7. DC CIRCUITS
You will study the basic concepts of resistance, DC voltage and current and acquire some
experience in using a high quality digital multimeter.
Theory
Why Study Electricity?
As a subject of study “DC Circuits” is arguably the most important in physics, more important even than
mechanics, as direct current (DC) circuits and techniques figure in some way or other in nearly every
activity in science and technology.
An Electric Circuit, Voltage and Current
In order for an electric current to flow and be sustained, a loop or circuit is necessary. An example is
drawn in Figure 1. On the left is a battery which
supplies the electrical energy, and on the right is a
load device which consumes, or transforms, the
electrical energy. The load might be a resistor, a
light bulb, a radio and so on. The battery produces
a potential difference or voltage V (unit volts, abbreviated V) across its terminals which “drives” a
current I (unit amperes, abbreviated A) around the
circuit. 1 What is described as the conventional
current flows to the load through the wire along
the top of the figure and from the load through the
wire along the bottom of the figure (opposite to the
actual electron current flow). Should the circuit
open up or break at any point this current would
immediately drop to zero. This current has the
same value I at any point in the circuit. (The same
current I even flows through the battery!) There is
no “beginning” and no “end” to this current; and
no current is “used up”. (Something is used up,
however, as will be discussed in a moment.)
wire carrying conventional current to load
battery
I
+
source of electrical energy
V
V
load device
(resistor, light bulb,
radio, etc.)
consumer of
electrical energy
wire carrying conventional current from load
I
Figure 1. A simple direct current (DC) Circuit.
“An Electric Circuit is like a Loop of Water”
With care, an electric circuit can be thought of as a
loop of water as shown in Figure 2. The loop of
wire in Figure 1 is here represented by a hollow
pipe through which flows a continuous stream of
water. On the left is a pump (the equivalent of the
battery in Figure 1) and on the right is a valve (the
equivalent of a load). The pump, by virtue of the
driving force it possesses, drives water to the valve
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7 DC Circuits
through the pipe along the bottom of the figure
and pulls water back to itself through the pipe
along the top of the figure. The valve limits the
flow of water by means of constriction. The more
open is the valve (lower is the resistance) the
greater is the water flow (electron current) for a
given pump force (battery voltage). The water
flows continuously with the same mass per second
being transported in each part of the circuit, ie., no
water is “used up”. There is no beginning and no
end to the flow of water. If the pump stops the
water flow also stops, and immediately.
water flow
(electron current)
pump
(battery)
driving force
(voltage)
valve
(resistor)
water flow
(electron current)
Figure 2. A water loop analogue of the electric circuit in Figure 1.
Ohm’s Law
We return now to Figure 1. For a given voltage the
current that is made to flow in a circuit is determined by the resistance of the load. The resistance
is defined as the ratio of voltage to current
R=
V
I
…[1]
and is usually denoted R (unit ohm, abbreviated Ω).
Thus 1 Ω = 1 V.A –1. The resistance of a load can
always be defined as in eq[1] for any arbitrary V
and I. However, if R is constant, independent of V
and I, eq[1] is called Ohm’s Law. Any device whose
resistance is constant is said to obey Ohm’s Law or
is said to be an ohmic, or linear, device. One example is the carbon composition resistor you will be
using in this experiment. Conversely, a device
whose resistance depends on V or I is said to be
non-linear. An example is the semiconductor diode.
Electrical Energy and Power
Neither voltage nor current is consumed in an
electric circuit, but electrical energy is. In each
second a fraction of the battery’s energy is
“transferred to the load”, and there transformed
into the kind of energy desired or work (which is
the same thing). This transformed energy may be
internal energy in the case of a resistor, sound
energy in the case of a radio, or mechanical work
B7-2
in the case of a motor. In time, of course, the
battery’s energy will be used up, the battery will
“die” and will need to be replaced.2 The point is
that the quantity consumed here is energy, not
current and not voltage. It can be shown that if a
current I flows through the load when a voltage V
exists across the load then the energy E transformed in t seconds is
E = IVt
P = IV
…[2]
(unit Joule, abbreviated J). Power is the time rate of
change of energy, or the amount the energy
changes in one second. Thus from eq[2] the power
P is
DC Circuits 7
…[3]
= I2R ,
(unit Watt, abbreviated W) if the device obeys
Ohm’s Law.
The Carbon Resistor, its Color Code and Power Rating
The load in this experiment is a carbon composition resistor (Figure 3). The resistor’s resistance is
specified by the manufacturer in the form of a
color code painted in four bands on the resistor’s
body. 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 or accuracy. The values
for each color are listed in Table 1. As an example,
let us suppose the bands are grey, red, yellow and
silver, in that order, where grey is the band closest
to one end. Then the resistance works out to be
Carbon composition resistors are rated as to the
electrical power they can successfully dissipate in
ambient air before undergoing significant changes,
either by the resistance changing unacceptably
beyond the manufacturer’s specification, or by the
resistor becoming so hot it burns up. Generally, the
bigger the resistor’s mass and surface area the
better the resistor can transfer heat to the surrounding air; and therefore the more power it can
handle. Ratings of typical resistors is 1/4 W, 1/2 W
and 1 W. The power rating of a resistor can usually
be determined by inspecting the resistor’s body
size, as is evident in Figure 4.
(82 x 104 ± 10%) ohms.
lead
1st significant figure
2nd significant figure
multiplier of 10
tolerance
(a)
insulation
R
compressed carbon
(b)
(c)
Figure 3. A carbon resistor shown (a) from the outside, (b) from the inside, and (c) symbolized
Table 1. Resistor Color Code
Bands 1, 2, 3
Black
0
Brown
1
Red
2
Orange
3
Yellow
4
Green
Blue
Violet
Grey
White
5
6
7
8
9
Band 4
Gold
Silver
No Color
5%
10%
20%
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7 DC Circuits
Figure 4. Approximate body sizes and power
ratings of carbon composition resistor.
1/2 W
2W
1/4 W
1W
Resistors in Series and in Parallel
Resistors can be connected together in various
ways. If connected “head to tail” as in Figure 5a
they are said to be in series; if connected “head to
head” and “tail to tail” as in Figure 5b they are in
parallel. You should be able to show that the total
resistance for series resistors equals the sum of the
component resistances; and for resistors in parallel,
the inverse of the total resistance equals the sum of
the inverses of the component resistances. One of
the objects of this experiment is to wire up the
circuits of Figure 5 to test the validity of the
relationships with your multimeter.
V
V
V1
R1
V2
V3
R2
V
I1
R1
I2
R2
I3
R3
R3
V
I
Figure 5a. Three resistors in series:
R = R1 + R2 + R3 .
I
Figure 5b. Three resistors in parallel:
1/R = 1/R1 + 1/R 2 + 1/R 3 .
The Voltmeter and the Ammeter
You will be using a digital multimeter (DMM) to
measure current, voltage and resistance. A multimeter set to function as an ammeter has a very
small internal resistance. When set to voltmeter
function it has a very large internal resistance. A
multimeter’s internal resistance is important as it
determines how and where it is used in a circuit.
For example, two multimeters might be used to
find the resistance of a resistor with the simple
circuit shown in Figure 6. Having measured I and
V, you can calculate R from eq[1].
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In Figure 6 the multimeter must be set to the
proper function and must be placed in the circuit
correctly. When in ammeter function it is connected in series with the resistor. This is because a
current must flow through an ammeter to make
the instrument work properly. Because of the
ammeter’s very small internal resistance only a
very small voltage drop occurs across the ammeter. (You will be measuring this voltage drop in
this experiment.) Ideally the ammeter in a circuit
has little effect on the circuit’s characteristics.
DC Circuits 7
In contrast to ammeter function, when in voltmeter
function the multimeter is connected in parallel
with the resistor, since by its nature a voltmeter
displays a potential difference between two points.
Since the voltmeter has a very large internal resistance, a negligible current flows through it. Ideally
the voltmeter placed where it is in Figure 6 has
little effect on the circuit’s characteristics.
This situation will change, however, if R is
very large, say of the order of the internal resistance of the voltmeter. The value of R that is
calculated by measuring V and I with the circuit in
Figure 6 and then dividing re eq[1] will yield an
incorrect result (one that does not agree with the
A
the ammeter
is connected
in series
value given by the color code). One of the tasks of
this experiment will be to examine this situation,
explain why an incorrect value for R results and to
deal with it. At the furthest extreme, if the multimeter intended to be used as an ammeter in Figure
6 were accidently set to voltmeter function, then
the multimeter would disrupt the characteristics of
the circuit to the extent that the display of the
multimeter would have little useful meaning. You
will also briefly examine this situation, described
as an “intentional mistake” in the exercises. It is
useful to experience these cases because they are
the cause of many “bad” measurements made in
the electronics lab.
A
V
+
V
the voltmeter
is connected
in parallel
V
R
Figure 6. A simple circuit showing how multimeters are connected to measure the current flowing through, and the
voltage developed across, a resistor. The symbol for a battery with an arrow through it stands for a variable voltage
power supply, the energy source used in this experiment.
The Ohmmeter
A DMM set to ohmmeter function can be used on
its own to measure a resistance. The resistor is
connected directly to the DMM as shown in Figure
7. In essence the multimeter acts like an active
device here supplying its own voltage across R and
thereupon measuring the corresponding I. The
number that gets displayed is the ratio eq[1]. This
is by far the most convenient use of the DMM and
the one you will investigate first.
Figure 7. The digital multimeter
set to ohmmeter function is
connected directly to the resistor.
R
Ω
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7 DC Circuits
The Experiment
Exercise 0. Preparation
Orientation
Identify the apparatus supplied: one Heathkit
Model EUW-17 variable voltage power supply (to
be used instead of a battery); two Digitek Model
DT-890D 3 1/2 digit multimeters3; one box containing four resistors; and connectors of various
lengths. (These connecting wires are hanging up in
the lab. You will have to find them.)
Checklist
Carry out the following cold start checks:
3 If ON, turn any and all instruments OFF.
3 If present, clear away any and all connecting
wires from all apparatus.
3 Examine the power supply. Turn the voltage
control on the supply to zero. This control
should always be zeroed before turning the
supply ON or OFF. This supply is capable of
delivering only a small amount of power:
maximum output rating is 30 volts at about
300 mA. Keep these figures in mind.
Ô Examine the four resistors issued and deduce
their resistance from the color code in Table 1.
Deduce their power rating from Figure 4.
Ô In what follows you’ll be required to wire up
circuits. Therefore, after wiring up your circuit
and before turning the power supply ON, have
your TA check your circuit for you. Don’t
worry, the voltages here are not dangerous.
Exercise 1. A First Look at the Multimeter
A digital multimeter (DMM) tends to intimidate
students at first. This exercise is therefore a warmup to enable you to identify the various inputs and
controls and check the battery. No circuits will be
wired up yet. Have your DMM in hand or consult
Figure 8 as you work your way through the
following:
Buttons
Identify the rectangular and oval buttons on the
upper left and upper right corners of the control
panel of the DMM. These are the POWER and
AC/DC buttons, respectively. At the moment the
POWER button should be “out” or OFF (the LCD
screen should be blank), and the AC/DC button
should be “out” for DC. If necessary, change the
position of these buttons now.
FUNCTION/RANGE Switch
Identify the rotary switch in the center of the
control panel of the DMM. This is the FUNCTION
and RANGE selector. This switch enables you to
B7-6
select the FUNCTION or kind of measurement you
wish to make (resistance, voltage, current, and so
forth), and the RANGE of the measurement. Note
for example there are 7 ranges of resistance (200 Ω,
2 kΩ, 20 kΩ, 200 kΩ, 2 MΩ, 20 MΩ and 2000 MΩ).
(Remember 1 kΩ = 10 3 Ω and 1 MΩ = 10 6 Ω.) On
the 200 Ω range the DMM will display a maximum
of 200 Ω; if you attempt to measure a resistance
greater than this it will display an overrange (a “1”
in the highest digit).
Sockets
Identify the four connection sockets (“20A”, “A”,
“COM” and “V/Ω”) arrayed along the bottom
sector of the multimeter. The “20A” socket will not
be used in this experiment. The “COM” socket, the
COMMON or GROUND connection, will be used
in all measurements. To measure current for example you would use the “COM” and “A” sockets, to
measure voltage you would use the “COM” and
“V/Ω ” sockets.
DC Circuits 7
ON/OFF
Push the POWER button to turn the meter ON. If
the battery is weak a “LOW BAT” sign will show.
If the “LOW BAT” sign does appear call your
TA—the battery will have to be replaced. If not
turn the meter OFF for the time being.
Specifications of the DMM
The DT-890D digital multimeter is claimed to have
good voltage- and current-measuring characteristics; that is, as a voltmeter, it has a very large
internal resistance and as an ammeter it has a very
small internal resistance. Scan Table 2 and locate
the claimed input resistance (impedance) for DC
voltage ranges.
Figure 8. The Digitek Model DT-890D digital multimeter. You should be able to identify the power switch in the
upper left corner, the AC/DC switch in the upper right hand corner and the range switch in the center. Note also the
sockets along the bottom. The socket labelled “COM” for “common” is used for all measurements.
Exercise 2. A Study of Resistance
Arguably resistance is the easiest and most immediately useful application of a DMM. To measure resistance with the DT-890D multimeter you:
¬ Set the function switch to the appropriate “Ω ”
position.
Á Connect the resistor whose resistance you wish
to measure directly to the “VΩ ” input and
“COM” sockets.
 Turn the POWER ON, rotate the range selecÃ
Ä
tor, if necessary, and read the number from the
display.
Multiply this number by the multiplier for the
range selected.
Using the range as a guide find the uncertainty
in the measurement from Table 2.
B7-7
7 DC Circuits
Table 2. DC Specifications of Digitek Model DT-890D Digital Multimeter.
Accuracies are ± (% of display + No. of digits in the least significant place of display) *
Temperature for guaranteed accuracy: 23 ˚ C ± 5 ˚ C. Less than 75% RH.
Function
Range
200 mV, 2 V, 20 V, 200 V
1000 V
→ Input Impedance 10 MΩ on all ranges
Accuracy
DC Voltage
± 0.5% of display + 3 ls digits
± 0.8% of display + 2 ls digits
DC Current
± 0.8% of display + 1 ls digit
± 1.2% of display + 1 ls digit
± 2% of display + 3 ls digits
200 µ A, 2 mA, 20 mA
20 0mA, 2 A
20 µ A, 20 A
Maximum input current: 20 A
200 Ω
± 0.8% of display + 3 ls digits
2 kΩ , 20 kΩ , 200 kΩ , 2 MΩ
± 0.8% of display + 1 ls digit
20 MΩ
± 1.0 % of display + 2 ls digits
2000 MΩ
± 5.0 % of display + 5 ls digits
Open Circuit Voltage: 200 Ω , 2000 MΩ range: 3.2 V Max; Other ranges: 0.3 V Max
Resistance
* For example, suppose the DC voltage display is 1.234 volts on the 2 V range. The accuracy or
error is given by (see the first line of Table 2):
1.234 x 0.005 (0.5 %) = 0.006 + 0.003 (3 ls digits) = 0.009.
Thus the measurement would be written (1.234 ± 0.009) V.
Single Resistors
For practice measure the resistance of the four
resistors issued you, one at a time. How well do
the experimental values, the values you measure,
agree with the theoretical values, the values given
by the color codes? Take account of the manufacturer’s tolerance and of the accuracy of the DMM
when used as an ohmmeter, as given in Table 2.
For an example calculation and interpretation see
Table 3.
How well do the resistances of the three resistors whose resistance is given as nominally equal
by the manufacturer compare? Are they equal to
within the 5% or 10% tolerance factor? Examine
them carefully. Do any show signs of having been
overheated?
B7-8
Resistors in Series
Connect two of the smaller-valued resistors in
series and measure the total resistance. How well
does the experimental value compare with the expected theoretical value calculated from the series
relation, Figure 5a? Repeat for three resistors. For a
sample error calculation for two resistors see Table
4.
Resistors in Parallel
Connect two of the smaller-valued resistors in
parallel and measure the total resistance. How well
does the experimental value compare with the
expected theoretical value calculated from the
parallel relation, Figure 5b? Repeat for three
resistors.
DC Circuits 7
Table 3. A Example Interpretation of Resistance Measurement
A resistor selected was found to have the color code: red, black, brown, gold.
Therefore the resistance deduced is: (1000 ± 5%) Ω = (1000 ± 50) Ω.
The same resistor was measured with the DMM to be: 985 Ω.
The range used was 2kΩ, with accuracy (Figure 2) of 0.8% + 1 digit.
Thus the uncertainty in the measurement is: 985 x 0.008 + 1 = 9 Ω.
The experimental value is therefore to be written as (985 ± 9) Ω.
The ranges of the theoretical and experimental values, (1000 ± 50) Ω and (985 ± 9) Ω overlap.
Thus it can be said that the two values agree to within the experimental error and tolerance.
Table 4. An Example Interpretation of Two Resistors Measured in Series
Two resistors were selected whose color codes were: red, black, brown, gold.
Therefore the resistance deduced for both is: (1000 ± 5%) Ω = (1000 ± 50) Ω..
The resistance expected for the two in series is: 2000 Ω,
with an uncertainty given by:
∆ R total =
502 + 502 =
2 50 Ω
= 70 Ω (rounded to one significant digit)
Therefore the theoretical value is given by (2000 ± 70) Ω
The resistance of the two resistors in series was measured with the DMM to be: 1988 Ω.
The range used was 2kΩ, with accuracy (Figure 2) of 0.8% + 1 digit.
Thus the uncertainty in the measurement is: 1988 x 0.008 + 1 = 20 Ω.
The experimental value is therefore to be written as (1990 ± 20) Ω.,
when rounded correctly.
The ranges of the theoretical and experimental values, (2000 ± 70) Ω and (1990 ± 20) Ω overlap.
Thus it can be said that the two values agree to within the experimental error and tolerance.
Exercise 3. Basic Studies of Current and Voltage
Some students are skeptical that electric current flows continuously and that a multimeter
can have its own internal resistance. Therefore we ask you to try the following:
Current Study
Hook up the circuit shown in Figure 9a using one
of the small-valued resistors. Set the two multimeters to ammeter function. Go slowly. Ask your
TA for help if you have trouble interpreting the
circuit. When you are ready turn the power supply
ON and increase the output by about a quarter
turn. Do you read the same current on the two
meters (within the uncertainty given in Table 2)?
Pay attention to the ranges. Locate the positive
connection to the power supply and disconnect
and reconnect it a few times and note the meter
B7-9
7 DC Circuits
readings. Do they both respond in unison? Should
they respond in unison?
the other in ammeter function. When you are
ready turn the instruments ON and increase the
output of the power supply about a quarter turn.
Measure single values of I and V and calculate the
resistance of the ammeter. Repeat this for a few
higher settings of the power supply output. Can
you conclude that the resistance of the ammeter is
small (in relation to the resistance of the voltmeter)? Do you get the same value of resistance for
the ammeter on different current ranges?
à NOTE
To allow for the fact that one meter may be “faster
acting” than the other try repeating the above with
the meters interchanged.
Ammeter Resistance
Hook up the circuit shown in Figure 9b, again
using one of the small-valued resistors. This time
one multimeter must be in voltmeter function and
V
+
–
A
A
+
+
R
V
R
V
–
+
A
(a)
(b)
Figure 9. A circuit (a) to study some aspects of the continuity of current and (b) to measure the resistance of the
multimeter in ammeter function.
An intentional mistake
Hook up the circuit shown in Figure 10a using a
multimeter in voltmeter function in a position
where it really should be set to ammeter function.
Turn the power supply ON and record your
+
V
observations for a few settings of the supply
output. Is anything meaningful displayed on the
multimeter? Explain.
–
+
+
V
R
V
R
A
(a)
Figure 10. Two examples in which multimeters are used incorrectly. Do not hook up circuit (b)!
B7-10
(b)
DC Circuits 7
Exercise 4. Measuring a “Small” Resistance Using the VI Method
Sometimes you want to know a resistance with
greater accuracy than is possible with a DMM. In
this case you may use the “VI” method. This
method consists of measuring I for a series of
different V and then finding R from the slope of
the V versus I graph. The graph method is superior
to measuring a single datapair because it yields a
better overall average value.
Go ahead and apply this method to find the
resistance of one of the small-valued resistors. In
doing this record the errors ∆V and ∆I from the
specifications in Table 2. Calculate also the power
dissipated. Plot your V vs I graph using pro Fit.
Does your result agree with the manufacturer’s
specifications for the resistor to within your experimental error? Comment on the accuracy of this
method. To assist you an example output and
interpretation is given in Figures 12 and 13. Do
you observe any evidence of equalling or exceeding the power rating of the resistor?
Figure 11. The Polynom dialog box.
Ohm's Law Data
Voltage (V)
1.0
0.8
0.6
0.4
0.2
0.04
0.08
0.12
0.16
Current (A)
Figure 12. A sample output from pro Fit for a VI method experiment.
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7 DC Circuits
Iterations:
7
------------------------------------------Chi squared =
0.1326
Parameters:
Standard deviations:
deg
=
1.0000
const =
0.0000
a1 = 7.1274
∆a1 = 0.2665
Goodness of fit:
=
0.9877
Figure 13. The results from the fit shown in Figure 11. Why does this function not include a constant term? The
resistance written correctly is (7.1 ± 0.3)Ω.
Exercise 5. Measuring a “Large” Resistance With the “Wrong” Circuit
You must be careful using the circuit of Figure 6.
To demonstrate this, substitute the large-valued
resistor for the small used in Exercise 4. Calculate
R from one measurement of V and I. Do your
results agree with the manufacturer’s specification
to within your experimental error?
In fact, a circuit like Figure 6 will yield increasingly inaccurate values of R as R’s value approaches the internal resistance of the voltmeter. Why?
Explain this effect in more detail than was done in
the Theory section.
B7-12
You will therefore require a different circuit to
measure this resistance! Design such a circuit by
modifying Figure 6. Think about it. Consult your
TA only as a last resort. (Hint: The circuit need be
modified by moving only one connection.) Using
your new circuit find R from single measurements
of V and I. Does your result now agree with the
manufacturer’s specification to within your experimental error? How well does your result agree
with the value you measured in Exercise 2?
DC Circuits 7
Addendum
Table 5. DC Specifications of Model 700T True RMS Digital Multimeter.
Accuracies are ± (% of display + No. of digits in the least significant place of display) *
Operating Temperature: 0 ˚ C to –40 ˚ C.
Function
Range
DC Voltage
200 mV, 2 V, 20 V, 200 V, 1000 V
→ Input Impedance 10 MΩ on all ranges
Accuracy
± 0.05% of display + 2 ls digits
DC Current
200 µA, 2 mA, 20 mA
200 mA, 2 A, 20 A
Maximum input current: 20 A
± 0.3% of display + 3 ls digit
± 0.5 % of display + 3 ls digit
200 Ω, 2 kΩ, 20 kΩ, 200 kΩ
2 MΩ
20 MΩ
Open Circuit Voltage: < 3.2 V.
± 0.1% of display + 2 ls digits
± 0.25% of display + 2 ls digit
± 0.5 % of display + 5 ls digits
Resistance
* For example, suppose the DC voltage display is 1.2341 volts on the 2 V range. The accuracy or
error is given by (see the first line of Table 5):
1.2341 x 0.0005 (0.05 %) =
0.0006 + 0.0002 (2 ls digits) = 0.0008.
Thus the measurement would be written (1.234 1± 0.0008) V.
Physics Demonstrations on LaserDisc
from Chapter 42 Resistance and DC Circuits
Demos 17-18 to 17-27 Resistance Wires, Ohm’s Law, Series/Parallel Resistors etc.
from Chapter 43 Voltage Drops and I2 R Losses
Demos 18-01 to 18-07 Voltage Drop Along Wire, Sum of IR Drops etc
Activities Using Maple
(Under construction)
Stuart Quick 94
EndNotes for DC Circuits
1
The direction of current shown here is the conventional direction, the direction that appears in most university
physics texts and that used by professional physicists. The electrons are actually moving in the opposite direction,
however. To avoid confusion we shall use the terms “conventional current” most of the time, and “electron current”
for the actual current when it is seen to be important.
2
This will happen if the energy source is a battery. In this experiment, however, the energy source is a power
supply whose primary source is Ontario Hydro. Ontario Hydro is not expected to “die” anytime soon.
3
If your meter is yellow it is a DT-890D type. If black it is a Model 700T True RMS Digital Multimeter. If black use
the specifications listed in Table 5.
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