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**EKT112 Principles of Measurement and Instrumentation Weeks 2-3**

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**Current, Voltage & Resistance Measurement**

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Topics Outline 1.0 Device for Current Measurement 1.1 Analog ammeter 1.2 Galvanometer 2.0 Device for Voltage Measurement 2.1 Analog voltmeter 2.2 Oscilloscope 2.3 Potentiometer 3.0 Device for Resistance Measurement 3.1 Ohmmeter 3.2 Wheatstone bridge circuit 4.0 Digital Multimeter

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2.0 Voltage Measurement

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2.1 Voltmeter A voltmeter is an instrument used for measuring the potential difference between two points in an electric circuit.

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A voltmeter is placed in parallel with a circuit element to measure the voltage drop across it and must be designed to draw very little current from the circuit so that it does not appreciably change the circuit it is measuring. To accomplish this, a large resistor is placed in series with the galvanometer. Its value is chosen so that the design voltage placed across the meter will cause the meter to deflect to its full-scale reading. A galvanometer full-scale current is very small: on the order of milliamperes.

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**Voltmeter – Principle of Operation**

The moving coil galvanometer is one example of this type of voltmeter. It employs a small coil of fine wire suspended in a strong magnetic field. When an electrical current is applied, the galvanometer's indicator rotates and compresses a small spring. The angular rotation is proportional to the current that is flowing through the coil. For use as a voltmeter, a series resistance is added so that the angular rotation becomes proportional to the applied voltage.

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**D’Ársonval Meter Movement Used In A DC Voltmeter**

The basic d’Ársonval meter movement can be converted to a dc voltmeter by connecting a multiplier Rs in series with the meter movement The purpose of the multiplier: is to extend the voltage range of the meter to limit current through the d’Arsonval meter movement to a maximum full-scale deflection current. Fig 2-1 The basic d’Arsonval meter Movement Used In A DC Voltmeter

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Cont. To find the value of the multiplier resistor, first determine the sensitivity, S, of the meter movement.

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Example 1-4 Calculate the value of the multiplier resistance on the 50V range of a dc voltmeter that used a 500A meter movement with an internal resistance of 1k.

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**Solution: Sensitivity,**

Multiplier, Rs = S X Range – internal Resistance = (2k X 50) – 1k = 99k

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**Voltmeter Loading Effects**

When a voltmeter is used to measure the voltage across a circuit component, the voltmeter circuit itself is in parallel with the circuit component. Since the parallel combination of two resistors is less than either resistor alone, the resistance seen by the source is less with the voltmeter connected than without. Therefore, the voltage across the component is less whenever the voltmeter is connected. The decrease in voltage may be negligible or it may be appreciable, depending on the sensitivity of the voltmeter being used. This effect is called voltmeter loading. The resulting error is called a loading error.

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Example 1-5 Two different voltmeters are used to measure the voltage across resistor RB in the circuit of Figure 2-2. The meters are as follows. Meter A : S = 1k/V, Rm = 0.2k, range = 10V Meter B : S = 20k/V, Rm = 1.5k, range=10V Calculate: Voltage across RB without any meter connected across it. (b) Voltage across RB when meter A is used. (c) Voltage across RB when meter B is used (d) Error in voltmeter readings. Fig. 2.2

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Solution: (a) The voltage across resistor RB without either meter connected is found Using the voltage divider equation:

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**Cont. (b) starting with meter A, the total resistance it**

presents to the circuit is The parallel combination of RB and meter A is Therefore, the voltage reading obtained with meter A, determined by the voltage divider equation, is

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**Cont. (c) The total resistance that meter B presents to the circuit is**

RTB = S x Range = 20k/V x 10 V = 200 k The parallel combination of RB and meter B is Re2 = (RB x RTB)/(RB + RTB) = (5kx200k)/(5k+200k) = 4.88 k Therefore, the voltage reading obtained with meter B, determined by use of the voltage divider equation, is VRB = E(Re2)/(Re2+RA) = 30 V x (4.88k)/(4.88k+25k) = 4.9 V

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**Cont. (d) Voltmeter A error = (5 V – 3.53 V)/5 V x (100% = 29.4%**

Voltmeter B error = (5 V – 4.9 V)/5 V x (100%) = 2 %

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**Five principal meter movements used in ac instrument**

1. Electrodynamometer 2. Iron Vane 3. Electrostatic 4. Thermocouple 5. D’Arsonval with rectifier

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**Application of meter movements:**

DC Use AC Use Applications Electrodynamometer YES Standards meter, wattmeter, frequency meter “Indicator” applications such as in automobiles Iron Vane Electrostatic Measurement of high voltage when very little current can be supplied by the circuit being measured Thermocouple Measurement of radio frequency ac signal D’Arsonval YES with rectifier Most widely used meter movement for measuring direct current or voltage and resistance

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PMMC Instrument on AC The PMMC instrument is polarized (terminals +ve & -ve) - it must be connected correctly for positive (on scale) deflection to occur. When an AC with a very low frequency is passed through a PMMC, the pointer tends to follow the instantaneous level of the AC As the current grows positively, the pointer deflection increases to a maximum at the peak of the AC As the instantaneous current level falls, the pointer deflection decreases toward zero. When the AC goes negative, the pointer deflected (off scale) to the left of zero This kind of pointer movement can occur only with AC having a frequency of perhaps 0.1Hz or lower

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PMMC Instrument on AC At 50Hz or higher supply frequencies - the damping mechanism of the instrument and the inertia of the meter movement prevent the pointer from following the changing instantaneous levels. The average value of purely sinusoidal AC is zero. Therefore, a PMMC instrument connected directly to measure 50Hz AC indicates zero average value. It is important to note that although a PMMC instrument connected to an ac supply may indicating zero, there can actually be very large rms current flowing in its coils

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**Two types of PMMC meter used in AC measurement :**

1. Half wave rectification 2. Full wave rectification

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**D’Arsonval meter movement used with half wave rectification**

To convert alternating current (AC) to unidirectional current flow, which produces positive deflection when passed through a PMMC, the diode rectifier is used. Several types of rectifiers are selected such as a copper oxide rectifier, a vacuum diode, or semiconductor or “crystal diode”.

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Cont… For example, if the output voltage from a half wave rectifier is 10Vrms so the dc voltmeter will provide an indication of approximately 4.5V dc Therefore, the pointer deflected full scale when 10V dc signal is applied. When we apply a 10Vrms sinusoidal AC waveform, the pointer will deflect to 4.5V This means that the AC voltmeter is not as sensitive as DC voltmeter. In fact, an AC voltmeter using half wave rectification is only approximately 45% as sensitive as a dc voltmeter.

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Cont… Actually, the circuit would probably be designed for full-scale deflection with a 10V rms AC applied, which means the multiplier resistor would be only 45% of the value of the multiplier resistor for 10V dc voltmeter. Since we have seen that the equivalent dc voltage is equal to 45% of the rms value of the ac voltage. Sac = 0.45Sdc

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Cont.. Commercially produced ac voltmeters that use half wave rectification also has an additional diode and a shunt as shown in Figure below:

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Cont… The additional diode D2 is reverse biased on the positive half cycle and has virtually no effect on the behavior of the circuit. In the negative half cycle, D2 is forward biased and provides an alternate path for reverse biased leakage current that would normally through the meter movement and diode D1. The purpose of the shunt resistor Rsh is to increase the current flow through D1 during positive half cycle so that the diode is operating in a more linear portion of its characteristic curve. Although this shunt resistor improves the linearity of the meter on its low voltage ac ranges, it also further reduces the AC sensitivity.

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**Fig. 1: AC voltmeter using half wave rectification**

Example 1-6 Compute the value of the multiplier resistor for a 15Vrms ac range on the voltmeter shown in Fig. 1. RS Ifs = 1mA Ein = 15Vrms Rm = 300Ω Fig. 1: AC voltmeter using half wave rectification

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**Solution: Method 1 The sensitivity of the meter movement,**

Rs = Sdc × Rangedc – Rm = 1k × - Rm = 1k × 0.45(10) – 300 = 4.2k

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**Cont. Sac = 0.45Sdc = 0.45(1k) = 450/V Method 2**

The AC sensitivity for half wave rectifier, Sac = 0.45Sdc = 0.45(1k) = 450/V Rs = Sac × Rangeac – Rm = 450 × 10 –300 = 4.2k

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Cont. Method 3 Rs = = 4.2k =

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Example 1-7 Calculate the ac and dc sensitivity and the value of the multiplier resistor required to limit the full scale deflection current in the circuit shown in Fig above.

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**D’Arsonval meter movement used with full wave rectification**

Fig. 2: Full bridge rectifier used in an ac voltmeter circuit During the positive half cycle, currents flows through diode D2, through the meter movement from positive to negative, and through diode D3. The polarities in circles on the transformer secondary are for the positive half cycle. Since current flows through the meter movement on both half cycles, we can expect the deflection of the pointer to be greater than with the half wave cycle, which allows current to flow only on every other half cycle; if the deflection remains the same, the instrument using full wave rectification will have a greater sensitivity.

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**Consider the circuit shown in Fig. 1-2**

Fig. 1-2: AC voltmeter using full wave rectification

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**Cont. When the 10Vrms of AC signal is applied to the circuit**

above, where the peak value of the AC input signal is And the average full wave output signal is Therefore, we can see that a 10Vrms voltage is equivalent to 9Vdc for full-scale deflection.

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Cont. Or This means an ac voltmeter using full wave rectification has a sensitivity equal to 90% of the dc sensitivity Sac = 0.9 Sdc

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**Fig. 1-2: AC voltmeter circuit using full wave rectification**

Example 1-8 Compute the value of the multiplier resistor for a 10Vrms ac range on the voltmeter in Figure 1-2. Fig. 1-2: AC voltmeter circuit using full wave rectification

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**Solution 1-8 The dc sensitivity is The ac sensitivity is**

Sac = 0.9Sdc = 0.9 (1k) = 900 /V

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**Cont. Therefore the multiplier resistor is Rs = Sac x Range – Rm**

= 900 x 10Vrms – 500 = 8.5k

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Cont. Note: Voltmeters using half wave and full wave rectification are suitable for measuring only sinusoidal ac voltages.

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2.2 Oscilloscope An oscilloscope is a piece of electronic test equipment that allows signal voltages to be viewed, usually as a two-dimensional graph of one or more electrical potential differences (vertical axis) plotted as a function of time or of some other voltage (horizontal axis Perform some computations using data taken from the voltage waveform that is displayed such as: * Rms value * Average Amplitude * Peak-to-peak Amplitude * Frequency

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Oscilloscope An oscilloscope is easily the most useful instrument available for testing circuits because it allows you to see the signals at different points in the circuit. Using for signal/wave display – Winamp Music Player, Electrocardiogram,

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2.3 Potentiometer A potentiometer is a variable resistor that functions as a voltage divider It is a simple electro-mechanical transducer It converts rotary or linear motion from the operator into a change of resistance, and this change is (or can be) used to control any volume.

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Potentiometer Schematic symbol for a potentiometer. The arrow represents the moving terminal, called the wiper. Usually, this is a three-terminal resistor with a sliding contact in the center (the wiper) - user-adjustable resistance If all three terminals are used, it can act as a variable voltage divider If only two terminals are used (one side and the wiper), it acts as a variable resistor

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**Potentiometer Circuit**

Any current flow through the Galvanometer, G, wpuld be a result of an imbalance in the measured voltage, Vm and the voltage imposed across points A to B, VAB. If Vm is not equal to VAB, a current will flow through the galvanometer, G. Galvanometer detects current flow due to imbalance in voltage Vm and VAB. When Vm = VAB, there is a balance and no current, means no displacement in Galvanometer.

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**Potentiometer – Application**

In modern usage, a potentiometer is a potential divider, a three terminal resistor where the position of the sliding connection is user adjustable via a knob or slider. For instance, when attached to a volume control, the knob can also function as an on/off switch at the lowest volume Potentiometers are frequently used to adjust the level of analog signals (e.g. volume controls on audio equipment) and as control inputs for electronic circuits (e.g. a typical domestic light dimmer).

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**3.0 Resistance Measurement**

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3.1 Ohmmeter The purpose of an ohmmeter, is to measure the resistance placed between its leads. This resistance reading is indicated through a mechanical meter movement which operates on electric current. The ohmmeter must then have an internal source of voltage to create the necessary current to operate the movement, and also have appropriate ranging resistors to allow just the right amount of current through the movement at any given resistance.

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Ohmmeter The original design of an ohmmeter provided a small battery to apply a voltage to a resistance. It used a galvanometer to measure the electric current through the resistance. The scale of the galvanometer was marked in ohms, because the fixed voltage from the battery assured that as resistance decreased, the current through the meter would increase. A more accurate type of ohmmeter has an electronic circuit that passes a constant current I through the resistance, and another circuit that measures the voltage V across the resistance.

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Ohmmeter The standard way to measure resistance in ohms is to supply a constant voltage to the resistance and measure the current through it. That current is of course inversely proportional to the resistance according to Ohm's law, so that you have a non-linear scale. The current registered by the current sensing element is proportional to 1/R, so that a large current implies a small resistance.

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**The Ohmmeter (Series ohmmeter)**

The ohmmeter consists of battery, resistor and PMMC. The full-scale deflection current, Fig. 2-7 Basic ohmmeter circuit function of Rz and Rm are to limit the current through the meter

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Cont. Rz = variable resistor Fig. 2-8 Basic ohmmeter circuit with unknown resistor,Rx connected between probes. To determine the value of unknown resistor, Rx, The Rx is connected to terminal X and Y. Fig 2-8 shows the basic ohmmeter circuit with unknown resistor, Rx connected between probes.

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The circuit current, The ratio of the current, I to the full-scale deflection current, Ifs is

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**3.2 Wheatstone Bridge Circuit**

A Wheatstone bridge is a measuring instrument invented by Samuel Hunter Christie (British scientist & mathematician) in 1833 and improved and popularized by Sir Charles Wheatstone in It is used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. Its operation is similar to the original potentiometer except that in potentiometer circuits the meter used is a sensitive galvanometer. Accurately measures resistance and detect small changes in resistance. Sir Charles Wheatstone (1802 – 1875)

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Wheatstone Bridge Definition: Basic circuit configuration consists of two parallel resistance branches with each branch containing two series elements (resistors). To measure instruments or control instruments Basic dc bridge used for accurate measurement of resistance: Fig. 5.1: Wheatstone bridge circuit

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**How a Wheatstone Bridge works?**

The dc source, E is connected across the resistance network to provide a source of current through the resistance network. The sensitive current indicating meter or null detector usually a galvanometer is connected between the parallel branches to detect a condition of balance. When there is no current through the meter, the galvanometer pointer rests at 0 (midscale). Current in one direction causes the pointer to deflect on one side and current in the opposite direction to otherwise. The bridge is balanced when there is no current through the galvanometer or the potential across the galvanometer is zero.

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**Cont. I1 = I3 and I2=I4 (3) At balance condition;**

voltage across R1 and R2 also equal, therefore (1) Voltage drop across R3 and R4 is equal I3R3= I4R4 (2) No current flows through galvanometer G when the bridge is balance, therefore: I1 = I3 and I2=I4 (3)

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**Cont. Substitute (3) in Eq (2), I1R3 = I2R4 (4) Eq (4) devide Eq (1)**

R1/R3 = R2/R4 Then rewritten as R1R4 = R2R3 (5)

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**Example 1 Figure 5.2 consists of the following, R1 = 12k, R2 = 15 k,**

R3 = 32 k. Find the unknown resistance Rx. Assume a null exists(current through the galvanometer is zero). Fig. 5-2: Circuit For example 5-1

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Solution 1 RxR1 = R2R3 Rx = R2R3/R1 = (15 x 32)/12 k, Rx = 40 k

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**Sensitivity of the Wheatstone Bridge**

When the bridge is in unbalanced condition, current flows through the galvanometer, causing a deflection of its pointer. The amount of deflection is a function of the sensitivity of the galvanometer.

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Cont. Deflection may be expressed in linear or angular units of measure, and sensitivity can be expressed: Total deflection,

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**Unbalanced Wheatstone Bridge**

Fig. 5-3: Unbalanced Wheatstone Bridge Fig. 5-4: Thevenin’s resistance Vth = Eab Rth = R1//R3 + R2//R4 = R1R3/(R1 + R3) + R2R4(R2+R4)

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Thévenin’s Theorem An analytical tool used to extensively analyze an unbalance bridge. Hermann von Helmholtz (1821 – 1894) German Physicist Léon Charles Thévenin ( ) French Engineer Thévenin's theorem for electrical networks states that any combination of voltage sources and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors. The theorem was first discovered by German physicist Hermann von Helmholtz in 1853, but was then rediscovered in 1883 by French telegraph engineer Léon Charles Thévenin ( ).

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**Thevenin’s Equivalent Circuit**

If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is where Rg = the internal resistance in the galvanometer

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**Example 2 R2 = 1.5 kΩ R1 = 1.5 kΩ Rg = 150 Ω E= 6 V R3 = 3 kΩ**

Figure 5.5: Unbalance Wheatstone Bridge Calculate the current through the galvanometer ?

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**Slightly Unbalanced Wheatstone Bridge**

If three of the four resistors in a bridge are equal to R and the fourth differs by 5% or less, we can developed an approximate but accurate expression for Thevenin’s equivalent voltage and resistance.

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Cont.. To find Rth: An approximate Thevenin’s equivalent circuit

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Example 3 10 V 500 Ω 525 Ω Use the approximation equation to calculate the current through the galvanometer in Figure above. The galvanometer resistance, Rg is 125 Ω and is a center zero μA movement.

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Kelvin Bridge The Kelvin Bridge is a modified version of the Wheatstone bridge. The purpose of the modification is to eliminate the effects of contact and lead resistance when measuring unknown low resistances. Used to measure values of resistance below 1 Ω .

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**Fig. 5-6: Basic Kelvin Bridge showing a second set of ratio arms**

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Cont. It can be shown that, when a null exists, the value for Rx is the same as that for the Wheatstone bridge, which is Therefore when a Kelvin Bridge is balanced

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Example 5-4 If in Figure 5-6, the ratio of Ra and Rb is 1000, R1 is 5 and R1 =0.5R2. What is the value of Rx.

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**Solution The resistance of Rx can be calculated by using the equation,**

Rx/R2=R3/5=1/1000 Since R1=0.5R2, the value of R2 is calculated as R2=R1/0.5=5/0.5=10 So, Rx=R2(1/1000)=10 x (1/1000)=0.01

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4.0 Digital Multimeter A multimeter or a multitester is an electronic measuring instrument that combines several functions in one unit. The most basic instruments include an ammeter, voltmeter, and ohmmeter

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**4.1 Digital Multimeter – Capabilities**

DC Voltage Measurements AC Voltage RMS Measurements DC and AC Current Measurements Resistance Measurements Capacitance/Inductance Measurements Frequency/Period Measurements Diode Measurements

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