Lab IV: Voltage, Current, and Resistance
George Wong
Instructor: Geoffrey Ryan
Experiment Date: 6 March 2012
Due Date: 20 March 2012
1
Objective
The objective of this laboratory was to learn about voltages and currents as generated by
the SWS suite and to investigate relationships between voltage and current in resistors of
various sorts.
2
Theory
Electric potential (voltage measured in volts) is related to the energy required to move
a charge from one point to another within an electric field. Because electric field is not
necessarily constant between two points, in order to properly find the voltage difference, the
path must be integrated along.
∆V = −
RB
A
E · dl
While voltage is one ‘part’ of electricity, another part is current which is a measure of charge
per time (units C/s or amps).
For ohmic materials, the following relationship can be said to exist: V = IR where V is
voltage measured in voltages, I is current measured in amps, and R is resistance measured
in ohms.
When resistors are added in series, the total equivalent resistance is given by the sum of the
component resistances: Req = R1 + R2 + · · · + Rn .
Further, when resistors are placed in parallel, equivalent resistance is given by the following
formula: R1eq = R11 + R12 + · · · R1n
3
Set Up
We primarily used the SWS program to generate the voltages and currents that were to be
‘ran through’ resistors etc., various potential differences and currents being measured as the
initial voltage/current was “affected” by the circuit(s).
There were two different types of voltmeter used in this lab: one that is analogue, the other
digital through SWS. Each voltmeter has affixed two leads that are ‘attached’ to a circuit
such that the voltmeter runs in parallel to the circuit.
Similarly, there was both an analogue and a digital ammeter (SWS) used in the lab. Opposite
the voltmeter, however, the ammeter(s) were ‘attached’ to the circuits such that they ran in
series.
A pre-fabricated piece of circuitry complete with several resistors, an LED, and an incandescent light bulb was also used for the experiment. The board was made in such a way
as to allow wires to be plugged into various ports (daisy-chained, as well) to create certain
1
circuits of varying complexity. In a way, the board acted as a assembled breadboard with
components readily in place.
4
Procedure
• The SWS interface was set to generate 5V DC power with the leads going through the
analogue voltmeter device.
• The voltmeter was set to measure at various different scales, it was observed how the
change in the voltmeter scale affected the readings.
• With the positive ramp option selected, the supply was set to various frequencies; the
reported values (on the voltmeter) were observed and trends noticed.
• Within the SWS software program, interfaces were set up to examine the voltage that
the machine ‘thought’ it was giving. These values were compared to the values presented
by the analogue devices.
• The settings on the analogue device were changed, with the voltage/current being
measured by the SWS suite and other analogue devices. The way that trending one
setting (on the device) affected others was noted.
• The resistors on the board were placed into various configurations with the voltmeters
(both analogue and SWS), ammeters, and power devices. The way that the different
components of the circuit interacted with the measurements (or were reflected in the
measurements) was noted.
• The power being generated was alternated between DC and AC current, with different
patterns (for the AC current). Again, the readings/manipulation of the devices on the
board were observed.
• The case where power was ran through a resistor was first observed, with special note
to Ohm’s law.
• The alternative case of a filament (not ohmic) was observed. The existence of hysteresis
was observed/considered.
• Various configurations of resistors (series/parallel) were hooked up on the board/power
and their effective resistances were measured (by voltage/current measurements).
5
Data
Section 10
Resistor 1: brown, black, brown, gold — 1 0 × 10 = 100 ± 5%
Resistor 2: brown, green, brown, gold — 1 5 × 10 = 150 ± 5%
2
Section 11
volts σv (volts) amps σa (amps)
5.00
± 0.03
0.0509 ± 0.0001
0.0305 ± 0.0001
3.00
± 0.03
1.04
± 0.01
0.0102 ± 0.0001
Section 15
volts σv (volts) amps σa (amps)
series
5.00
± 0.01
0.0378 ± 0.0005
± 0.01
0.198
± 0.001
parallel 5.00
Section 16
1 amp
0.1 amps
0.01 amps
Vnative volts Vresistor volts
0.103
0.083 ± 0.006
0.103
0.048 ± 0.005
0.103
0.024 ± 0.005
Inative amps
Iresistor amps
0.00830 ± 0.0004 0.010 ± 0.002
0.0046 ± 0.0004 0.082 ± 0.002
0.0026 ± 0.0003 0.219 ± 0.002
* Note that the uncertainty in the voltage produced is considered to be zero because it is
negligible compared to the reported voltage.
Section 17
v resistor (V)
value
7.40
unc
± 0.01
6
v supply (V)
9.010
± 0.001
I SWS (A)
0.746
± 0.001
I supply (A)
0.740
± 0.002
I analogue (A)
0.721
± 0.002
Calculations
* Note that many of the questions deal explicitly with calculations, so the two sections should
somewhat be considered similar.
R = V /I
16140 ohms = 5.003 volts / .00031 amps
32710 ohms = 5.004 volts / .00015 amps
Section 11
volts σv (volts) amps σa (amps)
5.00
± 0.03
0.0509 ± 0.0001
3.00
± 0.03
0.0305 ± 0.0001
1.04
± 0.01
0.0102 ± 0.0001
resistance (ohms)
98.2
98.4
102
Section 16
1 amp
0.1 amps
0.01 amps
Vnative volts Vresistor volts
0.103
0.083 ± 0.006
0.103
0.048 ± 0.005
0.103
0.024 ± 0.005
Inative amps
Iresistor amps
0.00830 ± 0.0004 0.010 ± 0.002
0.0046 ± 0.0004 0.082 ± 0.002
0.0026 ± 0.0003 0.219 ± 0.002
3
native resis (ohms)
10.3
1.26
0.470
7
Error Analysis
* The above note for the data section applies here as well.
R = V /I ⇒ dR = dV /I − V /I 2 · dI
Section 11
volts σv (volts) amps σa (amps)
5.00
± 0.03
0.0509 ± 0.0001
3.00
± 0.03
0.0305 ± 0.0001
0.0102 ± 0.0001
1.04
± 0.01
Section 16
Vnative V
1a
0.103
0.1 a
0.103
0.01 a
0.103
8
Vresistor V
0.083 ± 0.006
0.048 ± 0.005
0.024 ± 0.005
resistance (ohms)
98.2
98.4
102
Inative A
0.00830 ± 0.0004
0.0046 ± 0.0004
0.0026 ± 0.0003
ohms uncertainty
0.4
0.7
0.02
Iresistor A
0.010 ± 0.002
0.082 ± 0.002
0.219 ± 0.002
res (ohms)
10.3
1.3
0.470
unc res
0.1
0.9
3.0 (?)
Questions
Section 7
Increase the sensitivity of the meter to the 5 V scale. What is the meter reading? How does
it compare with the programmed voltage? While MON is still active, try changing the DC
voltage to another value.
The meter reading is at 5.02 ± 0.02V (on a 10V scale). This is only slightly off from the
reported voltage of 5.000V by the SWS program. As the DC voltage on SWS is changed, the
meter-read voltage changes accordingly.
If your voltmeter was center zero (rather than left zero) and you put a 100 Hz sinusoidal
voltage across the meter would it read anything?
In the above situation, because the frequency of the voltage change, the meter would be
centered at zero and would therefore (most likely) read nothing.
Sketch the voltmeter reading as a function of time. Change the frequency to 100 Hz. Describe
the meter reading and explain your observations.
See the included sketch of changing voltmeter needle positions.
Section 8
Program the interface output for 5 V DC and click MON. How do the readings of the output
voltage digital display, the voltmeter, and the programmed voltage, compare? What is the
current through the analog voltmeter on the 25 V and 5 V scales?
The voltages were all about the same, around 5.00V. The output voltage reported was between 5.003 and 5.004 V with an uncertainty of approximately 0.001 V. The other displays/scales showed roughly the same value, with a slight oscillation attributable to er4
Figure 1: Sketch of voltmeter reading with time
ror/uncertainty. The current through the voltmeter was 0.00031 A and 0.000153 A for the
two scales (25 V and 5 V). Essentially, it should be noted that the reported voltages are all
about the same, and they do not change regardless of the scale of measurement, whereas
the the reported current significantly changes. As the scale increases, the reported amperage
goes down.
What is the internal resistance of this voltmeter on these scales? (The internal resistance is
the voltage across the meter divided by the current through the meter.)
At the 25V scale, the resistance was approximately 16 kohms; at the 5V scale, the resistance
was approximately 32 kohms.
QUESTION. How do you think turning the sensitivity knob of the analog voltmeter changes
the sensitivity of the meter?
Turning the the sensitivity knob essentially changes the internal equipment such that the
circuit that is being completed has a different deflection (different resistance changes the way
the system interacts with itself (physically)).
Section 10
Verify these resistance values from the color. What is the range of acceptable resistance values
for these two resistors?
See the data section above. The acceptable resistance values were 100 ohms with 5% tolerance
and 150 ohms with 5% tolerance. These are, respectively: 95 – 105 ohms; 143 – 157 ohms.
Section 11
Is the resistor warm?
The resistor was not noticeably warm. It was left to potentially warm itself for several
minutes. After the several minutes, it was obvious that the resistor was somewhat warmed;
however, there the temperature difference was not appreciable.
Divide the measured voltage across the resistor by the measured current through the resistor
5
to obtain the resistance to 3 significant figures. Repeat for voltages of 3 V and 1 V. Compare
your 3 resistance values. Is there a trend you can explain?
When the voltage is changed, the current changes in a way that suggests there is a direct
relationship between the two. It turns out that the constant of proportionality is equal to
the reported resistance and does not (significantly) vary from one case to the next. In this
case, the ‘resistance’ was ≈ 100 ohms.
Section 13
Open a new graph display with output voltage on the vertical axis and output current on the
horizontal axis. The data you just took will appear on this graph. Does it look like Ohms law
is being obeyed?
It does appear that Ohm’s law is being obeyed. There is a direct relationship between the
two variables/quantities and the plot seems to go directly through the point (0,0) which is
the only one that would make sense (given there is no y-intercept other than zero).
Using python, perform a linear fit analysis on the voltage and current data. Determine the
slope, y-intercept, and their respective uncertainties. Are they what you expect? Create a plot
of the data with the line of best fit. Is Ohms law being obeyed? What is the intercept and
slope of the fitted line. Are they what you expect?
From python: slope= 99.092 ± 0.003; y-intercept= −0.016 ± 0.091. A plot is included at
the end of this report. It appears that Ohms law is being obeyed here, as there is a direct
relationship (see the graph) and there is relatively no y-intercept. (There should be no
y-intercept—V = IR—there is no extra value being added in there.)
The slope is approximately 100, which directly relates to the value for the resistance (in
ohms). The y-intercept is essentially 0, which corresponds to the nonexistent y-intercept in
the Ohms law equation. They are the values that one would expect to get (as the resistance
of the resistor was reported to be 100 ohms).
Section 14
Discuss your results. On the graph of output voltage vs. output current you will see hysteresis.
Can you explain this? Can you figure out which branch of the curve is taken with increasing
voltage magnitude and which is taken with decreasing voltage magnitude?
The hysteresis is due to the fact that the relationship between current and voltage is not as
clear-cut as with ohmic materials. As the filament in the bulb heats up, its effective resistance
changes and this constant change can be somewhat seen on the graph. When the current
first starts to ‘change’, it seems as if the change is getting ‘warmed up’, but it eventually
reaches a certain point at which its effective resistance seems to stop changing. At this point,
the graph distinctly changes shape.
Take a plot of output voltage vs output current at a frequency of 0.01 Hz. How is the hysteresis
affected? Explain.
At 0.01Hz, the hysteresis is much less noticeable. It is possible that this lesser degree is due
to the fact that it could be ‘stretched out’; however, I personally believe that this is not the
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case. Rather, I posit, there is less hysteresis. As the change is much slower at 0.01Hz, the
variables involved change less quickly, and the heat of the filament can change in a somewhat
direct way. This leads to the illusion (or not-so-much illusion) that there is less (or no)
hysteresis.
In both these cases, consider how long you need to take data.
In order to properly observe the hysteresis, it is required that enough of the standard cycle be
had such that any changes (along the course of the cycle) may properly manifest themselves.
As such, we might say that one full cycle is needed. This can be found by inverting the
Hz-rate number and counting that as the number of seconds required.
Section 16
For the various sensitivities of the analog current meter record the voltage output of the
interface, the voltage across the resistor, and the currents measured by the analog meter and
the digital meter. Discuss your results. Does the analog current meter affect the current in
the circuit?
The current meter does change the current in the circuit. According to the data, as the meter
becomes more sensitive by factors of ten, the current running through the resistor increases
by between factors of eight and six. Overall in the circuit, the current seems to halve from
one to the next.
What is the resistance of the analog current meter on its different scales? How does turning
the sensitivity knob change the sensitivity of the current meter?
From 1.0 A to 0.1 A to 0.01 A, the resistance goes from 10.3 to 1.3 to 0.47 ohms in the circuit
(which is assumeably the same current as that in the ammeter that is connected in series).
As the meter is made more sensitive, resistance seems to go down.
Section 17
The voltage across this resistor give the current. Does this sensor affect the current in this
circuit?
Ultimately, the sensor does not affect the current in the circuit. Current will be the same
through any devices that are hooked up in series; therefore, no matter how the sensor would
otherwise ‘affect’ the circuit, so long as it is in series, it will not affect the current.
Design and construct a single circuit that will allow you to simultaneously measure the same
current in the analog current meter and the SWS current sensor. Do these two meters agree?
The two meters do agree. They are off by a slight amount; however, the discrepancy is only
about 1% of the actual value. (See the diagrams attached at the end of this report for the
designed circuit.)
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9
Conclusion
One of the ‘standard’ relationships between voltage, resistance, and current that was observed
during the lab was Ohms Law, namely that V = IR. Obviously, though, Ohms Law does
not work in all cases—it only works in the cases where ohmic materials are being considered.
(The case of the incandescent light bulb was one example of a non-ohmic material.)
As could be seen on the graph (produced in PYTHON for section 13), there does appear to
be a direct relationship between voltage and current, moderated by some constant, which we
see fit to call resistance.
In the case of the incandescent bulb however, it was obvious that—if there were indeed
a relationship between voltage and current present—the relationship was dependent upon
several variables. This would indicate hysteresis. Interestingly, for the case of the bulb, it
seemed that the plot would start out ‘curved’ and then become more flat-line. This seemed
to indicate (to me) that the relationship began in a very non-ohmic nature and became ohmic
as time went on.
The ammeter and voltmeter both worked on using the electricity to affect some device within
them; and the best ways to measure current/voltage turned out to depend on how they were
hooked into a circuit. The ammeter is best in series whereas the voltmeter is best in parallel.
As it turns out, the different sensitivity scales on the two meters affected the amount of change
they caused to the circuit overall (e.g. in the case of the ammeter, increased sensitivity seemed
to lower effective resistance).
Because much of the lab was done using computerized equipment, there was much possibility
for error to be had in the measurements. If the computer were not properly calibrated (or it
were doing calculations incorrect), then error would appear in the displays, be recorded, and
summarily affect all of the data. This would NOT necessarily change the fact that there is
a reported direct relationship between voltage and current, though.
Not considered in the experiment was the fact that the wires/leads connecting the various
components have resistances of their own. This resistance was considered to be negligible;
however, due to the fact that several measurements were taken and considered to be in the
same state (as in no change in resistance), technically these values are incorrect. The exact
amount of effect the inherent wire resistance had upon the circuit is unknown (though still
considered slight).
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
def LineFit(x, y):
’’’ Returns slope and y-intercept of linear fit to (x,y) data set’’’
xavg = x.mean()
slope = (y*(x-xavg)).sum()/(x*(x-xavg)).sum()
yint = y.mean()-slope*xavg
8
Figure 2: Assembled Circuit Board
Figure 3: Current vs Voltage for V = IR
Figure 4: Various types of AC Voltage cycles
9
Figure 5: Two ammeters measuring one Amperage
return slope, yint
# Finds the y-intercept and slope of line of best fit of data x,y,u
def ChiSqFit(x,y,u):
’’’ Returns slope and y-int of linear fit to (x,y,u) data set ’’’
xhat = (((x/(u**2))).sum())/(((1/(u**2))).sum())
yhat = (((y/(u**2))).sum())/(((1/(u**2))).sum())
slope = (((x-xhat)*(y/(u**2))).sum())/(((x-xhat)*(x/(u**2))).sum())
yint = yhat - slope*xhat
s_b = np.sqrt(1/(((x-xhat)*(x/(u**2))).sum())) # calculates sigma_b
s_a = np.sqrt((s_b**2)*((((x**2)/(u**2)).sum())/(1/(u**2)).sum()))
return slope, yint, s_a, s_b
# returns m, b; sig_a, sig_b
volts, current = np.loadtxt(’13.txt’, skiprows=3, unpack=True)
m,b = LineFit(current,volts)
dev = []
for n in range(len(current)):
dev.append(volts[n]-m*current[n]+b)
# get stddev and print
sum = 0
mean = np.average(dev)
for a in range(0,len(dev)):
sum += (dev[a]-mean)**2
stddev = np.sqrt(sum / (len(dev)-1))
unc = []
for k in range(len(volts)):
unc.append(stddev)
10
m,b,s_m,s_b = ChiSqFit(current,volts,np.array(unc))
print ’slope’, m, s_m
print ’y-int’, b, s_b
predictvolts = current*m+b
plt.title(’Current vs. Voltage’)
plt.ylabel(’Voltage (volts)’)
plt.xlabel(’Current (amps)’)
plt.plot(current,volts,markersize=.1)
plt.plot(current,predictvolts,’-k’)
plt.show()
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