -), then
power is absorbed and will be a positive quantity
→ P is negative if device supplies energy
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 11
Passive Sign Convention for Power
Methodology:
(1) Choose an arbitrary direction of current flow.
(2) Label polarities of all active elements (voltage & current sources).
(3) Assign polarities to all passive elements (resistors & other loads);
for passive elements, current always flows into the positive
terminal.
(4) Compute the power dissipated by each element according to the
following rule:
→ If positive current flows into the positive terminal of an element,
then the power dissipated is positive (i.e., element absorbs power).
→ If the current leaves the positive terminal of an element, then the
power dissipated is negative (i.e., element delivers power).
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 12
6
Passive Sign Convention for Power
Power dissipated
= v(-i) = (-v)i = -vi
Power generated = vi
Power dissipated = vi
Power generated
= v(-i) = (-v) i = -vi
• The polarity of the voltage across the source and the direction of
the current through it indicate that the source is doing work in
moving charge from a lower potential to a higher potential.
• The load is dissipating energy because the direction of the
current indicates that charge is being displaced from a higher
potential to a lower potential.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 13
Ideal Sources
• To maintain a potential drop and flow of charge, an external
energy source called an electromotive force (EMF) is needed.
→ Examples: battery, power supply, signal generator
→ Without an EMF, the internal electric field arising due to the
movement charge will cancel the external electric field that caused
the flow of charge, resulting in zero current.
• An ideal voltage source maintains a constant voltage regardless
of the current that it must put out.
• An ideal current source maintains a constant current regardless
of the voltage needed.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 14
7
Schematic Diagrams
• A schematic diagram consists of idealized circuit elements that
each represent some property of the actual circuit.
• Symbol conventions
→ v and i - generic voltage and current sources
→ v(t) and i(t) - time-varying voltage and current
→ V and I - constant or DC voltage and current sources
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 15
Practical Voltage and Current Sources
• The ideal voltage and current sources discussed previously do
not account for the internal resistance of real sources.
• Consider an ideal voltage source:
→ as the load resistance R decreases, the source
must supply increasing current to maintain voltage
vs across terminals A and B (since i(t)=vs(t)/R)
→ therefore the ideal voltage source must provide an infinite amount
of current to the load in the limit as the load resistance tends to 0
→ this is impossible: there is a limit to the current that can be provided
by a practical voltage source
• The limitations of practical sources can be approximated by
considering the internal resistance of a source.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 16
8
Model of a Practical Voltage Source
• Model of practical voltage sources consists of an ideal voltage
source vs in series with a (small) resistance rs.
• rs provides a limit to the maximum current that vs can provide.
rS
iS
Practical
voltage
source
vS
+
_
vL
vs
rs + RL
lim is =
R L →0
iS
max
vs
rs
The maximum (short circuit)
current which can be supplied
by a practical voltage source is
+
+
_
RL
–
rS
vS
is =
+
vL
–
v L = iSRL = v s
(from Rizzoni Figure 2.38)
is (max) =
vs
rs
RL
rs + RL
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 17
Model of a Practical Current Source
• Model of a practical current source consists of an ideal current
source vs in parallel with a (large) resistance rs.
• rs provides a limit to the maximum voltage that is can provide.
• As the load resistance → infinity (an open circuit), the output
voltage of the current source approaches its limit.
A model for practical current sources
consists of an ideal source in parallel
with an internal resistance.
Maximum output voltage for
practical current source with
open-circuit load: v
=i
s (max)
+
iS
rS
vS
RL
–
+
srs
iS
rS
vS
–
(from Rizzoni Figure 2.39)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 18
9
Ground
• Strictly, voltage is the potential difference measured across an
electrical device. However, it is common to refer to the voltage at
a point; in this case the voltage is measured relative to a
reference point called the ground.
• Ground is assigned a potential of zero Volts for convenience,
although grounds in different circuits may not be at the same
potential.
• Three types of ground:
→ Earth - an infinite electrical sink
that can accept or supply large amounts of charge
→ Chassis - the metal framework/cover of an instrument
→ Common - reference point shared between circuits
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 19
Kirchoff’s Laws
Kirchoff’s Voltage Law (conservation of energy):
The net voltage around a closed circuit is zero.
Kirchoff’s Current Law (conservation of charge):
The sum of the currents at a node must equal zero.
N
∑v
n
=0
n=1
N
∑i
n
=0
n=1
Methodology:
(1) Define the currents and voltages on a diagram.
(2) Apply Kirchoff’s Laws to loops and nodes.
(3) Write down a set of linear algebraic equations.
(4) Solve for the unknown parameters.
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 20
10
Application of KCL to an Automotive Circuit
Ibatt
Equivalent electrical circuit
Ihead
Itail
Istart
Ifan
Ilocks
Idash
+
Vbatt
–
KCL gives: Ibatt-Ihead-Itail-Istart-Ifan-Ilocks-Idash=0
Current supplied by battery is divided among the circuits.
(from Rizzoni Figure 2.4)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 21
Application of KVL to an Electrical Vehicle
Battery Pack
Vbatt1 Vbatt2
Vbattn
12 V 12 V 12 V
Equivalent electrical circuit
12 V 12 V
+
DC-AC converter
(electric drive)
vbatt1
AC motor
31
KVL gives:
∑v
n=1
batt n
+
–
vbatt2 vbatt3 vbatt31
–+ – + –
+
Power
converter
and motor
vdrive
–
− v drive = 0
Electric drive is supplied by 31x12 V = 372 V battery pack.
(from Rizzoni Figure 2.8)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 22
11
Resistors in Series
• Two or more circuit elements are in series if the identical current
flows through each of the elements.
• For resistors connected in series, apply KVL to get:
N
N
n=1
n=1
N
v = ∑ v i = I∑ Ri
Equivalent resistance = REQ = ∑ Ri
n=1
R1
+
1.5 V
–
v
1
+
+
_
v
i
–
v
R2
R1
R 2
R3
Rn
RN
+
3
R
2
–
R EQ
3
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 23
Voltage Dividers
Voltage dividers are related to the concept of resistors in series.
R1
v
in
A
R2
v
out
B
• Voltage across input source: v in = (R1 + R 2 ) i
• Voltage across output between terminals A and B: v out = R 2 i
R2
• Output from the voltage divider is thus: v out =
v in
R1 + R 2
• In general, for a circuit with N series resistors and a voltage
source:
Rn
v in
vn =
R1 + R 2 + ... + Rn + ... + RN
PHY305F - Electronics Laboratory I,Fall Term (Kim Strong)
Section 1, Page 24
12
Resistors in Parallel
• Two or more circuit elements are in parallel if the
identical voltage appears across each of the elements.
• For resistors connected in parallel, apply KCL to get:
N
N
v
n=1 R i
i = ∑ ii = i∑
n=1
iS
Equivalent resistance :
i1
i2
i3 +
R1
R2
R3 v
N
1
i
1
= =∑
REQ v n=1 Ri
R1 R2 R3 Rn RN
REQ
–
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 25
Current Dividers
Current dividers are related to the concept of resistors in parallel.
A
i
in
R1
R2
i
out
B
• Source current is divided between resistors: iin = i1 + i2 =
• Current at the output between A and B: v = ioutR 2
• Output current from the current divider is thus:
iout =
v
v
+
R1 R 2
R1
iin
R1 + R 2
• In general, for a circuit with N parallel resistors and
a current source:
1 Rn
iin
in =
1 R1 + 1 R 2 + ... + 1 Rn + ... + 1 RN
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 26
13
Ohmmeters
• Ohmmeter - measures resistance of a circuit element when
connected across it.
→ The resistance of an element can be measured only when the
element is disconnected from any other circuit.
Ω
R
Ω
Symbol for
ohmmeter
Circuit for the
measurement of
resistance R
(from Rizzoni Figure 2.40)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 27
Ammeters
• Ammeter - measures current flowing through a circuit element,
when connected in series with it.
→ The ammeter must be placed in series with the element whose
current is to be measured (e.g., resistor R2).
→ The ammeter should not restrict the flow of current (i.e., cause a
→
voltage drop) or else it will not be measuring the true current
flowing in the circuit.
An ideal ammeter has zero internal resistance.
R1
R1
A
A
vS _+
R2 vS +_
i
Symbol for
ideal ammeter
(from Rizzoni
Figure 2.41)
A series
circuit
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
i
R2
Circuit for the measurement
of the current i
Section 1, Page 28
14
Voltmeters
• Voltmeter - measures voltage across a circuit element, when
connected in parallel with it.
→ The voltmeter must be placed in parallel with the element whose
→
→
voltage is to be measured.
The voltmeter should draw no current away from the element
whose voltage it is measuring or else it will not be measuring the
true voltage across that element.
An ideal voltmeter has infinite internal resistance.
R1
A series
circuit
_
vS +
R1
+
v2 R2
i –
V
vS +
_
Circuit for the
measurement
+ of the voltage v2
+
v2 R2 V v2
–
i –
(from Rizzoni Figure 2.42)
Ideal voltmeter
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 29
Practical Ammeters and Voltmeters
• A practical ammeter will contribute some series resistance to the
circuit in which it is measuring current.
• A practical voltmeter will not act as an ideal open circuit but will
draw some current from the measured circuit.
• However, these practical restrictions do not necessarily pose a
limit to the accuracy of the measurements obtainable as long as
the internal resistance of the measuring devices is known.
A
rm
V
Practical voltmeter
rm
Practical ammeter
(from Rizzoni Figure 2.43)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 30
15
Wattmeters
• Wattmeter - measures power dissipated by a circuit element.
→ It is a combination of a voltmeter and an ammeter.
→ The Wattmeter measures the current flowing through the load and
the voltage across it, simultaneously. Then it multiplies the two to
provide the power dissipated by the load.
i
R1
i
R1
W
A
+
vS _+
+
vS +_
v2 R2
–
Measurement of the power
dissipated in the resistorR2:
P2 = v2 i
V
v 2 R2
–
Internal wattmeter connections
(from Rizzoni Figure 2.44)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 31
Definition of Elements and Branches
• An element is a resistance or EMF.
• A branch is any portion of a circuit with two terminals connected
to it. It may consist of one or more circuit elements. In practice,
any circuit element with two terminals connected to it is a branch.
a
i
+
A
Branch v
voltage
BranchR
current
–
A branch
b
(from Rizzoni Figure 2.45)
rm
Ideal
resistor
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
A battery
Practical
ammeter
Section 1, Page 32
16
Definition of a Node
• A node is the junction of two or more branches (the junction of
only two branches is sometimes called a trivial node). In practice,
any connection that can be made by soldering various terminals
together is a node.
Node a
Node c
Node a
iS
vS
Node
Node b
Node b
(from Rizzoni Figure 2.47)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 33
Definition of a Loop
• A loop is any closed connection of branches.
Note how two different loops
in the same circuit may include some of the same elements or branches.
Loop 1
Loop 2
Loop 3
R
R1
R2
vS
iS
1-loop circuit
3-loop circuit
(How many nodes in
this circuit?)
(from Rizzoni Figure 2.48)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 34
17
Definition of a Mesh
• A mesh is a loop that does not contain other loops. Meshes are a
useful tool for some analysis methods. They are simpler to
visualize than loops.
How many loops can you identify in this four-mesh circuit?
R4
R3
Mesh 3
vS
+
_
R1
Mesh
Mesh
3
R
2
1
Mesh
4
R5
iS
(from Rizzoni Figure 2.49)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 35
Definition of a Mesh
• A mesh is a loop that does not contain other loops. Meshes are a
useful tool for some analysis methods. They are simpler to
visualize than loops.
How many loops can you identify in this four-mesh circuit? (Answer: 14)
R4
R3
Mesh 3
vS
+
_
R1
Mesh
Mesh
3
R
2
1
Mesh
4
R5
iS
(from Rizzoni Figure 2.49)
PHY305F - Electronics Laboratory I, Fall Term (Kim Strong)
Section 1, Page 36
18