Note 01
DC Circuit Elements
Most courses in electricity and magnetism commence with the study of electric charges at rest. This topic,
called electrostatics, will not concern us here. In this note we pick up the story of electricity with current
electricity: the basic elements of DC circuits, electric current, resistance, work, power, elementary circuit
analysis, and the ammeter and voltmeter. In an appendix we describe the instrument you will be using in
this course, the Radio Shack Manual/Auto Range digital multimeter.
Sources of Electrical Energy
The sources of electrical energy you will most likely
use in any science lab are the battery and the power
supply.
various values of current (Figure 1-2). The circuit
symbol for a chemical cell is drawn in Figure 1-3.
Chemical Cell/Battery
The most common small-scale source of electrical
energy is the chemical cell. Chemical cells are constructed from various materials, usually of two chemically dissimilar elements, like carbon and zinc. Forming a cathode and an anode, they are separated by a
liquid or a paste medium called an electrolyte. The
anode serves as the source of electrons that are driven
by chemical action through the electrolyte to the
cathode. The anode takes on a positive potential, the
cathode a negative potential. The internal structure of
the carbon-zinc type is drawn in Figure 1-1. Common
D, C, and AA cells, available in any convenience store,
are of this type.
Figure 1-2. At the top are shown common consumer-type
chemical cells of 1.5V. The batteries (bottom) of 6 and 9V
actually consist of two or more cells connected internally in
series or in parallel and encapsulated in a single practical
container.
Figure 1-3. Circuit symbol for a chemical cell.
Figure 1-1. Internal structure of the carbon-zinc cell.
A chemical cell is designed to produce an electromotive force (emf) of 1.5 volts and to have a size and a
shape appropriate to the energy requirements of the
kind of device for which it is intended. Cells are
connected internally in series and in parallel to form
batteries of 9 volts, 12 volts etc., capable of delivering
Cells and batteries are designed to have a charge capacity expressed in ampere-hours (Ah) or milliamperehours (mAh). Capacities for typical cell types are
listed in Table 1-1. The larger the current drawn from
a battery the shorter is its lifetime. Charge capacity is
related to the amount of chemical mass in the cell and
therefore to the cell’s volume. A cell for a low-power
digital watch or a hearing aid might be tiny whereas a
battery in a nuclear submarine might be as large as an
average refrigerator. Research is being carried out in
major corporations to develop cells of ever-increasing
capacity and lifetime. 1
1
A good source of information on cells and batteries is Enercell
Battery Guidebook (Master Pub. Inc.) available at most RadioShack
stores.
N1-1
Note 01
Table 1-1. Charge Capacities of Some Common Cell Types.
Type
D
C
AA
AAA
Use
General
General
General
Heavy
Composition
Carbon-Zn
Carbon-Zn
Carbon-Zn
Zn Chloride
Capacity (mAh)
1500
700
300
120
Example Problem 1-1
Cell Lifetime
Ordinary flashlights use D cells. A fresh D cell has a
typical charge capacity of 1500 mAh. If 25 mA are
drawn from the cell continuously, how long in hours
should the cell last?
Solution:
The number of milliampere-hours can be written I x
t, where I is in mA and t is in hours. Thus
t = 1500 (mA-hours)/25 (mA) = 60 hours.
The D cell should be expected to last 60 hours under
continuous use.
The Power Supply
Next to the chemical cell the most common source of
electrical energy in a lab is a power supply (Figure 1-4).
A power supply converts the input from the mains at
110 V AC to some DC voltage at a (possibly variable)
DC current. It is usually equipped with a panel control to enable the user to vary the output voltage or
current.
Figure 1-5. Drawings of a few circuit elements and their
corresponding symbols.
The Idea of a Complete Circuit
To function as intended, an electric circuit must be
closed or complete. That is to say, it must form a
closed loop (Figure 1-6). The physical components are
drawn at the top of the figure, the corresponding
circuit diagram at the bottom. Wires are shown as
lines. On the left is shown the source of electrical
energy, on the right the target load (consumer) of
electrical energy.
Figure 1-4. Circuit symbol for a power supply. The arrow
symbolizes a variable voltage or current output.
Circuit Elements and Symbols
We have seen two examples of circuit symbols.
Electric circuits are described by means of a schematic
diagram with symbols or icons representing the
physical circuit element. A few more examples are
illustrated in Figure 1-5. Many of these symbols are
self-explanatory. We shall encounter other circuit
elements and symbols in due course.
Figure 1-6. Any electrical circuit must be closed or complete.
The source, because of its intrinsic emf (electromotive
force) ε, “drives” electrons from its negative terminal
along the lower wire to the bottom connection of the
N1-2
Note 01
load, through the load, and back to the positive terminal of the source via the upper wire. Although we
know that in a solid (such as a wire, source and load)
the electrons are the particles that actually move, we
shall imagine (for reasons we will enlarge on presently) that it is the positive charges that do the moving.
Clearly, if the wire is broken at any point then the
flow of charges must stop (charges cannot move—at
least easily—through air across a break in the wire).
The flow of charge constitutes what is called an
electric current. If a charge Q flows past any point in
the circuit in an elapsed time ∆t then the current is
I=
Q
,
Δt
…[1-1]
(units: coulomb per second or C.s–1). One C.s–1 is given
the special name ampere, or a m p for short, and
denoted A.€ Our assumption about positive charge
movement means that our imagined current flows
around the circuit in the figure in a clockwise
direction. This direction in which positive charges
would flow if they were free to move is called the
conventional direction of current. At any instant the
same current I appears at any point in either wire, at
any point in the source and at any point in the load. 2
As we shall see in what follows the potential difference or emf ε that appears across the terminals of the
source also appears across the terminals of the load (if
the connecting wires are good conductors). The current that flows in the circuit is limited or set by an
attribute of the load called resistance. The resistance R
of any load is defined as the simple ratio:
V ε
R= = ,
I I
…[1-2]
where ε or V is the potential difference across the load
and I is the current through the load. The units of R
are V.A –1.€One V.A–1 is given the special name Ohm,
denoted Ω.
is called Ohm’s law. If a circuit element has a constant
resistance it is said to obey Ohm’s law or to be a linear
or ohmic element. As we shall see a carbon composition resistor is an ohmic element whereas a PN
junction diode is not (Note 07). It is important to keep
in mind that whereas the resistance of an element can
always be calculated from single measurements of V
and I via eq[1-2] the element does not necessarily obey
Ohm’s law.
Resistance: More Details
The resistance of a material is attributed to the scattering of charge carriers (conduction electrons and/or
holes) by lattice defects, impurity atoms, and
thermally-excited lattice vibrations. If an electric field
is not present in the material then the charge carriers
move around at random, following straight line paths
between collisions and having no net velocity in any
particular direction. There is, in short, no net electric
current in any direction.
A net electric current is set up in a material in a
certain direction by a net electric field in that direction
or by a potential difference between two points in that
direction (which amounts to the same thing). When a
potential difference is applied between the ends of a
conductor the motion of the charge carriers is still
predominantly random. But because of the electrostatic force now exerted on the carriers, they take on a
small drift velocity v d in the direction of the field. This
drift of the charge carriers in response to the field is
the electric current.
More Definitions
If there are n charge carriers per unit volume in a
wire, each of charge q and travelling with drift velocity vd , then the current in the wire is
I = nqv d A ,
where A is the cross sectional area of the wire.
The current density J is the current per unit area of
cross section:
€
Ohm’s Law
In the event that R as calculated by eq[1-2] is constant,
independent of the values of V and I then the
relationship
V = IR (R const)
…[1-3]
J=
I
= nqv d ,
A
…[1-5]
(units: A.m –2). The current density J is related to the
electric field E via the resistivity ρ:
€
J=
2
In physics texts the direction of current in a conductor is taken to
€ to the direction of electron movement. Be aware that
be opposite
textbooks aimed at the technical market take the direction of
current to be the direction of electron flow.
…[1-4]
E
.
ρ
…[1-6]
The units of ρ are Ω.m (see below). Resistivity is a
€
N1-3
Note 01
basic property of matter whereas resistance is not.
The resistivity is a constant for a particular material
at a given temperature, but can vary widely between
different materials and as a function of temperature,
as seen in Table 1-2. The smaller the resistivity the
better the conductor.
Table 1-2. Resistivities of a few substances
Substance
Copper
graphite
glass
Temp (K)
300
300
300
Resistivity (Ω.m)
1.7 x 10–8
3500 x 10–8
1011
 A
I = JA = ΔV   .
 ρL 
…[1-7]
Eq[1-7] is the definition of resistance R where R = ρL/A.
Thus the resistance of an ohmic conductor depends on
the material
of which it is made (via ρ) and on its
€
geometry (via L/A). If ρ is a constant, independent of
V and I, then R is a constant.
Example Problem 1-2
An Example of Ohm’s Law
Ohm's Law: More Details
A relationship can be obtained between the resistance
of a certain length of wire of definite resistivity and
cross sectional area.
To see this consider a segment of wire of length L
and uniform cross section A (Figure 1-7). Suppose a
potential difference ∆V exists between the ends of the
wire a and b. The electric field E produced inside the
wire because of this potential difference has the same
value at every point. The potential difference and the
field are related by:
A potential difference of 4.00 V is applied between the
ends of a copper wire of length 100.0 m and cross sectional area 1.00 x 10–6 m2. How much current flows?
Solution:
Taking the resistivity of copper from Table 1-2 as 1.7 x
10–8 Ω.m the resistance of the wire is
R=
ρL 1.7 x 10−8 (Ω.m) x 100.0(m)
=
A
1.00 x 10−6 (m 2 )
= 1.7 Ω .
€
E
a
The current is given by the ratio
I=
b
L
V 4.00(V )
=
= 2.35 A .
R 1.7(Ω)
Figure 1-7. A length of conductor in the form of a wire.
Electrical Work and Power
€
b
ΔV = Vb − Va =
∫
r r
E • ds = EL .
a
E=
Thus
€
ΔV
V.m–1 .
L
The current density in the wire is, from eq[1-6]:
€
J=
E ΔV
=
,
ρ ρL
where ρ is the resistivity of the material making up
the wire. The total current in the wire is the current
density times
€ the cross sectional area:
The energy source supplies energy to a system (load).
This is the same as saying that the source does work
on a load. The rate of doing work is power.
When a charge q moves between two points that
differ in electric potential ∆V it loses potential energy
q∆V. Thus when a charge q moves from the positive
terminal to the negative terminal of a battery of
voltage V, there is a loss of potential energy equal to
qV. The potential energy lost appears as (is converted
into) some other form of energy. The battery might
drive a motor doing mechanical work, or the energy
might be dissipated as heat in the load. The rate at
which energy is lost is termed the power developed by
the battery. In an elapsed time ∆t the power is
P=
qV
= VI Watts.
Δt
…[1-8]
N1-4
€
Note 01
Note that one watt (abbreviated W) is equivalent to
one V.A or one J.s–1. Using the definition of resistance,
eq[1-2], this equation can be written three ways:
P = VI = I 2 R =
V2
Watts.
R
Color Code of Carbon Composition Resistors
The resistance value of a carbon composition resistor
is indicated by a color code painted in four bands on
the resistor’s body (Table 1-3).
…[1-9]
Table 1-3. Resistor Color Code
The kind of load you will encounter most often in this
course is the resistor.
Bands 1, 2, 3
Black
0
Brown
1
Red
2
Orange 3
Yellow 4
€
Resistors
A resistor is a circuit element whose purpose is to
produce a desired voltage or a desired current from
an existing voltage or current. A byproduct of this
function is the conversion of a certain amount of
electrical energy into heat. When a current I flows in a
resistor R, the rate of production of heat is equal to the
electrical power P. In other words, a resistor converts
electrical energy into heat with 100% efficiency.
Variable Resistance Box
Physics labs are commonly equipped with variable
resistance boxes that enable the user to select by
decade switches any resistance value over a wide
range in increments as small as 1 Ω . The resistors
inside the boxes are wire wound, of uniform cross
section and of various lengths. Uncertainty in the
values is typically 1%. Resistance boxes are designed
for low current applications only.
Carbon Composition Resistor
A carbon composition resistor is fabricated from an
amount of carbon matrix compressed to a certain
shape and dimension, usually cylindrical (Figure 1-8).
The value of the resistance is determined by the
physical dimensions of the pellet of compressed
carbon matrix located at the core of the resistor’s body
and of its composition.
Green
Blue
Violet
Grey
White
5
6
7
8
9
Band 4
Gold
Silver
No Color
5%
10%
20%
Beginning with the band closest to one end of the
resistor, they give, respectively, the first significant
digit, the second significant digit, and the multiple of
ten. The fourth band gives the manufacturer’s tolerance. The tolerance is the manufacturer’s estimate of
the uncertainty in the resistance, based on the manufacturer’s quality control. Let us consider an example
in reading a color code.
Example Problem 1-3
Reading a Resistor’s Color Code
A resistor has color bands in the order: grey, red,
yellow and silver. What is the resistance?
Solution:
The numbers corresponding to the colors are: 8, 2, 4
and 10%. According to the code the resistance is:
(82 x 104 ± 10%) ohms. 3
In this example the manufacturer claims that if the
resistance is measured with a high-quality instrument,
then the result will fall within ±10%, or ±8 x 104 Ω, of
the value specified by the color code. Resistors of 1 %
and 0.5 % tolerance are available on the market, but at
higher cost than 10% tolerance resistors.
Power Rating
A resistor can transfer only so much heat to the surrounding air at room temperature before undergoing
an unacceptable self-heating and change in resistance.
Carbon composition resistors are rated as to the max-
Figure 1-8. A cutaway view of a carbon composition resistor.
3
The resistance expressed in the more preferable standard form is
(82 ± 8) x 10 4 Ω. For more information on the subject of the correct
handling of uncertainties see Note 03.
N1-5
Note 01
imum power they can dissipate without the resistance
drifting outside the range specified by the manufacturer. The ratings most commonly available off the
shelf are 1/8, 1/4, 1/2, 1, 2, 5, and 10 watts. The rating
is largely a factor of the resistor’s volume and surface
area (body size) (Figure 1-9).
following. Because charge cannot accumulate anywhere in the circuit, the current is the same everywhere in the circuit. A potential difference IR1 appears
across R 1, and so on so the potential difference across
the combined resistors is I(R 1 + R2 + R3).
I
R1
2W
R2
V
€
1W
R3
I
€
Figure 1-10. Three resistors in series.
1/2 W
€
1/4 W
Figure 1-9. Examples of resistors having the same resistance
but different body sizes and power ratings.
The larger the surface area the greater the power
dissipation. Should a manufacturer’s rating be exceeded a resistor can heat up sufficiently to self-destruct.
Forced air cooling will increase the effective power
dissipation.
Higher-power resistors are also available, though
often not needed in modern low power applications.
These resistors are nearly always wire-wound and
have a large surface area. Other types of resistors are
the carbon film and metal film types that are designed
to produce low levels of electrical noise for computer
applications. Much research is ongoing to develop
resistors from new materials that are smaller, stabler
and electrically quieter.
Elementary Circuit Analysis
A good part of the study of basic electricity is concerned with circuit analysis. By this is meant reducing
the complexity of a circuit in order to solve for certain
unknowns, such as the current flowing through an
element or the voltage across an element. The simplest
circuit reductions involve resistors in series and in
parallel.
Series Resistor Circuit
A series circuit is one in which all charges follow the
same path (Figure 1-10). Put simply, the “tail” of the
one resistor is connected to the “head” of the one
This potential difference is maintained by the battery
and must be equal to the potential difference V
between its terminals. Therefore V = I(R 1 + R 2 + R3). So
the three resistors in series have an effective combined
resistance of
R = R1 + R2 + R3 .
…[1-10]
The general rule for combining N resistors in series is
N
€
R = ∑ Ri .
i=1
Parallel Resistor Circuit
A parallel€circuit is one in which all the circuit
elements are in different current paths (Figure 1-11).
Put simply, the “heads” of all the resistors are connected together, and the “tails” of all the resistors are
connected together. The total current in the circuit is
therefore
I = I1 + I2 + I3 .
…[1-11]
I
€
I1
I2
I3
V
R1
€
€ I
€
R2
R3
Figure 1-11. Three resistors in parallel.
€
€
€
N1-6
Note 01
Since the resistors are in parallel, the same potential
difference V appears across each resistor. So by the
definition of resistance:
I1 =
V
V
V
, I2 = , I3 =
.
R1
R2
R3
…[1-12]
1
1
1
I = V +
+ .
 R1 R2 R3 
9
9
9
= 0.6 A; I2 = = 2.25 A; I3 = = 1.125 A
15
4
8
Thus the total current is
I = I1 + I2 + I3 = 0.6 + 2.25 + 1.125 = 3.98 A .
…[1-13]
Method 2. In this method we calculate the resultant
resistance and then the total current drawn. If R is the
resultant resistance then
€
By comparing eq[1-13] with the definition of a single
resistance R, it follows that the three resistors combine
according
€ to the rule
1 1
1
1
= +
+ .
R R1 R2 R3
1 1 1 1
= + + = 0.0667 + 0.25 + 0.125 = 0.4417 .
R 15 4 8
Thus
€
Thus the general rule for any number N of resistors is
€
I1 =
€
Thus substituting eqs[1-12] into [1-11] we get
€
three branches:
1 N 1
=∑ .
R i=1 Ri
R = (0.4417)−1 = 2.264 Ω
I=
and
€
V
9(V )
=
= 3.98 A ,
R 2.264(Ω)
the same answer as in method 1. These calculations
are illustrated in Figure 1-13.
€
Let us consider an example of resistors in parallel.
€
Example Problem 1-4
An Example of a Parallel Resistor Circuit
Calculate the total current supplied by the 9 V battery
in the circuit of Figure 1-12.
Figure 1-13. Illustrating two ways of solving the problem.
Circuits can consist of resistors in series and parallel
combinations, making for a more complicated case of
circuit reduction. Let us consider an example.
Example Problem 1-5
An Example of a Series-Parallel Resistor Circuit
Figure 1-12. A typical parallel resistor circuit.
Solution:
We can solve this problem two ways.
Method 1. In this method we calculate the currents
flowing in each branch of the circuit and then add the
currents. Since the same 9 V appears across each of
the three resistors we have for the currents in the
Calculate the total current in the circuit shown in
Figure 1-14 (top).
Solution:
This kind of problem is best solved in a series of
reductive steps.
Step 1: In this step we reduce the two resistors in
parallel to their equivalent resistance. If R is the
equivalent then
N1-7
Note 01
3Ω
3Ω
real battery must therefore be represented as shown in
Figure 1-15 (bottom).
6Ω
10V
3Ω
6Ω
Figure 1-14a. A combination series-parallel resistor circuit
(top). The first step in reducing the circuit (bottom).
1
1
1
=
+
; thus R = 2 Ω .
R 3(Ω) 6(Ω)
Step 2: In this step we find the resultant of the two
resistors in series and then calculate the current
€ (Figure 14b). The result is:
I=
€
V 10 (V )
=
= 2 A.
R 5 (Ω)
3Ω
2Ω
Figure 1-15. An ideal battery (top) and real battery (bottom).
We can calculate the maximum power that can be
drawn from this real battery by an external resistor R
(Figure 1-16). The power P dissipated by the load
resistor R is
I
P = I 2R =
10V
Figure 1-14b. The circuit is reduced to two resistors in
series.
V2
2 R.
( r + R)
The value of R that results in maximum power transfer can be found by performing the calculation dP/dR
= 0 and
€ solving for R. The result is 4
R = r.
Internal Resistance
The energy source we have considered thus far is in
fact what is regarded as an ideal source (Figure 1-15
top). No battery can maintain a constant potential
difference across its terminals irrespective of the current drawn by the load resistor. A real battery behaves
electrically as an ideal voltage source in series with a
finite internal resistance r. The equivalent circuit of a
…[1-14]
…[1-15]
Substituting eq[1-15] into eq[1-14], the maximum
power in the load resistor is found to be
€
Pmax =
V 2r V 2 V 2
=
.
2 =
(2r) 4r 4R
…[1-16]
4
We give here only the end result as you are asked to do this
yourself in Experiment 01, “DC Circuits and Measurements”.
€
N1-8
Note 01
modern electrical instruments.
The Galvanometer
Figure 1-16. These two circuits are equivalent.
The moving coil of a galvanometer consists of a coil of
wire to which is attached a pointer, suspended in the
magnetic field of a permanent magnet (Figure 1-17). If
the current to be measured is passed through the coil
then the magnetic field produced by the current
interacts with the magnet’s magnetic field to produce
a torque on the coil causing it to rotate. The deflection
of the coil and pointer is proportional to the current.
Thus if a calibrated scale is attached to the galvo the
value of current can be read from the scale. 5
Maximum power is transferred when the load resistor
equals the internal resistance of the battery. The
power developed in the load resistor is never greater
than V2/4R, which is also the rate at which electrical
energy is transformed into thermal energy in the battery. So, as a source of energy, a battery is never more
than 50% efficient. Of course, in many applications it
is not desireable that the source deliver maximum
power, for in so doing the source lifetime would be
drastically shortened.
We shall continue with other aspects of this topic in
Note 02.
Measuring Current and Voltage
Before continuing with the study of DC circuit
analysis we need to survey the instruments that are
used to measure current and voltage in the modernday lab. The instruments that are used to measure
current and voltage are called, respectively, the
ammeter and the voltmeter. These may be analogue or
digital in nature. The ammeter and voltmeter are
circuit elements that display, and sometimes record,
electrical measurements. The elements, though
perhaps highly-sophisticated and expensive, are not
entirely ideal. In order that you use them without
error in real applications you need to understand their
limitations.
In the modern day lab the functions of ammeter and
voltmeter are commonly combined in a single instrument called a multimeter. Digital multimeters (DMMs)
are pretty much the norm. Multimeters measure many
other quantities—resistance, sound level, and capacitance, to name a few. In spite of (or because of) the fact
that DMMs are complicated instruments, with their
own built-in amplifiers and digital circuitry, it is
useful for purposes of understanding to imagine that
the core of the instrument is a simple analogue
current measuring device or galvanometer. The
galvanometer or galvo, for short, is the ancestor of all
Figure 1-17. Sketch of the low-sensitivity d’Arsonval meter
movement found in many common ammeters and voltmeters.
An ideal galvo would have an internal resistance of
zero, but a real galvo has a small resistance due to the
wire making up the coil. Typical galvos have internal
resistances of about 100 Ω. Thus a galvo should be
represented by an ideal galvo element in series with a
resistor (Figure 1-18).
10 µA
R i ~ 100 Ω
A
V = 1 mV
Figure 1-18.A real galvanometer shown within the dashed
outline has the symbol A, or sometimes G. It has an
effective series resistance Ri .
5
You will be using an instrument of this type in Experiment 01,
“DC Circuits and Measurements”, and a far more sensitive type in
Experiment 04, “The d’Arsonval Galvanometer”.
N1-9
Note 01
A galvo is designed to give a maximum or full scale
deflection (abbreviated FSD) for a certain current. A
typical value is 10 µA. FSD pins the pointer on the peg
of the galvanometer and should therefore not be
exceeded. Thus the voltage drop across the galvo in
Figure 1-18, for a current of 10 µA flowing through it,
is 1 mV. This means too that, in principle, the device
could also be used to measure a maximum voltage of
1 mV! However, a galvo is far too sensitive a device to
be used alone. A resistor is usually added to the galvo
to make a working ammeter and voltmeter from it.
Constructing an Ammeter from a Galvanometer
An ammeter can be made from a galvo and a single
resistor. One case might be to use the galvo just
described to make an ammeter with an FSD of 50 µA.
How must the resistor be connected to the galvo and
what is its value? The solution is sketched in Figure 119.
50 µA
R i ~ 100 Ω
A
Constructing a Voltmeter from a Galvanometer
Suppose we wish to add a single resistor to the galvo
just described to make a voltmeter that will give an
FSD of 10 mV. How must this resistor be connected
and what is its value? The solution is sketched in
Figure 1-20.
1 mV
Ifsd = 10 µA
Ri ~ 100 Ω
R
10 mV
Figure 1-20.A working voltmeter is made from a real galvo
by adding a resistor R in series with it.
At FSD the voltage drop across the galvo is 1 mV.
Thus the resistor must be connected in series with the
galvo so that at FSD the voltage drop across the galvoresistor combination is 10 mV. The current IFSD = 10 µ
A flows through both elements. Thus
Ifsd = 10 µA
10 x 10−6 (100 + R) = 10 x 10−3 ,
R
Figure 1-19.A working ammeter is made from a galvo by
adding a parallel shunt resistor Rg.
A
so that
R = 900 Ω .
Thus in order to make a voltmeter with an FSD of 10
€mV a resistor of 900 Ω must be placed in series with
The galvo’s properties remain unchanged by the
added resistor (the current that produces FSD remains
the same at 10 µA). Therefore, the resistor must clearly
be placed in parallel with the galvo to bypass an
amount of current
Ig = 50 µA −10 µA = 40 µA .
the galvo.
We have seen that an ammeter and a voltmeter can
be made from a galvo by placing a resistor of the
appropriate value in series or in parallel with it. It
should also be evident that you can make a voltmeter
from an ammeter and vice versa by a similar procedure. Finally, these procedures of adding resistors
would work even if the galvo itself were a digital
device.
The voltage drop across the galvo at FSD remains the
same:
€
Thus
€
Vg = 10 x 10−6 x 100 = 10−3 V .
Rg =
Vg
Ig
=
10−3 (V )
= 25 Ω .
40 x 10−6 (A)
Therefore, in order to construct an ammeter with an
FSD of 50 µA a resistor of 25 Ω must be placed in
€parallel with the galvo. This resistance is commonly
referred to as a shunt resistance as it effectively shunts
current around the galvo.
N1-10
Note 01
Appendix A
Quick Start for the
RadioShack Manual/Auto Range Digital Multimeter
The most basic type of ammeter is the moving-coil or
d’Arsonval galvanometer. But today, both current and
voltage are commonly measured with a generalpurpose instrument called a multimeter, and then
mostly with a digital multimeter (DMM). The DMM
you will use in this course is the RadioShack
Manual/Auto Range DMM.
The Radio Shack (RS) Manual/Auto Range DMM
(Figure A-1) is one of the least expensive DMMs on
the market that has reasonable accuracy (1-2%) and a
serial port. 6
measurement you wish to make and the RANGE of
the measurement. There are positions for at least
seven types of function: resistance (OHM area),
capacitance (LO, HI), voltage (V), current (A) and so
on. There are 7 ranges of resistance (200 Ω, 2 kΩ, 20
kΩ, 200 kΩ, 2 MΩ, 20 MΩ and 2000 MΩ). On the 200 Ω
range the DMM will display a maximum of 200 Ω; if
the resistance exceeds this value the DMM will print
an “OL” in the display, meaning overrange. There are 5
ranges of voltage and current.
Sockets
Four sockets designed to accept a standard banana
plug are arrayed along the bottom sector of the panel.
These are labelled “20A”, “mA”, “COM” and “V/Ω”.
You will not be using the “20A” socket in this course.
On the other hand, you will always use the “COM”
socket, otherwise known as the COMMON or
GROUND connection, in all measurements you make
(excepting capacitance). To measure current use the
“COM” and “mA” sockets; to measure resistance or
voltage use the “COM” and “V/Ω” sockets. There are
also two sockets for measuring capacitance.
Specifications
Radio Shack claims the DMM has good voltage- and
current-measuring characteristics, meaning that
Figure A-1. The RS multimeter (Former Cat. No. 22-168A)
Controls
The two most important buttons are the POWER
button (colored red) and the DC/AC button. These
buttons are located on the upper left and upper right
hand corners of the DMM’s control panel just below
the display area.
The rotary switch in the center of the DMM’s control
panel is the FUNCTION and RANGE selector. With
this switch you select the FUNCTION or kind of
6
DMMs are quickly being superseded with newer models. Already as this note was being written this instrument (manufactured
by the OEM Metex Corporation) was dropped from the RadioShack
catalogue. And ironically RadioShack has just been taken over by
Circuit City.
• when operated as a voltmeter, it has a very large
internal resistance (10 MΩ) and
• when operated as an ammeter it has a very small
internal resistance (10 Ω - 1000 Ω depending on the
range). For more specifications see Table A-1 below.
Quick Start
To turn the instrument on push the POWER button.
The display should come alive. If the battery is weak a
LOW BAT sign will appear in the display. If the LOW
BAT sign does appear call your instructor—the
battery will need replacing.
The instrument has a number of modes which are
selected by the Function button. To see what these
modes are do the following:
•
Push the Function button slowly about 10 times
and observe the mode names as they appear each
time on the LCD screen. The modes are “A-H”,
“D-H”, “MIN”, “MAX”, “REL”, “MEM”, “RCL”,
“DUAL”, “COM”, “CMP”. After “CMP” the
N1-11
Note 01
•
meter will revert back to the “A-H” mode.
Put the DMM into any mode you like and then
turn the DMM OFF and ON. Observe that the
DMM will always revert to the “A-H” mode on
boot up. Most of these modes will not concern us
here, and so we restrict our description to the
most important:
A-H:
DMM shows in its secondary display the
reading taken 4 seconds earlier. This is the
power on mode and therefore the mode you
will use most often in this course.
• REMEMBER: To quickly reset your DMM to the
“A-H” default just turn it OFF and back ON again.
A-H stands for Auto-Hold . In this mode the
Table A-1. Selected specifications of the RS DMM. Input Impedance is 10 MΩ on all DC and AC voltage ranges.
Function
DC Voltage
Range
200 mV, 2 V, 20 V, 200 V, 1000 V
Accuracy
± 0.8% of display + 1 ls digits
DC Current
200 µA, 2 mA
20 mA, 200 mA, 2 A
20 A
± 1.0% of display + 1 ls digit
± 1.5% of display + 1 ls digit
± 2.5% of display + 5 ls digits
Resistance
200 Ω
2 kΩ, 20 kΩ, 200 kΩ, 2 MΩ
20 MΩ
2000 MΩ
± 1.0% of display + 3 ls digits
± 1.0% of display + 1 ls digit
± 1.5% of display + 2 ls digits
± 5.5% of display + 5 ls digits
AC Voltage
200 mV, 2, 20, 200, 1000 V
750 V
± 1.2% of display + 3 ls digits
± 1.5% of display + 3 ls digits
AC Current
200 µA-2 mA
20 mA-200 mA
20 A
± 1.5% of display + 3 ls digits
± 2.3% of display + 5 ls digits
± 3.5% of display + 7 ls digits
Frequency
2 kHz – 20 kHz
± 2.5% of display + 3 ls digits
Capacitance
200 pF – 200 nF
20 µF – 200 µF
± 2.5% of display + 3 ls digits
± 4.5% of display + 5 ls digits
N1-12
Note 01
Appendix B
Quick Start for the
Agilent Model 34401A Digital Multimeter
General Description
The Agilent Model 34401A DMM (Figure B-1) is a
research-grade 6-1/2 digit instrument with an accuracy in the 0.003% range. It is claimed to employ “a
continuously integrating, multislope III ADC”. It
provides measurements of DC and AC voltage, DC
and AC current, resistance, continuity, Diode Test,
DC:DC Ratio measurements, period and frequency. It
will perform a number of MATH operations and can
store up to 512 readings in internal memory. This is
one of the best instruments of its kind on the market
that is within the budget of a teaching laboratory.
Figure B-1. The front panel of the Agilent Model 34401A digital multimeter at a glance. This shows  Measurement function
keys,  Math operation keys,  Single Trigger/AutoTrigger/Reading Hold key,  Shift/Local key,  Front/Rear Input
Terminal Switch,  Range/Number of Digits Displayed keys,  Menu Operation keys.
N1-13
Note 01
Specifications
The instrument can be operated at three precision
levels: 4-1/2 digits, 5-1/2 digits and 6-1/2 digits. The
precision selected determines the measurement speed,
the more precision the slower the speed. Measurement speed depends on the function as well as the
resolution. A selection of measurement speeds is
listed in Table B-1. The instrument is also rated as to
transfer speed, the speed at which it can transfer data in
bulk to a controlling computer. Transfer speed is
faster than measurement speed. A selection of transfer
speeds is given in Table B-2. As would be expected,
transfer speed via GPIB is greater than for RS-232.
Quick Start
The front panel controls are grouped by function
(Figure B-1). When the instrument is turned ON it
enters DC Voltage mode automatically. You are
advised to ensure that the Front/Rear Input Terminal
switch  is set for “Front”.
A command is entered in response to a menu via
push buttons on the front panel. The menu is organized in a top down tree structure with three levels
(Figure B-2). Once into the menu you can move one
item horizontally right or left by pressing the “>” or
“<” buttons, one item down or up by pressing the “∨”
or “^” buttons, respectively.
Table B-1. A Selection of Measurement Speeds.
Function
DCV, DCI,
Resistance
Digits
6 1/2
6 1/2
5 1/2
5 1/2
4 1/2
Readings/s
0.6 (0.5)
6 (5)
60 (50)
300
1000
Figure B-2. Tree structure of the menus.
Practice Problems
Table B-2. A selection of Transfer Speeds. This refers to the
transfer of data from internal memory.
Mode
ASCII Readings to...
Rate (#/sec)
DC
RS-232
HP-IB
RS-232
HP-IB
RS-232
HP-IB
55
1000
50
50
55
80
AC
Freq & Period
N1-14