course 300111 natscilab unit electrical engineering i 0 ... · ammeter v a rsh load since the basic...

First Experiment

Uwe PagelNovember 16, 2018

Using a Voltmeter

The Multimeter

Multimeter Errors

Error Propagation

Course 300111NatSciLab Unit Electrical Engineering I

Introduction to the first experiment


Department of EECSJacobs University Bremen

Instructor - Uwe Pagel

e-mail - [email protected] tel.: +49 421 200 3114

Website - http://www.faculty.jacobs-university.de/upagel/

First Experiment


Using a Voltmeter

The Multimeter

Multimeter Errors

Error Propagation

Outline

1 Using a Voltmeter

2 The Multimeter

3 Errors Using a Multimeter

4 Error Propagation

First Experiment


Using a VoltmeterVoltage Measurement

Extend the Voltage Range

Current Measurement

Extend the Current Range

Resistor Measurement

The Multimeter

Multimeter Errors

Error Propagation

Voltage MeasurementBasic

To measure voltage the instrument has to be connected in parallel tothe circuit element to be characterized.Attention: The voltmeter acts as an additional load!! If the internalresistance of the voltmeter is low compared to the internal resistance ofthe circuit, the circuit is changed (current divider)!!

VU

LoadVoltmeter+/-199.9mV

In the shown case the range is ±199.9mV and the resolution is0.1mV . ±199.9mV , or ±399.9mV are typical basic voltmeters with aninternal resistance of 1GΩ.

First Experiment




Current Measurement



The Multimeter

Multimeter Errors

Error Propagation

Voltage MeasurementExtend the Range

VU Load Range+/-199.9mV

Rsh19MOhm

Rsh21MOhm

To measure higher values avoltage divider is used.Disadvantage : Resistance ofthe whole system is lowered!For a standard multimeterRsh1 + Rsh2 is typically 10MΩ.

±Uin

(Rsh1 + Rsh2)∗ Rsh2 = ±UMeter = ±199.9mV

For a range of 2V we solve the equation for Rsh2:

Rsh2 =200mV ∗ 10MΩ

2V= 1MΩ and Rsh1 = 10MΩ − Rsh2 = 9MΩ

We get a reading of ±199.9 for ±1.999V .In a general purpose instrument we have many ranges and the resistor Rsh1 ischanged by a switch (turnwheel or automatic). Usually there are ranges up to1000V.

First Experiment




Current Measurement



The Multimeter

Multimeter Errors

Error Propagation

Current Measurement

100Ohm10V

I ATo measure current the the instru-ment has to be connected in serieswith the load.

U

AmmeterV

ARsh

Load

Since the basic instruments measu-res voltage and has a high input re-sistance we need to convert the cur-rent. A resistor acts as a Shunt andthe current is determined byI = U/R.

To get different ranges for the input current RSH needs to be adjusted!Attention: The RSH acts as an additional load and as voltage divider!! Ithas to small compared to the load resistance. If not the circuit ischanged!!

First Experiment




Current Measurement



The Multimeter

Multimeter Errors

Error Propagation

Current MeasurementExtend the Range

If the voltmeter has a range of 200mV RSH is calculated as follows:

U

AmmeterV

ARsh

Load

Rsh = VMeter/IRange

We want to measure ±200mA:

Rsh =199.9mV199.9mA

= 1Ω

The reading is direct in mA and theresolution is ±0.1mA. The commais already at the right position!

In a multimeter RSH is changed by a switch. In a general purpose instrumentthere a ranges up to 20A. Rsh has values between 0.01Ω and 1kΩ. The lowerthe range the higher the resistor. (because the voltage drop is larger)An ammeter is extremely sensitive to overload because the powerdissipation over the internal shunt resistor. So the current path of amultimeter is normally protected by a fuse.

If you use an ammeter, start measuring in the highest range!!

First Experiment




Current Measurement



The Multimeter

Multimeter Errors

Error Propagation


A constant current is injected into the device under test. The resistance ismeasured indirectly as voltage drop.

V RxIconst

1mA

The voltmeter has again the 200mV range so it is possible to calculate theresistance:

Rx =U

Iconst=

199.9mV1mA

= 199.9Ω

For different ranges the current source may be varied, or the range of the DMImay be increased like described in the previous slide.

First Experiment


Using a Voltmeter

The MultimeterUsing the Multimeter

Multimeter Errors

Error Propagation

The MultimeterMultimeters Used in Lab

Since it is not very easy to handle several instruments the describedmeasurements are integrated in one instrument. The so called multimeter.

• In the Elabo multimeter all ranges has to be set manually.

• The Tenma has selectable small ranges. For higher values it is autoranging.

The DMI shows positive values when the common input is more negative thanone of the other inputs.

First Experiment


Using a Voltmeter


Multimeter Errors

Error Propagation


• Typical internal resistance for current is 0.01Ω to 1000Ω

• Typical internal resistance for voltage is 10MΩSometimes there is a high impedance mode, like in the Tenma with 1GΩ.Except for these special modes the internal resistance is constant.

• The range for resistance measurements is between 1Ω and 10MΩ.

For the exact specifications it is mandatory to look at the data sheet beforeusing the instrument!

• Carefully select the input terminals!

• Whenever using a multimeter start in the highest available range to avoidoverloading.

• Mind the polarity, especially when testing an unknown circuit! The signbelongs to the number and tells you a lot about the properties of yourcircuit or signal!

First Experiment


Using a Voltmeter

The Multimeter

Multimeter ErrorsErrors in General

Instrument Errors

Methodical Errors

Error Propagation

Errors Using a MultimeterErrors in General

There are different kind of errors:• Systematic instrument errors• Methodical errors because of misuse of an instrument.• Personal errors -

are erroneous/bad setups and reading errors.• Random errors (statistical erors) -

are generated by the changing environment. To find this error it isnecessary to take a lot of values and calculate a statistics. In EElab this is not done. In our case only the error propagation andthe maximum error are of interest.

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors

How to Use the Formulas

Errors Dependent ofRange

Methodical Errors

Error Propagation

Errors Using a MultimeterInstrument Errors

The accuracy of an instrument may be defined in different ways.

a) as the percentage of the read value

b) as the percentage of the range

c) as the number of units of the resolution

d) as the absolute value

For a typical multimeter the error is given as a combination of a) andc), or a) and b).

The absolute error (Eabs,∆E) of the most voltage ranges of theinstruments in lab is:

• Tenma ∆E = ±(0.06%rdg + 3dig) – ∆E in [V ]

• Elabo ∆E = ±(0.03%f .Value + 0.01%f .Range) – ∆E in [V ]

For the current and resistor ranges these formulas are different!

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors



Methodical Errors

Error Propagation

Errors Using a MultimeterInstrument Errors

To compare error values the ’Relative Error’ (Erel ,Erel%,E%) is used. Thegeneral formula is:

Erel =Valmeas − Valtrue

Valtrue– E% =

Valmeas − Valtrue

Valtrue∗ 100%

Valmeas - is a measured value.Valtrue - is the known true value.

To get the relative error from the multimeter we take

Valmeas − Valtrue ≡ ∆E the error value from the formula andValtrue ≡ reading from multimeter

Erel =∆ Erdg

– E% =∆ Erdg

∗ 100%

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors



Methodical Errors

Error Propagation

Errors Using a MultimeterInstrument Errors - How to Use the Formulas

You measure with the Tenma and the Elabo. The Tenma is in range 1 (4V) andthe Elabo is in the 2V range! Tenma reading is 1.500V. Elabo reading is1.5000V. Mind the digits after the decimal point!!!

Calculation for the Tenma, rdg = 1.500V and 1dig = 1mV :

∆E = ±(0.06%rdg + 3dig) = ±(

0.06 ∗ 1.500V100

+ 3 ∗ 1mV)

= ±0.0039V

E% = ±(

∆Erdg

∗ 100%

)= ±0.26%

Calculation for the Elabo, rdg = 1.5000V and Range = 2V :

∆E = ±(0.03%f .Value + 0.01%f .Range)

= ±(

0.03 ∗ 1.5000V100

+0.01 ∗ 2V

100

)= ±0.00065V

E% = ±0.043%

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors



Methodical Errors

Error Propagation

Errors Using a MultimeterInstrument Errors - Errors Dependent of Range

Using a wrong range affects the error quite a lot. I.e. measure 10mV in the 4VRange (reading 0.010V).

Emax = ±(0.06%rdg + 3dig) Emax = ±(0.03%f .Value + 0.01%f .Range)

Relative error of Tenma in range 1 (4V)

0

5

10

15

20

25

30

35

40

45

50

0.001 0.01 0.1 1 10

Reading in V

E in

%

Relative error of Elabo in 2V range

0

5

10

15

20

25

30

35

40

45

50

0.0001 0.001 0.01 0.1 1 10

Reading in V

E in

%

These two charts show how the error in one range changes with the readvalue. The behavior for the other ranges is the same!

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors

Methodical Errors

Current Measurement

Voltage Measurement

Error Propagation

Errors Using a MultimeterMethodical Errors

Methodical/ systematic errors are induced because of:• misuse of instruments

• influence of the instrument into the circuit

• influence of the setup into the circuit and into the multimeter input (straycapacitance, induced noise, thermoelectric voltage)

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors

Methodical Errors

Current Measurement

Voltage Measurement

Error Propagation

Errors Using a MultimeterMethodical Errors - Current Measurement

To measure current the the circuit needs tobe opened because the instrument has to beconnected in series with the load.

Itrue =10VRload

=10V100Ω

= 100mA100Ohm10V

I A

Instrument, contacts, and connecting wiresare an extra load shown as extra resistors!If too high in comparison to the real load thefunction of the circuit is changed and the ta-ken values are wrong.

Imeas =U

Rreal=

10V103Ω

= 97.1mA

100Ohm10V

Rmeter1Ohm

Rwire + R contact2Ohm

I

E% =Imeas − Itrue

Itrue∗ 100% =

97.1mA − 100mA100mA

∗ 100% = −2.9%

... and the load is -NOT- connected to the nominal voltage!

First Experiment


Using a Voltmeter

The Multimeter


Instrument Errors

Methodical Errors

Current Measurement

Voltage Measurement

Error Propagation

Errors Using a MultimeterMethodical Errors - Voltage Measurement

To measure voltage the instrument is con-nected in parallel to the load.

Vtrue =10VRtot

∗ R1M =10V2MΩ

∗ 1MΩ = 5V 1MOhm10V

1MOhm

VV+

Instrument, contacts, and connecting wiresare extra resistances! Wires and contactsare usually negligible here. If the voltmeterresistance is low compared to the rest of thecircuit the function is changed and the valuesare wrong.

1MOhm10V Rmeter10MOhm

Rcontact

Rwire

1MOhm

Vmeas =10V

R1M + R1M ||10MΩ∗ R1M ||10MΩ =

10V

1.91MΩ∗ 0.91MΩ = 4.76V

E% =Vmeas − Vtrue

Vtrue∗ 100% =

4.76V − 5V

5V∗ 100% = −4.71%

... and the circuit is -CHANGED- !

First Experiment


Using a Voltmeter

The Multimeter

Multimeter Errors

Error PropagationExample - Addition

Example - Multiplication

Error Propagation

Read the "Errorbooklet_PhysLab_F2011.pdf" from the course web site!

The following topics are mandatory to know!!!!• Error propagation

• Error propagation - Special cases

• Presentation of data analysis

• and of course terms like ’Accuracy’, ’Precision’ ...

Whenever measured values or values with a tolerance are used in calculationsthe error is propagated. In almost all cases the accuracy of the result is muchlower then the accuracy of the single values!

First Experiment


Using a Voltmeter

The Multimeter

Multimeter Errors



Error Propagation

When using measured values in a formula the error of the result will depend onthe individual errors of the values. To calculate the final error, the influence ofthe individual errors to the final error has to be determined by errorpropagation.

Given is a function x = f (a, b, c, ...). The maximal error ∆E is calculated

∆Emax =

∣∣∣∣∣(δfδa

)b,c

∗ ∆a

∣∣∣∣∣ +

∣∣∣∣∣(δfδb

)a,c

∗ ∆b

∣∣∣∣∣ +

∣∣∣∣∣(δfδc

)a,b

∗ ∆c

∣∣∣∣∣ + ...

Simple cases are• sums and difference.

For sums and difference the absolute error ∆E adds up.

• products and ratios.For products and ratios the relative error E% adds up.

First Experiment


Using a Voltmeter

The Multimeter

Multimeter Errors




Two resistors with tolerance in series :

R = R1 + R2R1 = 100Ω ± 5%R2 = 100Ω ± 10%

∆R =

∣∣∣∣∣(δRδR1

)R2

∗ ∆R1

∣∣∣∣∣ +

∣∣∣∣∣(δRδR2

)R1

∗ ∆R2

∣∣∣∣∣The solution of this equation is :

∆R = ∆R1 + ∆R2

So absolute errors add up

∆R = 100Ω ∗5

100+ 100Ω ∗

10100

= 5Ω + 10Ω = 15Ω

and the relative error becomes

E% =∆RR

∗ 100% =15Ω

200Ω∗ 100% = 7.5%

First Experiment


Using a Voltmeter

The Multimeter

Multimeter Errors



Error PropagationExample - Multiplication

Ohm’s law:

U = R ∗ I R = 100Ω ± 5%I = 1A ± 10%

∆U =

∣∣∣∣( δUδR)

I∗ ∆R

∣∣∣∣ +

∣∣∣∣( δUδI)

R∗ ∆I

∣∣∣∣The solution of this equation is :

∆U = I ∗ ∆R + R ∗ ∆I

If this equation is divided by R ∗ I = U we get the relative error

∆UU

=I ∗ ∆RR ∗ I

+R ∗ ∆IR ∗ I

=∆RR

+∆II

Here the relative errors add up E% = R% + I% = 5% + 10% = 15%

course 300111 natscilab unit electrical engineering i 0 ... · ammeter v a rsh load since the basic...

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