Date of Report: 09/01/2012Date of Experiment: 18/11/2011

Investigation into the limitations of electrical measurements

Experiment conducted by Sultan LeMarc.

Abstract

In this report limitations of devices used to make electrical measurements and their suitability for particular readings were investigated by experimentally determining the associated internal resistances. The discrepancies between stated and measured values of voltage and current were mathematically analysed to deduce the limitations of ammeters, voltmeters and both digital and analogue multimeters. By application of Ohm’s law, it was determined that the internal resistance of the analogue ammeter, the primary focus of the investigation was (0.504 ± 0.024)Ω. It was also determined that the V pp readings of the digital multimeter were the most accurate in representing the true value of an electrical quantity of a sinusoidal nature.

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Introduction

The aim of this experiment is to investigate the limitations of electrical measurement equipment and examine the fundamental principles of electricity and electric circuits. Electrical measurement equipment to be introduced in this investigation includes the analogue multimeter (AMM), the digital multimeter (DMM) and the A.C function generator. Their suitability for taking various readings of a range of electrical quantities will be analysed along with their associated electrical properties.

Electrical quantities to be measured in this investigation include voltage, current, resistance, internal resistance, electromotive force and resistivity of a material. This will be conducted through the use of various combinations of electrical circuits (series and parallel) and components such as variable resistors. In addition, statistical varieties such as Root Mean Square (RMS) and Peak-to-Peak (PP) will be introduced and compared in a quantitative analysis.

Knowledge of familiar quantities voltage and current will be extended to examine the internal resistance of an ammeter, resistivity of a material and the electro-motive force of a cell.

Electricity is a term used for the phenomena resulting from the presence and flow of electric charge. On a microscopic level, observation of electrical phenomena show that there are two kinds of electric charges; positive and negative. Electrons are the subatomic matter which are identified as having negative charge and are responsible for the transfer of charge and thus electric current.

Electric conductors are materials in which some of the electrons are free electrons that are not bound to atoms and can move relatively freely through the material. Electrical insulators are materials in which all electrons are bound to atoms and cannot move freely through the material.

Electric current is the rate of flow of charge that is in motion through a region of space (usually in a conductor). The amount of flow depends on both the material through which the charges are passing and the potential difference across the material. Whenever there is a net flow of charge through a region, an electric current is said to exist. (Serway, “Physics for Scientists and Engineers”, 2010)

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The average current, Iavg, is equal to the charge, Q, that passes through a point per unit time, t:

I avg=∆ Q∆ t

(1)

(Serway, “Physics for Scientists and Engineers”, 2010)

The SI unit of current is the ampere (A), where one ampere of current is equivalent to one coulomb of charge per second. It is conventional to assign to the current the same direction as the flow of positive charge. In electrical conductors, the current results from the motion of negatively charged electrons, therefore the direction of current is opposite to the direction of flow of electrons. A potential difference between the ends of a conductor creates an electric field which exerts forces on the conduction electrons causing them to move and create a current.

Voltage is the potential difference between two points in an electrical system such as an electric circuit. It is defined as the work done by the electric force when a unit charge moves from one point to the other. The voltage, V, can be defined by the relation between work done, W, and charge, Q:

V= ∆W∆ Q

=∆ WI ∆ t

(2)

(Serway, “Physics for Scientists and Engineers”, 2010)

The SI unit of voltage is the volt (V). One volt of potential difference is the equivalent to one joule per coulomb, as can be deduced from equation (2). An instrument that measures the difference of potential between two points is called a voltmeter.

Resistance is defined as the ability of a material to prevent or resist the flow of electric current. Resistance in electrical terms comes in the form of collisions between conduction electrons and atoms during a flow of charge, which slows down the electrons, and thus the flow of charge. It converts some of the electrical potential energy to heat energy, and in some cases, light energy. Some materials resist electrical current more than others and therefore have different resistance values.

The SI unit of resistance is the ohm (Ω), where one ohm is equivalent to one volt per ampere. Resistance can be measured by an instrument called an ohmmeter. Ohm’s law states that for many materials, the ratio of the current density to the electric field is a constant that is independent of the field producing the

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current. Materials that obey ohm’s law and demonstrate this are said to be ohmic. A relationship between current, voltage and resistance can be derived from Ohm’s law which defines the resistance as the ratio of the voltage across a conductor to the current in the conductor:

V=IR (3)(Serway, “Physics for Scientists and Engineers”, 2010)

Equation (3) defines resistance, R, for any conductor whether or not it obeys Ohm’s law, but only when R is constant this relationship is referred to as ohmic.

Figure 1: Current-voltage curves for an ohmic resistor (a) and a non-ohmic resistor (b). (Young and Freedman,

“University Physics” 13 th Edition, 2011)

As shown by Figure 1(a), ohmic materials and devices have a linear current-voltage relationship over a broad range of applied voltages. The two are directly proportional and the gradient of the I-V curve in the linear region yields a value for 1/R.

Non-ohmic materials and devices have a non-linear current-voltage relationship as shown by Figure 1(b). One such device is the semiconductor diode. The resistance of this device is low for currents in one direction (positive voltage) and high for currents in the reverse direction (negative voltage). For positive voltage, current increases exponentially with increasing voltage.

Most modern electrical devices, such as semiconductor diodes and transistors, have non-linear current-voltage relationships and their operation depends on the violation of Ohm’s law.

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2(b)

Most electric circuits use resistors to control the current in parts of the circuit. Every ohmic material has a characteristic resistivity that depends on the properties of the material and on temperature. The resistance of a sample depends on its dimensional geometry as well as on resistivity. Resistivity, ρ, is the measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. It is the reciprocal quantity of conductivity, the measure of a material’s ability to conduct an electric current. The SI unit of resistivity is the ohm meter (Ωm). Many resistors and conductors have a uniform cross-section with a uniform flow of electric current, made of one consistent material. Resistivity can be mathematically defined by its relationship with resistance, R, length of conductor, L, and cross-sectional area, A, of conductor:

ρ=RAL

(4)

(Serway, “Physics for Scientists and Engineers”, 2010)

The resistivity of a conductor nearly always increases with increasing temperature. As temperature increases, the ions of the conductor vibrate with greater amplitude making it more likely that a flowing electron will collide. This impedes the flow of electrons through the current and hence reduces the current. An exception of this is the resistivity of graphite which decreases with increasing temperature. Some materials, including metallic alloys and oxides, show a phenomenon called superconductivity, where resistance decreases to zero when below a critical temperature.

(a) (b) (c)

Figure 2: Resistivity-Temperature curves for (a) metal, (b) semiconductor and (c) superconductor. (Young and Freedman, “University Physics” 13 th

Edition, 2011)

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For most metals, resistivity is nearly proportional to temperature in a linear relationship as shown in Figure 2(a). However a non-linear region always exists at very low temperatures and the resistivity usually reaches a finite value as the temperature approaches absolute zero. This residual resistivity near absolute zero is caused primarily by the collision of electrons with impurities and imperfections in the metal. In contract, high-temperature resistivity (linear region) is predominantly characterised by collisions between electrons and metal atoms.

Figure 2(b) indicates that the resistivity decreases with increasing temperature. This behaviour is indicative of a class of materials called semiconductors, and is due to an increase in the density of charge carriers at higher temperatures. As the charge carriers in semiconductors often associate with impurity atoms, the resistivity of these materials is very sensitive to the type of concentration of such impurities.

There is a class of metals and compounds whose resistance decreases to zero when they are below a certain temperature T c known as the critical temperature. These materials are known as superconductors. Figure 2(c) follows that of a normal metal at temperatures above T c but below it the resistivity drops to zero.

The analysis of complex circuits is simplified using Kirchoff’s rules, which follow from the laws of conservation of energy and conservation of electric charge for isolated systems. Most of the circuits analysed are assumed to be in steady state, which means that currents in the circuit are constant in magnitude and direction. A current that is constant in direction is called a direct current (DC). A current which changes direction periodically is called an alternation current (AC).

Kirchoff’s rules provide to principles which can be used to analyse complex circuits. Kirchoff’s first rule regards the conservation of electric charge. All charges that enter a given point in a circuit must leave that point. This is known as the “Junction rule” which summarises as the sum of the currents at any junction must equal zero ∑I = 0. Kirchoff’s second rule is known as the “Loop rule” which states that the sum of the potential differences across all elements around any closed circuit loop must be zero ∑V = 0. (Giancoli, “Physics”, 2005)

When two or more resistors are connected together along a single patch, they are said to be connected in series. In a series combination the same amount of charge passes through all resistors in a given time interval and the currents are the same in all resistors. The voltage applied across the

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series combination of resistors divides between the resistors according to equation (3)

Figure 3: Resistors connected in series combination. (LeMarc, 2012)

The equivalent resistance of n resistors connected in series is given by the sum:

RT=R1+R2+R3+…+Rn ( 4 )(Young and Freedman, “University Physics” 13 th Edition, 2011)

When two or more resistors are connected parallel to each other are said to be connected in parallel. In a parallel combination the potential differences across the resistors are the same.

Figure 4: Resistors connected in parallel combination. (LeMarc, 2012)

In a parallel combination the current is divided so that the amount passing through each resistor is inversely proportional to the resistance value. In this configuration, the total resistance of n resistors is given by:

1R T

= 1R1

+ 1R2

+ 1R3

+…+ 1Rn

(5 )

(Young and Freedman, “University Physics” 13 th Edition, 2011)

The total resistance is always less than the smallest resistance in the configuration. Household circuits are always wired so that all appliances are connected in parallel. Each device operates independently of the others so that if one is switched off, the others remain on. In addition, this type of configuration allows all the devices to operate on the same voltage.

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A battery is generally used as a source of energy for circuits. A battery is a combination of electrochemical cells that convert stored chemical energy into electrical energy. As the potential difference at the battery terminals is constant in a particular circuit, the current in the circuit is constant in the magnitude and direction, and is called direct current.

A battery is also referred to as a source of electromotive force (emf). The emf, ε, is the maximum possible voltage the battery can provide between its terminals. When an electric potential difference exists between two points, the emf influences the current to flow from a lower to a higher potential. The SI unit of emf is the volt (V). Other examples of sources of emf include electric generators, solar cells, thermocouples and fuel cells. (Serway, “Physics for Scientists and Engineers”, 2010)

The positive terminal of a cell or battery is at a higher potential than the negative terminal. As a real cell is made of matter, there is resistance to the flow of charge within it. This resistance is called internal resistance, r.

Figure 5: Circuit diagram of a source of emf, ε, with internal resistance, r. (LeMarc, 2012)

For an idealised cell with zero internal resistance, the potential difference across it is called the terminal voltage and equals its emf. However, for a real cell the terminal voltage is not equal to the emf because as the current moves through the internal resistance it experiences an associated drop in potential equal to Ir. Thus the true terminal voltage of the source is given by:

∆ V =ε−Ir(6.1)(Yusaf, “First Year Physics Laboratory Manual, 2011)

From this expansion, it can be deduced that ε is equivalent to the open-circuit voltage i.e. the terminal voltage when current is zero.

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Figure 5 shows that the terminal voltage ∆V must equal the potential difference across the external resistor, R, often called the loan resistor.

Combining equation (2) and equation (6) gives the following expression:

ε=IR+ Ir (6.2)(Yusaf, “First Year Physics Laboratory Manual, 2011)

Solving for current gives:

I= εR+r

(6.3 )

(Yusaf, “First Year Physics Laboratory Manual, 2011)

Equation (6.2) shows that the current in the circuit depends on both the load resistance R external to the battery and the internal resistance r of the battery itself.An alternative form of this expression can be given in terms of the voltage:

V T=V R+V r (6.4 )(Stone, “Mathematics for Physicists”, 2009)

VT is the total voltage of the circuit, VR is the voltage across the load resistor R and Vr is the total voltage across the internal resistance of the source itself. In this case V r = Ir.The internal resistance of a cell can be determined experimentally where load resistance is varied and the voltage, current readings are taken, to which equation (6.4) can be applied. It can also be determined graphically.

Figure 6: Voltage-Current graph to determine ε. (LeMarc, 2012)

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Equation (6.1) can be rearranged in the form of a straight line equation y = mx + c, where y is the voltage, x is the current and the y-intercept is ε whilst gradient represents –r. (Yusaf, 2011)

An AC circuit consists of circuit elements and a power source that provides an alternating voltage. This time-varying voltage from the source is related by:

∆ V =∆ V max sinωt (7)(Stone, “Mathematics for Physicists, 2005)

∆Vmax is the maximum output voltage of the source, which is called the voltage amplitude. Possible sources of AC include generators and electrical oscillators. In domestic homes, each electrical outlet from the mains serves as an AC source.

As the output voltage of an AC source varies sinusoidally with time, the voltage is positive during one half of the cycle and negative during the other half. Likewise the current in any AC circuit also varies sinusoidally with time.

Figure 7: Sinusoidal function of AC on a Voltage-Period graph. (Young and Freedman, “University Physics” 13 th Edition, 2011)

The angular frequency of the AC voltage is given by:

ω=2 πf =2 πT

(8)

(Stone, “Mathematics for Physicists”, 2009)

The frequency of the source is f and T is the period. The source determines the frequency of the AC.A useful way to describe a quantity that can either be positive or negative such as the voltage of an AC circuit is through the Root Mean Square (RMS) value.

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The amplitude voltage Vmax is squared so all negative voltages become positive and the square root of the average values of V 2 is taken to be the root-mean square voltage, denotes as V rms:

V rms=V max

√ 2(9.1)

(Weber, “Mathematic Methods for Physicists”, 2004)

The Vrms corresponds to the DC voltage that would produce the same average power as the AC voltage of which the V rms value belongs to. RMS values are often more suitable to use when discussing alternating quantitative of a positive-negative variety. It provides an average without regard for positive or negative values, which proves especially useful for sinusoidal quantitates where the mean is zero.

An alternative way to describe an alternating quantity of the sinusoidal variety is through the Peak-to-Peak (PP) value. This is the value measured between the maximum positive and negative amplitudes (maxima) of the alternating quantity. In terms of AC circuits, the peak-to-peak voltage, V PP

is twice the amplitude of voltage Vmax:V PP=2 V max(9.2)

(Weber, “Mathematic Methods for Physicists”, 2004)

The root mean square value can be converted to peak-to-peak value by:

V PP=2 ×√ 2× V rms(9.3)(Weber, “Mathematic Methods for Physicists”, 2004)

Quantities such as potential difference, current and resistance are measured in common devices using a d’Arsonval galvanometer. A pivoted coil of fine wire is placed in the magnetic field of a permanent magnet. A spring is attached to the coil which points to zero in the equilibrium position when there is no current. When a current exists, the magnetic field exerts a torque on the coil that is proportional to current, thus allowing the magnitude of the current to be measured. (Serway, “Physics for Scientists and Engineers”, 2010)

A current measuring instrument is usually called an ammeter. An ammeter always measured the current passing through it. An ideal ammeter would have zero resistance, so including it in a branch of a circuit would not affect the current. In other words, it is a short circuit. However, no real ammeter can ever be ideal and there all ammeters have a relatively small internal resistance.

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A voltage measuring instrument is called a voltmeter. A voltmeter always measures the potential difference between two points and its terminals must be connected to these points. An ideal voltmeter would have infinite resistance, so connecting it between points in a circuit would not alter any of the currents. Real voltmeters always have a finite resistance but it is large enough that it does not change the currents appreciably.

A voltmeter and an ammeter can be used together to determine resistance of a circuit with the use of equation (3). Al alternative method for measuring resistance is by using a device called an ohmmeter.The measurement of the three quantities, voltage, current and resistance, can be done by a single device called a multimeter. A multimeter is a versatile instrument for measurement of different electrical quantities. It is equipped with the circuitry that allows it to be used as a voltmeter, an ammeter and an ohmmeter.

An analogue multimeter (AMM) uses the d’Arsonval meter to display readings using a pointer on a scale. When used as a voltmeter, it can be used to visually follow an oscillating or alternating quantity on the analogue display. When used as an ammeter, an analogue multimeter has a very low resistance. (Britannica, 2012)

In situations where high precision is required, instruments containing d’Arsonval meters have been supplanted by electronic instruments with direct digital readouts. A digital multimeter (DMM) is a device which converts the signal under test into a voltage which is then amplified and displays the measured quantity as a number on a display. For this reason, unlike the AMM, the DMM eliminates the parallax errors. It is better than an AMM when used as a voltmeter because usually its resistance is much higher.

Analogue multimeter are able to display an alternating reading more effectively from a visual perspective than a digital meter, which presents such data in a manner that can be difficult to follow especially at high frequencies.However, analogue meter movements are inherently more fragile physically and electrically than digital meters. They are also susceptible to damage if pointer is pushed passed its maximum level.

The way in which the meters are connected in a circuit affects the measurement of the circuit through and the voltage across the test resistance.

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Figure 8: Circuit diagrams for ammeter-voltmeter configurations (Young and Freedman, “University Physics” 13 th Edition, 2011)

The circuit arrangement in Figure 8(a) is the configuration of a voltage divider, with the voltmeter connected in parallel over both the resistor and the ammeter. The ammeter correctly reads the current flowing through the resistor but the resistor and ammeter in series act as a voltage divider i.e. there is a voltage drop across the ammeter due to its internal resistance. For this reason, voltmeters are always to be in a parallel configuration to give the required voltage reading.

The circuit arrangement in Figure 8(b) is the configuration of a current divider, with the voltmeter connected in parallel with the resistor only which, together, act as a current divider. The voltage reading is that of the voltage across the resistor only and not the ammeter. A generator is a device which converts mechanics energy into electrical energy. An A.C generator works on the principle of electromagnetic induction where an induced emf is produced by the rotation of a coil in a magnetic field. The flux linking the coil changes continuously hence a continuously fluctuating emf is created. An A.C generator can be used as a source of AC in experiments. (Britannica, 2012)

Although the main application for ammeters is the direct measurement of current, zero-centre ammeters are used for applications requiring current to be measured with both polarities, common in scientific and industrial equipment. Zero-centre ammeters are also commonly placed in series with a battery. In this application, the charging of the battery deflects the needle to one side of the scale (commonly, the right side) and the discharging of the battery deflects the needle to the other side. A special type of zero-centre ammeter for testing high currents in cars and trucks has a pivoted bar magnet that moves the pointer, and a fixed bar magnet to keep the pointer cantered with no current. The magnetic field around the wire carrying current to be measured deflects the moving magnet.

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While resistance usually increases with temperature, the reverse effect is observed with the internal resistance of a cell. The internal resistance of a galvanic cell is temperature dependent, decreasing as the temperature rises. The internal resistance of most cell chemistries also tends to increase significantly towards the end of the discharge cycle. For this reason, batteries are stored in cool temperatures as an effective way to preserve the energy.

Resistors are used as poor conductors to limit and control the flow of electricity through a circuit with a predetermined voltage to current ratio. Resistors are generally made of substances which only partially conduct electricity such as carbon, special alloys and some metal oxides. An example of where the property temperature dependence of resistance is utilised is in a resistance thermometer, which act as sensors to measure temperature by correlating resistance.

The primary focus of modern work done on low temperature resistances is the field of superconductivity, a phenomenon of zero electrical resistance and expulsion of magnetic fields occurring in certain materials below a characteristic temperature. In most cases, resistance is undesirable as it results to a loss of energy and reduces the efficiency of a device. Superconductors require very cold temperatures, often around -250 oC for conventional superconductors, such that materials which experience a superconductive state above -243.2oC are called ‘high’-temperature superconductors.

Applications of this low temperature resistance phenomenon include powerful electromagnets that are used in magnetically levitating transport systems, magnetic resonance imaging and nuclear magnetic resonance machines. Furthermore, they’re used in the production of sensitive magnetometers and high sensitivity particle detectors, as well as highly efficient electric generators. (superconductors.org)

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Figure 9: Standard symbols for circuit components used in circuit diagrams. (Young and Freedman, “University Physics” 13 th Edition, 2011)

Method

This experiment had four tasks. For the first task, the internal resistance of an ammeter was investigated. The apparatus required for this experiment included an AMM, of which the internal resistance was investigated, a power supply with a rectifier to provide a constant DC voltage, a variable resistor and a DMM to measure the relevant resistance and voltage directly.

Figure 9: Circuit diagram of experiment task 1 – investigating internal resistance of ammeter. (LeMarc, 2012)

The ammeter (AMM) was connected in series to the circuit with a variable resistor. The resistor was used to investigate voltages for various resistances. A DMM voltmeter is parallel to the resistor to measure the voltage across it, VR in accordance with equation (6.4). The DMM was used to verify the resistance value across the resistor and the voltage output of the supply to ensure that they match the stated values.

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Caution: the AMM meter was not be allowed to have its pointer pushed passed its maximum level as this could have damaged it. Awareness of the hazards to safety when dealing with electrical equipment was expected and the experiment was conducted with this consideration.

The independent variable in this task is the resistance of the load resistor. Resistances that were investigated were 2004, 3013, 4007, 5014, 6001 (Ω), giving five trials. The current was measured on a scale of mA by the AMM ammeter. The control variable of this task was the supply voltage which was kept constant at 9.013V for all five trials. This was verified by the DMM voltmeter for each individual trial.

Having established values for the total voltage, VT, voltage across the resistor, VR, equation (6.4) was used to find the voltage of the internal resistance of the ammeter, V r. The principles of Ohm’s law and equation (3) was used with the reading of the current on the ammeter, I, to find the corresponding value of its internal resistance, r. An average of the five values was taken.

Next, an extension to this investigation of internal resistance was explored, as well as the concept of the electromotive force. This time, the internal resistance of a cell as well as its emf was investigated. The same circuit configuration as Figure 9 was used but with a cell in place of a power supply.

Figure 10: Circuit diagram of experiment task 2 – investigating internal resistance and emf of a cell. (LeMarc, 2012)

The independent variable in this task was the resistance of the load resistor again. Resistances that were investigated ranged between 0-100Ω in intervals of 10Ω for the ten trials. A DMM was used as an ohmmeter to verify the resistance values of the load resistor. The current was measured by the same AMM ammeter of which the internal resistance value had been

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established in the previous task. The control variable was the voltage supply i.e. the cell, which was the same for each of the ten trials and measured at 5.500V on the DMM.

The two quantities of interest are the voltage across the resistance of the circuit, VR, and the current, I, through it, in order to apply equation (6.2) to establish the emf, ε, and the internal resistance, r, of the cell.

The findings of the previous task were incorporated into this task with the knowledge of the AMM ammeter being a source of resistance. Therefore, the total resistance of the circuit was considered to be of both the resistor and the ammeter’s internal resistance. This extended equation (6.2) to ε=I (R+r A)+ I r C where rA is the internal resistance of the ammeter and rc is that of the cell. The importance of incorporating this was reflected by the ammeter’s resistance as a relative proportion of the total circuit resistance, which was as high as 5.5% for the lower load resistance values.The data was plotted on a graph of voltage against current where the internal resistance and emf of the cell were obtained graphically, as shown by Figure 6.

In addition, the resistivity of conducting putty was investigated where conducting putty was used as a resistor in a circuit. The aim of this particular investigation was to measure the resistances of a geometrically altered resistor in accordance with the equation for resistivity, equation (4).Metal plates were embedded at both ends of the shaped putty and a ruler was used to measure the putty’s geometric dimensions. The conducting putty acts as a variable resistor, as its resistance can be altered by changing its geometry.

Figure 11: Diagram of the circuit used for the investigation of resistivity of conducting putty, symbolised as a resistor. (LeMarc, 2012)

The independent of variables in this investigation were the geometric dimensions of the putty, length and cross-sectional area. There were a total

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of five trials, each with its own length and cross-sectional area. The current in the circuit was measured by the same AMM ammeter used in previous tasks. The voltage was supplied by a battery pack, and as the control variable, the voltage output was verified for each trial by the DMM to be 0.2500V. The volume of the putty was also a control variable.

The conducting putty was moulded into a uniform cylindrical shape for each of the five trails but with different dimensions. The principles of Ohm’s law and equation (3) were used to find the corresponding resistance using the measured voltage and current readings. The internal resistance of the AMM ammeter in series with the circuit was incorporated into the data by subtracting the value established from task 1 from the resistance values calculated. This gave the resistance of the putty alone.

The formula for resistivity in equation (4) was then used to find the resistivity values for each of the five trials and an average was taken.

The last part of the investigation involved the experimentation with an A.C function generator in order to compare and contrast the readings of an AMM and a DMM to establish the suitability for taking readings. This was done by using the generator to provide voltages at different frequencies and converting the measurements of the voltages from the meters to find the actual PP voltage.

The DMM utilizes a rectifier circuit inside it to measure the average or peak absolute value of the voltage but is calibrated to show the RMS value for a sinusoidal waveform. For the AMM, both ends of the fluctuating readings were recorded and used to find the average voltage amplitude.

Figure 11: Circuit diagram of an A.C generator used to compare suitability of meters in reading true voltage. (LeMarc, 2012)

The meters were connected directly to the A.C function generator to give a reading of the voltage.

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The independent variables were the meters and the frequency of the generator. There were two trials for each meter, one at which the voltage frequency is 1Hz and the other at 1000Hz.

An average was taken for readings that fluctuated between a maximum and minimum. The readings were converted to the RMS and PP voltages.

Results

Table 1 shows the measurements taken for the investigation of an AMM ammeter’s internal resistance when a variable resistor is used to observe voltage across it and the ammeter.

Total Voltage VT

±5×10-4 (V)

Resistance RT

±5×10-1

(Ω)

Total Current

IT ±5×10-

2 (mA)

Resistor Voltage

VR

±5×10-4

(V)

Ammeter Voltage Vr

±5×10-5

(V)

Internal Resistance

r

±7×10-4

(V)

Uncertainty in r

∆r (V)9.013 2004 4.6 8.925 0.0023 0.5000 0.01229.013 3013 3.1 8.942 0.0016 0.5161 0.01829.013 4007 2.3 8.981 0.0011 0.4783 0.02419.013 5014 1.9 9.008 0.0010 0.5263 0.02979.013 6002 1.6 9.011 0.0008 0.5000 0.0349

xW = 0.5041 ∆xW = 0.0240Table 1: A table showing the measurements and calculations done for the

investigation of internal resistance of an AMM ammeter.

The quantity with the greatest relative error in the table is the internal resistance, r. The voltage across the resistor and ammeter was noted, as well as the current of the circuit. These three quantities were used in equation (6.4) to calculate the internal resistance, r. The uncertainty in the calculated values of r was found using error prorogation equation (Yusaf, 2011):

∆ r=|V r

I |√(∆ V r

V r)

2

+(∆ II )

2

(10)

This incorporates the instrumental errors of the two quantities. The errors increase at higher load resistances. The results show that the AMM ammeter has an average internal resistance of (0.504 ± 0.024)Ω.

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Table.2 shows the measurements taken for the investigation of a cell’s internal resistance and emf over ten trials of varying load resistances.

Table 2: A table showing the results of the investigation of internal resistance and emf of a cell.

The table shows the total cell voltage to be 5.500V, which was the value verified by a DMM of a 6V cell. The resistances is set by the variable resistor were also measured by a DMM. The total current of the of the circuit was measured by the AMM ammeter of the previous investigation which Table 1 shows to have its own internal resistance of (0.504 ± 0. 0240)Ω.

The results follow Ohm’s law with current decreasing as the load resistance increased. It must be noted that due to the ammeters internal resistance, r, the total current reading given by it is not representative of the true value. The internal resistance of the ammeter has been considered by adding it to the load resistance value to give a total resistance, R T, which is a more precise measure of the circuit’s resistance. All errors are instrumental errors so methods of error propagation were not used.

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Total Cell Voltage VT

±5×10-4

(V)

Load Resistance R

±5×10-4

(Ω)

Total Resistance RT

(R+rA)

±2.45×10-2

(Ω)

Resistor Voltage VR

±5×10-4

(V)

Total Current IT

±5×10-4

(A)5.500 9.649 10.153 5.736 0.5705.500 18.96 19.470 5.760 0.2905.500 27.50 28.004 5.820 0.2005.500 36.66 37.171 5.833 0.1505.500 48.67 49.177 5.832 0.1135.500 57.89 58.399 5.843 0.0955.500 67.07 67.577 5.848 0.0825.500 76.38 76.893 5.849 0.0725.500 85.93 86.442 5.846 0.0645.500 93.22 93.724 5.847 0.059

By rearranging equation (6.1), the internal resistance and emf of the cell was determined graphically, as shown in Figure 12.

Figure 12: A graph of voltage, VR, against current, IT, for determining the internal resistance (gradient) and emf (intercept) of a cell. Error bars were

too small to be represented on the graph for both voltage error and current error as they’re both on the 10 -4 scale.

The graph shows the ohmic relationship in accordance with equation (3), as the voltage across the resistor decreased i.e. resistance decreased, the current of the circuit increased.

The line of best fit for the ten data points was determined using the method of least squares fit. The equations of the least squares fit require the values of Sx, Sy, Sx

2 and Sxy of the data to be calculating using the following:

Sx=∑i=1

10

x10 , S y=∑i=1

10

y10 , S x2=∑i=1

10

x210 , Sxyx=∑

i=1

10

x10 y10(11)

(Yusaf, 2011)

For the data in Table.2, it was found that Sx= 1.695, Sy= 58.215, Sxy= 9.812, Sx

2= 0.513.

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The formulae for the gradient, m, and the intercept, c, using the vertical least squares method are:

m=n S xy−Sx Sy

nSx2−( Sx )2

=−0.241 (12.1 )

c=n Sy Sx2−Sx Sxy

n Sx2−(Sx )2=5.862(12.2)

(Yusaf, 2011)

With this line of best fit using the method of least squares fit, the internal resistance and emf can be deduced as it is a rearranged representation of equation (6.1) where the gradient, m, is equivalent to –r and the intercept, c, is equivalent to ε. Therefore, the internal resistance of the cell is 0.241Ω and the emf is 5.862V.

The errors of these two quantities can be calculated by finding the errors of the gradient and intercept. This is done using the following equations:

∆ m :2σm

√ N, where σ m=√ n SD 2

(n−2)¿¿¿

∆ c=2σ c

√N,where σc=σ m √ Sx2

(n)(13.2)

(Yusaf, 2011)

For the data in Table.2, it was found that SD2 = 0.1206. With this value it

was found that σm= 0.259 to 3 decimal places, giving an error of 0.082 at 65% confidence level. σc= 0.059 to 3 decimal places, giving an error of 0.019 at 65^ confidence level.

As a final result of this investigation, it is deduced that the cell’s emf is ε = (5.862 ± 0.082)V and its internal resistance is r= (0.241 ± 0.059)Ω.

Table.3 shows the measurements taken for the investigation of the conducting putty’s resistivity over five trials of varying its geometric dimensions to examine relationship of equation (4).

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Table3: A table showing the measurements for investigation of resistivity of a conducting putty of different geometries.

The table shows that the voltage across the putty increases as cross-sectional area decreases. The putty resistance, RP, is calculated using equation (3) in accordance with Ohm’s law. The resistance increases as the length increases, following the hypothesis in the introduction. The uncertainty in RP is calculated using the following error propagation equation:

∆ R=|VI |√( ∆V

V )2

+(∆ II )

2

(14)

(Yusaf, 2011)

The relative error in RP increases at higher voltages. The putty’s resistivity is calculated using equation (4) in accordance with the fundamental principles of resistivity with respect to a material’s geometry. The results show that varying geometries of the conductor change its resistance but preserve the material’s fundamental electric property of resistivity which remains constant. This is shown by the results of ρ not deviating more than 7% from the average.

The uncertainty of ρ is calculated by incorporating the instrumental errors of L and the derived errors of A and R in the following way:

∆ ρ=|RAL |√(∆ RA

RA )2

+( ∆ LL )

2

(15.1 )

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Total Voltage

VT

±5×10-5

(V)

Length L

±5×10-1

(mm)

Area A

(mm2)

Error in Area∆A

(mm2)

Total Current

IT

±5×10-2

(mA)

Putty Voltage

VP

±5×10-5

(V)

Putty Resistance RP (Ω)

Error in Putty

Resistance

∆RP (Ω)

Putty Resistivity

Ρ(Ωm)

Error in Putty

Resistivity ∆Ρ

(Ωm)0.2500 47 415.5 18.1 8.7 0.0160 1.84 0.01 0.0163 0.000030.2500 63 346.4 16.5 8.6 0.0193 2.24 0.01 0.0123 0.000040.2500 82 314.2 15.7 8.2 0.0273 3.33 0.02 0.0128 0.000050.2500 138 201.1 12.6 6.8 0.0627 9.22 0.07 0.0134 0.000110.2500 169 176.7 11.8 6.0 0.0848 14.13 0.12 0.0148 0.00016

xW = 0.0139 ∆xW = 0.0062

where ∆ RA=|RA|√( ∆ RR )

2

+(∆ AA )

2

(15.2 )

a s∆ A=|π|∆ r a2 (15.3)

(Yusaf, 2011)

Therefore, with the methods of error propagation, the results show the conducting putty’s resistivity is ρ = (0.0139 ± 0.0062)Ωm.

AC Voltage Output (V)

AC Frequency f (Hz)

AMM VRMS

±3.5×10-2 (V)AMM VPP

±1×10-2 (V)DMM VRMS

±4×10-4 (V)DMM VPP

±1×10-3 (V)

3 1 0.57 1.60 0.644 1.822

3 1000 0.57 1.60 0.899 2.543

The following table shows the measurements taken for the investigation of the suitability of meters for taking particular readings from an A.C generator as a test of how true the readings are.

Table 4: Results of the voltage measurements of A.C generator at varying frequencies.

The table shows four different voltage readings, from two kinds of meters, the AMM and the DMM.At the lower frequency of 1Hz, the AMM pointer is fluctuating significantly between a maximum and minimum voltage reading. The AMM V RMS is calculated using the reading of Vmax and Vmin to find the voltage amplitude of 0.8V. This is then used in the equation (9.1) to find the corresponding VRMS value. As the table shows, the 0.57V value for AMM VRMS is only 19% of the “true” voltage value output of the A.C generator.The AMM’s VRMS reading at 1000Hz proved to be easier to read as the fluctuations were minimal allowing for a single reading to be taken. The VRMS values agree to 0.57V for both frequencies and thus have the same V PP

values of 1.60V which are calculated using equation (9.3) to convert V RMS to VPP.

The DMM VRMS values were established in the same manner but give values much closer to the “true” value of 3V. The V RMS readings of 0.644V and 0.899V are 21% and 29.9% respectively of the true value. Unlike the AMM, the DMM VRMS values are difference for each frequency, with the reading for 1000Hz being more consistent and closer to the voltage output. The DMM VPP values are calculated in the same way as discussed above. The VPP values of 1.822V and 2.543V are the closest to the true value out of all eight values, with 2.543V being within 85% of the true reading.

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The errors of VRMS and VPP values are established using the following error propagation methods:

∆ V RMS=| 1

√2|∆ V max , ∆ V PP=|2√2|∆ V RMS(16)

(Yusaf, 2011)

It must be noted that these are optimistic errors. As the results show, the relative errors of AMM readings are higher than those of DMM readings. The readings of AMM also had parallax error, although this cannot be quantified.The results show that the meter and reading that is most consistent with the true voltage output is the DMM’s VPP reading of VPP= (2.543 ± 1x10-3)V.

Analysis

The biggest source of error in electrical measurement equipment is their internal resistance which affect the total load resistance of the circuit and thus the reading itself. In AMM, the additional parallax error exists in the reading of the measurement. As analogue displays have a pointer to display the reading on a graduated scale, they can be difficult to read especially at the smaller scale divisions. The parallax error does not exist in DMM, which gives a definitive digital reading.

The investigation supported the hypothesis that real meters are not ideal as they have an ammeter has its own internal resistance and a voltmeter does not have infinite resistance. Therefore they do not give an exact reading of an electrical quantity in a circuit. It also brought to light that equipment do not necessarily give the exact output they state to be giving, which was the case with the cells, batteries, power source, variable resistor and A.C generator. This was reflected by the discrepancies in the output reading on the DMM when stated values were being verified.

The method for determining the internal resistance of the AMM ammeter was more reliable in its result than the method for determining the same of the cell through graphical methods.

The internal resistance of the AMM ammeter is (0.504 ± 0.0240)Ω, giving a relative error of 4.8%. In contract, the internal resistance of the cell is 0.241 ± 0.059Ω, giving a relative error of 24% which is significantly larger than the former method. This suggests that determining the internal

25

resistance through the means of a direct calculated is more successful than using a graphical method.

A possible explanation for the large error in the graphical method is the method for the line of best fit. An improvement of this is to apply the weighted least squares fit method. However, the emf value which was obtained by the same graphical method has a much lower error as ε = (5.862 ± 0.082)V, giving a relative error of just 1.4%. This suggests that the large error in the value of internal resistance exists because it is a relatively small quantity which is more difficult to measure graphically.

One particular error that is to be highlighted is that of the DMM voltmeter measuring the cell to give a (5.500 ± 5 × 10 -4)V reading despite the readings across the load resistor in the 5.7-5.8V range, showing an inaccuracy in the reading.

The results of the investigation of the resistivity of the conducting putty gave a final calculated value for resistivity as ρ = (0.0139 ± 0.0062)Ωm. A significant source of error in this experiment is the measurement of the geometric dimensions i.e. length and diameter, as they are influenced by both the instrumental error of the ruler used to measure them and the parallax error in taking the reading. More importantly, the inconsistent shape of the putty meant that it did not have a constant cross-sectional area and thus does not simulate the model of equation (4). These errors contributed to two of the fundamental quantities in the equation, length, L, and the area, A.

Furthermore, due to the high order of the propagated error, as outlined by equation (15.1), contributed by the errors of area, resistance and length, the error in the resistivity is the highest relative error of any result in the entire investigation. The relative error of 45% means that this particular experiment is deemed unsuccessful.

However, as the conducting putty id predominantly calcium carbonate (whiting), approximate value of its resistivity is given to be approximately 0.10Ωm (www.iBotz.com), the experiment has been able to provide a relatively close value for its resistivity considering values can range from 10-8Ωm to 1015Ωm for materials.

An error analysis of meter readings of voltage values VRMS and VPP of an A.C generator is shown in Table 5 below.

AMM VRMS (V) AMM VPP (V) DMM VRMS (V) DMM VPP (V)

26

Instrument Error (V) ±0.05 ±0.05 ±0.0005 ±0.0005Value (V) 0.57 1.60 0.899 2.543

Calculation Error (V) ±0.035 ±0.01 ±0.00035 ±0.001

% error from value 6.14% 0.63% 0.04% 0.04%

% Error from Nominal Value:

3V 81.0% 47.7% 70.0% 15.2%

Table 5: Comparing VRMS and VPP values of AMM and DMM in the reading of a 3V output from an A.C generator at 1000Hz.

The error analysis in Table.5 compares the readings of the two meters. The readings for 1000Hz AC voltage are used for the comparison as the readings were not fluctuating as they were for at 1Hz, and thus proved easier to read and measure. For this reason, the results at this frequency are believed to be a better representation of the data. They are also the readings that happen to be closest to the nominal value.

The results show that the VPP readings of both the AMM and DMM were a more accurate measure of the true voltage output as they have smaller % errors than their VRMS counterparts. The % error in AMM’s VRMS from the nominal value is 81.0%, whereas AMM’s VPP has a 47% error from the true value of 3V. Likewise, the DMM’s VRMS value has a 70% error in contrast to the DMM’s VPP 15.2% error.

The results suggest that the VPP measure is the best suited reading of an AC voltage as it considers the sinusoidal variant in a direct manner. They also show that the DMM is the best suited meter as it has the smaller instrumental value, smaller calculation error and it gave the closest reading to the true voltage output.

Improvements to this entire investigation can be made by having stated values of the meter’s internal resistances, as well as the putty resistivity, cell emf and resistance. This will allow the experimental values to be compared with the true values of the equipment used.

Additional improvements can come in the form of data collection techniques, such as taking repeat measurements and more trials for a better average. Also the putty can be moulded into a more uniform shape with a more consistent cross-sectional area. A digital vernier calliper can be used to measure the dimensions of the putty to eliminate the parallax error and improve precision in the measurements of its dimensions. The graphical

27

method for finding emf and internal resistance of a cell can use weighted least squares fit method for the line of best fit.

Conclusion

In conclusion, the entire investigation has successfully supported the background physics outlined in the Introduction. It has confirmed that a significant source of error in measuring electrical readings is the associated internal resistance of meters. In this case, the experiment showed that in practical terms a real ammeter is not ideal and therefore has internal resistance which causes a voltage drop when current flows through it. This voltage burden subtracts directly from the source voltage and thus reduces the measured current, providing an inaccurate reading that is not representative of the true reading. The lower the source voltage, the greater the relative error caused by this internal resistance. Likewise, a real voltmeter is not ideal and does not have infinite resistance, allowing current to flow through it and similarly contributing to a reading that is not an exact measure of the true voltage.

In addition, the aim of comparing the limitations of electrical measurement equipment has been met. The investigation shows and compares the precision of analogue and digital meters. The investigation’s results also show that the DMM is the better suited type of meter to take readings. The advantages of it include the smaller instrumental error, elimination of parallax error, greater accuracy at smaller scales and are calibrated to consider its own internal resistance in the reading it gives. Its reading of the AC voltage by VPP proved to be closest to the true value. However, it must be noted that the AMM is easier to visually follow when fluctuating at a higher frequency.

Further extensions to this investigation include having a circuit configuration with a capacitive divider with the AC generator. Also, an oscilloscope can be used to measure the AC voltage to verify its output. An additional circuit configuration would be to have a thermocouple bonded with resistive heat element in series with the AC generator. Moreover, variables which affect resistivity such as temperature can be considered, as well as experimenting with conductors of different materials.

To summarise, the investigation has been successful in meeting its aims to investigate the limitations of electrical measurement equipment both experimentally and quantitatively. The results of the experiments followed the fundamental physics of the concepts and agreed with the hypothesis

28

formed from the relationships described. The inclusion of error propagation allowed the values to be compared and analysed to verify their validity. Finally, it has brought to light how choice of measurement equipment is a significant factor in the integrity of recorded data in experiments.

BibliographyYusaf, F. (2011). "First Year Laboratory Physics".

London: King's College London.

Young, H. (2011). "University Phyysics with Modern Physics". Pearson Education Inc.

Serway, R. (2010). "Physics for Scientists and Engineers – 8th Edition". Cengage Learning.

Stone, M. (2009). "Mathematics for Physics". Cambridge University Press

Weber, H. (2004). “Mathematic Methods for Physicists”Academic Press

Giancoli, D. (2005). "Physics". Pearson Education Inc.

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