Exp 01
DC Circuits and Measurements
To be Read: Notes 01, 02 and 03
This experiment deals with the basic concepts of DC circuits. It has four parts. In Parts A and B you will
measure the internal resistance and the deflection sensitivity of a common d'Arsonval type DC meter
movement. Using this movement you will design a voltmeter, and use it in Part C to measure the
Thevenin equivalent circuit of a two-terminal “DC black box”; you will then verify your measurements
using a professional quality digital multimeter (DMM) of high input impedance. By opening the DC
black box and tracing the circuit, you will calculate its Thevenin equivalent from the circuit constants. In
Part D you will determine the maximum power the black box can deliver and the load resistance that
makes this possible. You will then compare these values with the theoretical values you obtained from
circuit analysis.
Getting Started
The background material for this experiment is
covered in Notes 01 through 03. Even though your
instructor may not yet have covered this material in
lectures you should read the notes before going on.
It is good practice in any experiment to record the
serial and model numbers of the meters, instruments,
and black boxes you use. Then if you forget some key
measurement, you may be able to make it later
without having to repeat the whole experiment.
In this experiment you will need the following
apparatus:
•
•
•
•
•
1 common d’Arsonval meter movement
1 variable resistance box 10 kΩ maximum
1 variable resistance box 100 kΩ maximum
1 1.25V standard voltage source
1 Radio Shack Manual/Auto Range digital multimeter
• 1 10 MΩ carbon resistor
• 1 “DC black box”
Examine the meter movement supplied you. Handle it
gently! There should be a wire of very low resistance
connected across its terminals to provide damping
protection of the movement in transit.
REMOVE THE DAMPING WIRE/RESISTOR!
Now the meter is particularly vulnerable; handle it
very gently.
The d’Arsonval meter movement is described in
Note 01. The movement provides an angular deflection from its equilibrium position which ideally is
proportional to the current flowing through the meter
coil; deviations from linearity may be due to angular
variation of the permanent magnetic field or of the
restoring spring constant. In the meter supplied here,
the angular deflection is indicated by the motion of an
attached needle relative to a graduated scale.
Zero the pointer (at the left hand end or at the center,
as appropriate to your meter). The zero changes if you
change the meter attitude (horizontal, vertical, or at an
angle), so once you set the zero, do all of your subsequent work with your meter in the same attitude.
Part A
Measuring the Internal Resistance
of the Meter Movement
A d’Arsonval meter movement is a passive device
since it possesses no internal energy source of its own.
To measure its internal resistance RI you need to supply an external source. You can measure the internal
resistance using the circuit in Figure 1-1. Connect the
movement in series with a variable resistance box set
to at least 90,000 Ω and the 1.25 V standard voltage
source.
R
+
s
100kΩ
V
R
I
Meter
Movement
Figure 1-1. Circuit for measuring the full scale deflection
(FSD) of a common d’Arsonval meter movement.
*** CAUTION ***
To avoid damaging the meter movement, set the
variable resistance box to at least 90,000 Ω initially.
There are two kinds of variable resistance boxes set
out in the lab; use the one that will give you at least
this resistance. Have the instructor check your circuit before you connect both battery terminals to it!
E1-1
Exp 01
When you connect the standard voltage source, the
needle of your meter movement should initially
indicate less than full scale. Gradually decrease the
resistance R S until you get exactly full scale deflection
(abbreviated FSD). To get exactly FSD you may have
to employ the low-value decades (hundreds, tens or
even units). When you are satisfied you have FSD
record the value of R S and its uncertainty.
*** CAUTION ***
When decreasing the resistance of the resistance
box, you may wish to set a decade on the box to
zero. Before doing so, be sure to switch the next
lower decade switch to maximum!
Leaving the series resistance R S set to the value for
FSD, add to your circuit a 10 kΩ resistance box R p , in
parallel with the meter movement, as is shown in
Figure 1-2 .
R
+
V
Part B
Deflection Sensitivity and Linearity
The quality (and cost) of a d’Arsonval meter movement is determined by its deflection sensitivity and
linearity. Now that you have a numerical value for the
internal resistance R I of your meter movement, you
can measure the deflection sensitivity of the movement and check the linearity of the meter response by
using the voltage of the standard voltage source and
the resistance box as standards. 2
Remove the parallel resistance R p from your circuit
but leave the series resistance R S in place. The circuit
has now been returned to the configuration of Figure
1-1 and your meter movement should once again read
FSD. Now increase the value of the series resistance RS
in several steps and tabulate the meter deflection θ for
each value of R S. Don't forget to include the point R S =
∞, in which case I should be zero and θ should be
zero. Inform your instructor if the needle on your
meter does not go to zero for zero current.
s
Calculating Current Sensitivity
100kΩ
RI
10kΩ
Rp
Meter
Movement
Figure 1-2. Circuit for finding 1/2 FSD.
Adjust the resistance box R p until you find the parallel resistance Rp that decreases the meter deflection to
exactly one-half of FSD. When you are satisfied you
have one-half FSD, record the value of R p and its
uncertainty.
You should be able to show that you can calculate
the internal resistance of the meter movement from
the expression:
RI =
RS R p
.
RS − R p
…[1-1]
Also calculate ∆R I. When writing your report, prove
eq[1-1] by circuit analysis, and include the proof in
your report.
€ 1
To deduce the current sensitivity α of your meter
movement, refer again to the circuit in Figure 1-1.
According to Ohm's Law, the current flowing through
the meter is given by the battery voltage divided by
the total series resistance, I (amperes) = V (volts) / (RS
+ R I) (ohms). If the deflection θ is, indeed, linearly
dependent on the current I then
θ = αI = α
V
.
RS + RI
…[1-2]
(You can neglect the very small internal resistance of
the standard voltage source). If you plot the meter
deflection
€ θ against V/(RS + R I), you can get α from
the slope of the best straight-line fit to your data.
Express the meter sensitivity both as the current
required for FSD and as current per smallest scale
division. Do this work with a program like pro Fit that
encorporates uncertainties in the measurements and
provides a graph complete with error bars. Express
your result for α along with its computer-generated
uncertainty ∆α.
Linearity Check
Now check the linearity of the meter response.
Calculate the meter deflection corresponding to each
value of RS as
1
The resistance boxes contain precision wire-wound resistors of
uncertainty 1%. You may experience some uncertainty in reading
FSD and 1/2 FSD. Reading uncertainty is discussed in Note 03.
E1-2
2
The uncertainty in the voltage of the standard voltage source is
1%.
Exp 01
θcalc = α
V
.
RS + RI
…[1-3]
Measuring V eq
Plot, as a function of the calculated current,
€
Icalc =
V
,
RS + RI
…[1-4]
the difference between the deflection θ read on the
meter and the calculated deflection θ calc. Plot also the
error bars
€ on your graph. Is there any evidence of
significant nonlinearity in your meter movement? If
so, you may have to take nonlinearity into account,
and recalculate the sensitivity and internal resistance
of the meter movement.
Constructing a Voltmeter
The business of constructing a voltmeter from a
d’Arsonval meter movement is discussed towards the
end of Note 01. All you have to do is add a resistor to
the meter movement. Calculate the external series
resistance that is needed to give FSD when a voltage
of 2.5V is applied to the circuit. (Don't forget to
include R I!) Adjust the series resistance box RS to this
value. You now have a voltmeter with 2.5V FSD. You
will use this voltmeter in Part C.
Part C
The Thevenin Equivalent Circuit
Obtain a “DC black box” from your instructor. The
name of the box refers not to its colour but to the fact
that you don't know what is inside it!) It will have two
terminals and a switch. The switch is there only to
prevent the internal energy sources from discharging
when the box is not in use; turn it on and leave it on
during your experiment. When you have finished
with it turn it off again.
In contrast to a d’Arsonval meter movement the DC
black box is an active element. As you have seen from
Note 02, and perhaps from an elementary text in
electronics, a two-terminal DC network can be represented by a Thevenin equivalent circuit, as illustrated
in Figure 1-3.
+
Your task is to determine Veq and R eq of your black
box.
Measure the “open circuit” voltage of your black box
by connecting a digital multimeter (DMM) directly to
the black box’s terminals. Leaving the DMM connected to the terminals of the black box, measure the
voltage between its terminals with your 2.5 V FSD
homemade voltmeter. Record the DMM reading when
your homemade voltmeter is connected to the
terminals, and when it is not connected. Do the two
readings agree within the accuracy of the DMM? If
not, your homemade meter must be drawing a
significant current from the DC black box. 3
Measuring Req
Connect the 10 kΩ decade resistance box and also the
DMM (but not your home-made voltmeter) in parallel
with the terminals of the DC black box, and find the
value of the load resistance that decreases the terminal
voltage to one-half of its open-circuit value. By
assuming that the DMM has a large input impedance,
and hence does not draw a significant current from
the DC black box, deduce the Thevenin output resistance R eq of your DC black box.
Provided that the input resistance of the DMM is
much larger than the resistance values in the circuit to
which it is connected, the DMM will not draw a
significant current. Estimate the input resistance of the
DMM by connecting the 10 MΩ resistor supplied in
series with it and noting the voltage reading when
this series combination is connected to your DC black
box. Explain clearly the logic of doing this.
Checking Voltmeter Quality
Now that you know the Thevenin output resistance of
the DC black box, you can explain why connecting
your homemade voltmeter reduced the black box
terminal voltage; include the explanation and relevant
calculations in your report. Explain clearly which
voltmeter (your homemade voltmeter or the DMM)
gives the best value for the true open circuit voltage
Veq of the black box, and why.
R eq
V
eq
Figure 1-3. Thevenin Equivalent
3
For accuracies of the RadioShack Manual Auto/Range DMM see
Table A-1 of Note 01.
E1-3
Exp 01
Deducing the Thevenin Equivalent
from the Circuit Constants
Remove the cover of the black box, identify the
components in it, and draw a diagram of the circuit,
including the values and tolerances of all the components. Find the Thevenin equivalent of the circuit by
analysis and check how well this agrees with your
measurements of Veq and R eq above. To do this you
may wish to review the material on equivalent circuits
in Note 02.
Theoretical Analysis
You can derive a theoretical expression for P max and
determine from it what the corresponding value of R L
must be. Figure 1-4 shows the Thevenin equivalent
circuit of your DC black box connected to a load
resistance R L . The power dissipated in R L is
R eq
+
V
RL
eq
Figure 1-4. Circuit to measure power transfer.
Part D
Optimum Power Transfer to a Resistive
Load
It is instructive to determine the conditions under
which a black box will deliver a maximum power to a
load. (This is not so important in the case of a black
box here containing cells and batteries, but it is
important if the black box were a power source like a
solar cell.)
Measuring RL for P max
The power dissipated in a resistor of resistance R is P
= VI = V2R, where P is in watts when V, I, and R are in
volts, amperes and ohms respectively. Connect a
resistance box R L and a DMM in parallel across the
output of your DC black box, and measure the
terminal voltage as a function of the load resistance
R L . Calculate the power dissipated in the load resistance for each measurement, plotting your results as
you go; choose a range of resistance values such that
you obtain a well-defined maximum in the power
curve. From your graph of P vs R L find the maximum
power Pmax dissipated in the load and also the value of
the load resistance R L when the power dissipated in
the load is a maximum. Estimate the uncertainties in
these quantities.
E1-4
P = I 2 RL ,
I=
where
€
so
€
Veq
Req + RL
 V
2
eq
P = 
 RL .
 Req + RL 
…[1-5]
Differentiate P with respect to R L and find the value of
R L for which P = Pmax. Derive an expression for Pmax in
terms of
€ Veq and R eq. Does this expression predict the
values for RL and Pmax you measured in Part C?
CHALLENGE: Do an internet search for a similar
experiment done at another university, and from
which this experiment might be improved upon.
Tidying Up
Before leaving your station in the lab, turn off your
DC black box and all the other powered equipment.
Replace the shorting resistor on your meter movement. Put away all connecting wires so your work
station looks the same as when you found it. Thank
you.