Lab Report
Lab report MUST be uploaded to blackboard by 25
Sept. Also, please bring a paper copy to my office.
If you have trouble uploading, please email it
directly to me (eno@umd.edu)
Lab #1: Imperfections in Equipment
•  Measure
the internal resistance of a battery
•  Measure the input impedance of the oscilloscope
•  Measure the output impedance of the signal
generator
Theme: model the real by an ideal in series
with a resistor.
Basic electrical terms
Make sure you have a solid conceptual
understanding of the following terms, their
relations, and their differences
•  voltage
•  electric field
•  current
•  electric potential
•  resistance
Current
•  Current: amount of charge that passes a
point on the wire each second (amps =
columb/second)
•  Determined by number of charges and by
their speed
Basic Electrical Concepts
Conductors
Use a battery or some other emf to set a
voltage across an object. Chemical reaction
in the battery allows rearrangement so as to
maintain an (approximately) constant voltage
difference between the two terminals.
Terminal velocity depends on voltage, the
geometry of the materials, and the
properties of the material Resistivity
Ohmic materials:
A
I = ΔV where E=ΔV / d
ρl
Resistance
Material
Insulators
Mica
Glass
Rubber
Semi-conductors
Silicon
Germanium
Conductors
Carbon
Nichrome
Copper
resistivity at room temp (W-m)
2x1015
1012-1013
1013
2200
0.45
3.5x10-5
1.2x10-6
1.7x10-8
Kirchhoff’s Rules
From course work, remember how to use Kirchhoff’s rules to calculate
voltage and currents in circuits? If not, see lab writeup.
•  In going round a closed loop, the total change in potential must be zero
•  The flux of charge is conserved so that at any junction the current
flowing into the junction is equal to the current flowing out of the junction
Applying these rules to enough junctions and loops generally leads to
enough equations to solve for the number of unknown currents and
voltages.
Effective Resistance
Sometimes you can use these shortcuts instead.
Reff = R1 + R2
1
1
1
=
+
Reff R1 R2
When calculating currents and voltages in a circuit, can replace
these combinations by an “effective resistance” without altering
the current through and voltage across these “elements”
Circuits
When using these rules of the time, you neglected
to take into account the fact that the instruments
you use to measure the circuit can themselves alter
the performance of the circuit.
We will study this in the lab, see how big the effect
is, and from that get an idea of when this needs to
be taken into account when comparing results to
predictions.
Internal Resistance of a Battery
Imagine a simple circuit consisting of a battery and
a resistor.
If you varied the resistance R and
plotted V versus I, what would you
get?
V =ε
ε
A horizonal line
Simple circuit
A more realistic model
r
ε
If I plotted V vs I, what would
I get now?
V = IR
Have 3 variables in the equation (V,I,R).
Need to get rid of R.
ε − Ir − IR = 0
ε − Ir ε
R=
= −r
I
I
so
V=ε -Ir
I get a straight line.
What does the slope represent?
What does the intercept represent?
An even more realistic model
Will get rA and rV from meter manual and estimate effect on
estimate of r and ε
Multi-meter syst errors
Input/Output Impedences
A function generator, like a battery, is a voltage source.
An oscilloscope, like a multimeter, is a measuring instrument.
We will measure the “internal resistance” of each of these devices.
Estimating Errors: Review
• Systematic errors : sources of error that have the same size
effect on every measurement that is made (or a correlated effect)
•  a ruler that was not manufactured correctly
•  a consistently delayed reaction when using a stop watch
•  your inability to perfectly estimate the size of a stray
magnetic field from your computer that leaks into your
experimental area
•  Random errors : sources of error whose effect varies with each
measurement
•  precision of your measuring device
•  when using a stop watch, a reaction time that sometimes
anticipates the event, some times is in retard of the event.
Multi-meter syst errors
Will assume that the systematic error due to the factor calibration is in
the form
Vmeasured = λVtrue + b
λ = 1 ± σλ
b = 0 ± σb
Systematic Errors and fits
Last week, we learned
•  how to propagate errors in measured quantities to errors in quantities
calculated from them via a simple algebraic formula (both random and
systematic are handled the same way).
•  how to calculate the uncertainty on the fit slope and intercept from a
linear fit due to random errors in the x y variables
This week we’ll learn how to calculate the uncertainty on the fit slope
and intercept from a linear fit due to systematic errors in the x-y
variables.
Error on slope and intercept
σb = σ
∑x
2
j
N ∑ x − (∑ x j )
2
j
Note error on
intercept
scales with
root(N)
Fitting and syst errors
Suppose you are measuring V using a meter that has infinite
accuracy and that has no random errors, but that always reports a
voltage that is always off by 0.25V?
Adding points does not reduce the error. Previous
formula can not work for systematic errors
slope
How can slope be changed? If voltage is always off by a scale
factor, or if current is always off by a scale factor, slope is off by the
same factor. The error in the offset (b) does not cause an error in
the slope at all.
xmeasured = λ x xtrue + bx
ymeasured = λ y ytrue + by
σ = (m ⋅ σ λ x ) + (m ⋅ σ λ y )
2
m
2
2
intercept
What if the voltage is always off by a fixed,
constant amount?
(see “lectures” link of class web site,
kelly_SystematicErrors.pdf, for a more complete,
rigorous derivation of this result.)
Random and Sys errors
•  first, fit to a straight line using only random
errors
•  get the error on the fit m and b due to random
errors from the spreadsheet
•  calculate the errors on m and b due to
systematic errors as shown on previous 2 slides
•  take the error on m due to random errors and
the error on m due to systematic errors and add
them in quad
•  ditto for b
Fitting and Syst Errors
If you don’t understand this (how to calculate the
syst error on slope/intercept and then combine with
the stat error), don’t leave the room today until you
do! It’s important for this and future labs!
linearizing
This semester, we will often do a variable transformation in
order to get a linear dependence that we can easily fit.
1
1
1
=
RS +
VS VB RIN
VB
feed to fitter:
1
y→
VS
x → RS
get from fit m and b.
1
m→
VB RIN
1
b→
VB
Please show to yourself that
b
R IN =
m
Linearizing
When we transform variables, we also need to
recalculate the errors.
In this lab:
Rounding uncertainties
If your digital voltmeter says 3.02 V, the real measurement could be
between 3.015 and 3.025V with equal probability. What is the
uncertainty? -> want +- 1 sigma to include 68% of the measurements.
Δ /2
1=
∫
A dx where Δ is the lsb
− Δ /2
→ A =1/ Δ
choose "σ " so
σ
1
.68= ∫ dx
Δ
−σ
2σ
Δ
σ = 0.34Δ
.68 =
Sqrt(12)
When you have an LSB, what is the random error?
Imagine a step with width a centered at zero.
Remember:
Δ/2
x2 =
∫
−Δ/2
3
2
x dx
Δ
=
x
3
Δ/2
−Δ/2
Δ
RMS = Δ / 12 = .29Δ
Δ2
=
12
Changes to lab
•  experiment starts bottom of page 8
•  check that probes are not set on x10
•  check that ammeter is set on DC, not AC
•  Before starting section A, put your voltmeter directly across the battery while the battery
is not yet inserted into the circuit. Record the voltage you measure in your spreadsheet
(and label clearly).
• Do not do page 8, paragraph starting “Before”.
•  Never use the nominal value of a resistor. Always measure the resistance using an ohm
meter. Always remove the resistor from the circuit before measuring its resistance (why?)
•  All numbers should have units and be carefully labeled.
• Some of the resistors have values that drift with temperature. It is important to measure
V&I simultaneously. If you measure one, wait a minute, then measure the other, you’ll get
a bad result. There will be a random error from your ability to read the 2 meters at the
same time. How will you estimate this random error? (Drift is biggest when using smallest
resistor. Why?)
•  ignore the last paragraph of section part I, A (pg 9).
•
Lab changes
• For the last paragraph of section Part I, B (pg 9), the answer is yes, systematic
errors are much easier to handle if you use the same scale for all readings. Please
do it this way.
•  Ignore the last sentence just above Part II, A.
•  Be careful with grounds when measuring the output impedance of the signal
generator.
•  You need to quote errors on all measured numbers and all numbers calculated
from measured numbers.
What are we testing
•  Before you leave class, tell professor Eno
what this lab was testing (relevant for the
abstract of your lab report)
Bureaucracy
•  Please note lab report is due Sept 25.
Please upload to elms and bring a paper
copy to my office (slide under my door if I’m
not there)
•  No Class Oct 2
•  See you Oct 9
You must upload your spreadsheet before leaving class!