eeg383 measurement - chapter 5 - basic electric quantities measurement (new slides)

Prof Fawzy IbrahimEEG383 Ch.5 B. Measurements1 of 81

EEG 383 Electrical Measurements and Instrumentations

CHAPTER 5

Basic Electric Quantities Measurement

Prof. Fawzy Ibrahim

Electronics and Communication DepartmentMisr International University (MIU)


Chapter Contents5.1 Types of Meters

5.1.1 Analog Meters5.1.2 Digital Meters

5.2 Permanent Magnet Moving Coil (PMMC) Meter5.2.1 Construction5.2.2 Theory of Operation

5.3 DC Ammeter and DC Current Measurement5.3.1 Ammeter Full Scale Deflection (FSD) Extension5.3.2 Ammeter Disturbance due to Measurement5.3.3 Multi-Range Ammeter

5.4 DC Voltmeter and DC Voltage Measurement5.4.1 Voltmeter Full Scale Deflection (FSD) Extension5.4.2 Voltmeter Disturbance due to Measurement5.4.3 Multi-Range Voltmeter5.4.4 Ammeter and Voltmeter Accuracy5.4.5 Voltmeter Sensitivity


Chapter Contents5.5 Ohmmeter

5.5.1 Ohmmeter Construction5.5.2 Ohmmeter Design5.5.3 Multirange Accuracy

5.6 Multimeter

5.7 Digital Measuring Instruments 5.7.1 Digital Measuring Instruments Block Diagram5.7.2 Multirange Digital Ammeter5.7.3 Digital Voltmeter5.7.4 Multirange Digital Ohmmeter

5.8 Analog AC Measurement5.8.1 D’ Arsonval or PMMC meter on AC Measurements5.8.2 Root Mean Square (RMS) versus Mean of AC Signal5.8.3 Full-wave rectifier (AC) Ammeters and Voltmeters5.8.4 Precision Rectifier Circuits


Chapter Contents5.9 Resistance Measurement

5.9.1 Resistance Measurement using Ohmmeter5.9.2 Resistance Measurement using Wheatstone Bridge

5.10 Inductance Measurement 5.10.1 Inductor Equivalent Circuit 5.10.2 Inductor Quality Factor (Q)5.10.3 Inductance Measurement using Q-Meter5.10.4 Inductance Measurement using AC Voltmeter5.10.5 Inductance Measurement using AC Bridges

5.11 Capacitance Measurement 5.10.1 Inductor Equivalent Circuit 5.11.2 Capacitor Dissipation Factor (D)5.11.3 Capacitance Measurement using Q-Meter5.11.4 Capacitance Measurement using AV Voltmeter5.11.5 Capacitance Measurement using AC Bridges


5.1 Types of Meters

• Analog instruments use analog meters which are relatively simple andinexpensive. Analog meters are electromechanical devices that drive a pointeragainst a scale as shown in Fig. 5.1.

• All types of analog meters are basically modified forms of the analog ammeter,irrespective of the quantity that they are designed to measure.

• Analog meters are designed to measure quantities other than current usingappropriate electrical circuits convert voltage or resistance measurementsignals into current signals.

Advantages• Analog instruments cab be passive instruments that do not need a power

supply. They are relatively simple and inexpensive.• They suffer less from noise and isolation problems.

Disadvantages• They are prone to measurement errors from a number of sources that include

inaccurate scale marking during manufacture, bearing friction, bent pointersand ambient temperature variations.

• Human errors are introduced through parallax error (not reading the scale fromdirectly above) and mistakes in interpolating between scale markings.

• Quoted inaccuracy figures are between ±0.1% and ± 3%.

5.1.1 Analog Meters


5.1 Types of Meters

Fig. 5.1 Different shapes of analog meters

5.1.1 Analog Meters


5.1 Types of Meters

• Digital instruments use digital meter which are shown in Fig. 5.1.• All types of digital meter are basically modified forms of the digital voltmeter

(DVM), irrespective of the quantity that they are designed to measure.• Digital meters designed to measure quantities other than voltage are in fact

digital voltmeters that contain appropriate electrical circuits to convert current orresistance measurement signals into voltage signals.

• Digital multimeters are also essentially digital voltmeters that contain severalconversion circuits, thus allowing the measurement of voltage, current andresistance within one instrument.Advantages

• Digital meters have been developed to satisfy a need for higher measurementaccuracies and a faster speed of response to voltage changes than can beachieved with analogue instruments.

• Quoted inaccuracy figures are between ± 0.005%) and ± 2%.• They have very high input impedance (10 MΩ compared with 1–20 k Ω for

analog meters.

Disadvantages• They have a greater cost due to the higher manufacturing costs compared with

analogue meters.

5.1.2 Digital Meters


5.1 Types of Meters

Fig. 5.1 Different shapes of digital meters

5.1.2 Digital Meters


5.2 Permanent Magnet Moving Coil (PMMC) Meter• A moving-coil or D’ Arsonval meter is a very commonly used form of analog

instrument because of its sensitivity, accuracy and linear scale.• D’ Arsonval meter as shown in Fig. 5.5 consists of a rectangular coil wound

round a soft iron core that is suspended in the field of a permanent magnet.5.2.1 Construction1. Permanent magnet with two soft iron pole shoes to provide the magnetic field.2. Moving coil wounded over a core of a rectangular or cylinder shape former

which is pivoted on jeweled bearing. The frame is usually made of Al or iron toprovide the required electromagnetic damping. The coil rotates between thepole shoes.

3. Two control spiral springs are made of phosphor bronze hair springs areused to control the movement. They are also serve to lead the current in andout of the coil.

4. Pointer and Calibrated Scale: The pointer is of light weight constructioncarried by the spindle and moves over a graduated scale.


5.2 Permanent Magnet Moving Coil (PMMC) Meter (Continued)

Fig. 5.3 Construction of PMMC or D’ Arsonval meters

5.2.1 Construction



• The signal being measured is applied to the coil and this produces a radialmagnetic field.

• Interaction between this induced field and the field produced by the permanentmagnet causes a torque, which results in rotation of the coil.

• The amount of rotation of the coil is measured by attaching a pointer to it thatmoves past a graduated scale.

• A deflection instruments uses a pointer that moves over a calibrated scale toindicate a measured quantity.

• For this to occur, three forces are operating in the electromechanical movementas follows:

1. Deflecting Force: lets the pointer moves from its zero position when a currentflows.

2. Controlling Force:- Returns the pointer to its zero position when the current is disconnected.- Balances the deflecting force so that for constant current, the pointer

remains stationary at the appropriate position on the scale.

3. Damping Force: minimizes the oscillation of the pointer.

5.2.2 Theory of Operation



1. Deflecting Force and Deflecting Torque:• When a current I flows through a one-turn of the coil of length L situated in a

magnetic filed of a magnetic flux B, a force F is exerted on each side of thecoil, as shown in Fig.5.4, which is given by:

• Since any turn of the coil has a rectangular shape, then the same force will acton the other side. Therefore the total force per one turn is:

• And the total force for N turns, deflecting force, is:

• The Force on each side acts on each sideof the coil at a radius r (radius of thecylinder base), producing a deflecting torque:

Where, A is the surface area enclosed by the coil.


Fig. 5.4 The deflection force in PMMC or D’ Arsonval meters

LIBF (5.1)[Newton]

LIBF 2 (5.2)[Newton]

LNIBFD 2 [Newton] (5.3)

ANIBrNLIBrNLIBTD 22 (5.4)



1. Deflecting Force and Deflecting Torque:• Clearly the deflection torque, for a given PMMC, is directly proportional to the

coil current only.• Thus the deflecting angle (θ) is directly proportional to the coil current only.

2. Controlling Force and Controlling Torque:• The two spiral springs are used to produce the controlling and damping forces.• The controlling and damping forces cause controlling torque (TC).• TC is directly proportional to the deformation (or wind up) of the springs which

in turn proportional to actual angle of deflection (θ) of the pointer. Its given by:

• For a given deflection (current), the pointer gets to rest when:

• K1 is a constant. Then

• K2 is another constant. Clearly the deflection angle, θ, for a given PMMC, isdirectly proportional to the coil current only.

• Thus if the scale is calibrated by the amount of current that produce thecorresponding θ, the PMMC can be used to measure the current.


oC KT (5.5)

1KBINATT CD (5.6)

IK2 (5.7)



Notes:1. The current in the coil of the PMMC instrument must flow in one particular

direction to cause the pointer moves from its zero position over the scale.2. If the current is reversed, the pointer will rotate to left of the zero position (off

the scale). Thus the PMMC instrument can be used to measure the dc currentsonly.

3. A PMMC meter can be used to measure very small currents.4. For a PMMC meter to measure high current or quantities other than current

they are modified by using appropriate electronic circuits to convert voltage orresistance measurement signals into current signals as will be discussed in thenext sections.


• For a given PMMC instrument, there is a maximum rated current that producesfull-scale deflection of the indicator (FSD) as shown in Fig. 5.5 (a).

• A typical FSD current rating for a moving meters is IFS = 50 µA, with internalwire resistance, Rm = 1 kΩ.

5.3.1 Ammeter Full Scale Deflection (FSD) Extension• To increase the range of measured current, a shunt resistance (Rsh) is

connected in parallel to the instrument as shown in Fig. 5.5 (b).• Rsh must be small relative to the internal resistance of the PMMC instrument

(Rm) to allow a big portion of the measured current pass thought it.


5.3 DC Ammeter and DC Current Measurement

Fig. 5.5 DC Ammeter : (a) Basic circuit; (b) FSD Expansion

(a) (b)

5.3.1 Ammeter Full Scale Deflection (FSD) Extension• I is total current to be measured, IFS, Ish are the currents of Rm and Rsh

respectively and Vm is the voltage drop across Rm (i.e. it also across thePMMC) which is given by:

• Total current to be measured (I) is given by:

• Clearly I is linearly proportional with IFS, then the scale can be calibrated uponthe value of I instead of IFS.

• Note that, the shunt resistance required to provide an extended FSD (IFSD) isgiven by:

Notes:1. Ammeters are connected in series with the test circuit.2. Ideally they should have zero resistance, so that they cause no voltage drop.3. Practical ammeters should have internal resistance much lower than that of the

circuit being tested.


5.3 DC Ammeter and DC Current Measurement (Continued)

shm

m

shmmshFS RR

VRR

VIII//

11

(5.8)

FSEFS

msh

sh

mFSEFSsh II

VRRVIII

mFSm RIV

(5.9)

(5.10)

5.3.1 Ammeter Full Scale Deflection (FSD) ExtensionExample 5.1

using PMMC or D’Arsonval meter characterized by the full scale current,IFS = 100 µA and internal resistance, Rm = 100 Ω, design a DC Ammeter tohave an extended full scale rang IEFS = 1 mA:

SolutionFrom (5.8), Vm, the voltage drop across, Rm or PMMC is calculated as:

From (5.10), Rsh, the shunt resistancerequired to provide the extended FSD(IEFS) is given by:



11.119.0

101.01

10

mAmV

mAmAmV

IIVR

FSEFS

msh

mVxxRIV mFSm 1010010100 6

5.3.2 Ammeter disturbance due to measurementAmmeters are connected in series with the test circuit. Ideally they should havezero resistance, if , Rm zero Ω, the impact on the circuit to be measured orAmmeter disturbance is calculated as follows:Example 5.2: Determine the error in the reading of the currents in thecircuit shown in Fig. 5.6 if R1 = 3 , R2 = 1.5 , V = 2V and the internalresistance of the Ammeter is Rm = 0.5 .SolutionWith no ammeter connected:I1 (of R1) = 2/ 3= 0.667AI2= 2/ 1.5= 1.333AWith an ammeter, of Rm =0.5 ΩI1= 2/ 3.5= 0.571AI2= 1.333A (no error)δI1 =(0.667-0.57 )/0.667=14.4%With an ammeter, of Rm = 1.5 ΩI1= 0.667A (no error) and I2= 1A.δI2 =(1.333-1 )/1.333=25% Fig. 5.6




5.3 DC Ammeter and DC Current Measurement (Continued)5.3.3 Multi-range Ammeter

Several shunt resistances are connected to perform a Multi-range Ammeter asshown in Fig. 5.7 (a). Clearly

Fig. 5.7(a) Multirange Ammeter

FSEFSshshFSEFS IIIIIIIiiii

i

mmshishmmm R

RIIRIRIVii

FSEFS

mFS

FSEFS

mshi II

RIII

VRii

(5.12)

(5.11)

(5.13)

5.3.3 Multi-range Ammeter• For the previous multi-range ammeter, a make-before-break switch must be

used to avoid destroying the PMMC instrument due to high current that maypass through the coil during transition from position to another.

• Another method that avoid using make-before-break switch is shown in Fig.5.7(b).

)1(4321

1ssss

mFSs RRRR

RII



Fig. 5.7 (b) Multirange Ammeter

(5.15)

(5.16)

(5.14)tsh smFSm RIRIV ' )1(

'

st

mFSshFS R

RIIIIsi

)1(4

3214 R

RRRRII sssmFSs

)1(43

213

ss

ssmFSs RR

RRRII

)1(432

12

sss

smFSs RRR

RRII

(5.18)

(5.19)

(5.17)

5.3.3 Multi-range AmmeterExample 5.3

using PMMC or D’Arsoval meter characterized by the full scale current,IFS = 100 µA and internal resistance, Rm = 500 Ω, design a multirange DCAmmeter to measure current in the following ranges:(a) 0 to 1 mA (b) 0 to 10 mA (c) 0 to 100 mA

Solution(a) From (5.8), Vm, the voltage drop across, Rm or PMMC is calculated as:

From (5.10), Rshi, is given by:

(b)

(c)



55.551.01

501 mAmA

mVRsh


FSEFS

msh II

VR

05.51.010

502 mAmA

mVRsh

5.01.0100

503 mAmA

mVRsh

• The deflection angle of a PMMC instrument is directly proportional to thecurrent flowing through the moving coil.

• Coil current is directly proportional to the voltage across the coil (Vm=ImRm).• So to let PMMC instrument works as DC voltmeter, the scale of the PMMC

instrument could be calibrated to indicate the voltage across the coil, given by:

• Most meters are very sensitive. That is, they give full-scale deflection for asmall fraction of an ampere.

• A typical FSD current rating for a moving coil meters is IFS= 50 µA, with internalwire resistance of Rm = 1 kΩ. With no additional circuitry, the maximum voltagethat can be measured using this meter is VFS = 50 x 10-6x 1000 = 0.05 V asshown in Fig. 5.8 (a).

• Thus additional circuitry is needed for the measurement of high voltages(practical measurements).

• To increase the range of measured voltage a series resistance (Rmultiplier or Rs)is connected in series with the instrument as shown in Fig. 5.8 (b).

• This series resistance must be large relative to the internal resistance of thePMMC instrument (Rm) to let small portion of the measured voltage to bedropped across Rm.


5.4 DC Voltmeter and DC Voltage Measurement

mFSFS RIV (5.20)

5.4.1 Voltmeter Full Scale Deflection (FSD) Extension• The Extended Full Scale Deflection (EFSD) is then increased to:

• For a given required EFSD voltage, the multiplier resistance, Rs, is chosen as:

• e.g. to provide a voltmeter with EFSD reading of 10 V with the given meter(IFSD= 1 mA, Rm= 500 Ω):


5.4 DC Voltmeter and DC Voltage Measurement

Fig. 5.8 DC Ammeter : (a) Basic circuit; (b) FSD Expansion

(a) (b)

)( smFSEFS RRIV

mFS

EFSs R

IVR

kRs 5.95001010

3

(5.21)

(5.22)

5.4.1 Voltmeter Full Scale Deflection (FSD) ExtensionExample 5.4

using PMMC or D’Arsonval meter characterized by the full scale current,IFS = 100 µA and internal resistance, Rm = 100 Ω, design a DC Voltmeter tohave an extended full scale rang VEFS = 100 mV:

SolutionFrom (5.20), Vm, the voltage drop across, Rm or PMMC is calculated as:

From (5.22), Rsh, the series resistancerequired to provide the extended FSD(VEFS) is given by:


5.4 DC Voltmeter and DC Voltage Measurement (Continued)


kkk

mAmV

RI

VR mFS

EFSs

9.01.01

1001.0

100

5.4.1 Voltmeter Full Scale Deflection (FSD) ExtensionNotes:

1. Voltmeters are connected in parallel with the circuit under test.2. Current is drawn from the circuit through the voltmeter, and may affect the

voltage being measured.3. Ideally, a voltmeter has infinite resistance, so that no current is drawn from the

test circuit.4. Practically, voltmeter has finite high resistance, so that small current is drawn

from the test circuit and cause an error called loading error.5. Practical voltmeter should have a much higher resistance than that of the circuit

under test to decrease the loading error.6. There will always be some degree of loading even if the voltmeter resistance is

much larger than that of the circuit under test.7. Loading can be minimized by using electronic voltmeter.



5.4.2 Voltmeter disturbance due to measurementA Voltmeter are connected in parallel with the test circuit. Ideally they shouldhave zero resistance, if , Rm , the impact on the circuit to be measured orVoltmeter disturbance is calculated as follows:Example 5.5: Determine the error in the reading of the voltage across R2 in thecircuit shown in Fig. 5.9 if R1 = R2 = 250 M, V = 24 V, using PMMC with the fullscale current IFS = 100 µA and the internal resistance of the VoltmeterRm = 10 M.SolutionWith no Voltmeter connected:

With a Voltmeter of Rm =10 M Ωtotal resistance across VO:

δVO =(12-0.89 )/12 = 11.11V or 96.3%.This example is extreme in that the resistanceof the voltmeter is less than that of the test circuit. Fig. 5.9Repeat this example for a voltmeter has Rm =500 MΩ.



VVMMMVo 1224)250250/(250

MMMRo 615.910//250VVMMMVo 89.024)615.9250/(615.9

5.4.3 Multi-range VoltmeterMulti-range Voltmeter is obtained by connecting several series resistances asshown in Fig. 5.10 (a). Clearly the extended voltages are given by:

and the resistance Rsi iscalculated form (5.22) as:



Fig. 5.10(a) Multirange Voltmeter

(5.23))(

)( 11

simFSiEFS

smFSEFS

RRIV

RRIV

mFS

EFSisi R

IVR

5.4.3 Multi-range VoltmeterExample 5.6

using PMMC or D’Arsonval meter characterized by the full scale current,IFS = 1 mA and internal resistance, Rm = 500 Ω, design a multirange DCVoltmeter to measure voltages in the following ranges:(a) 0 to 1 V (b) 0 to 10 V (c) 0 to 100 V (d) 0 to 1000 V

Solution From (5.20), Vm, the voltage drop across, Rm is calculated as:

(a) From (5.22), Rsi, is given by:

(b)

(c)

(d)



50050011

4 mAVRs

VxxRIV mFSm 5.0500101 3

kmA

VRs 5.9500110

3

mFS

EFSisi R

IVR

kmA

VRs 5.995001100

2

kmA

VRs 5.9995001

10001

5.4.3 Multi-range Voltmeter• A more practical circuit is shown in Fig. 10 (b). For each successively higher

voltage range, more multiplier resistors are added in by switching.



Fig. 5.7 (b) Multirange Voltmeter

(5.24))( 212 ssmFSESD RRRIV

mFS

EFSs

FS

EFSsm R

IV

RI

VRR

11

11

)( 12

2 smFS

EFSs RR

IV

R

)( 213

3 ssmFS

EFSs RRR

IV

R

)( 11 smFSEFS RRIV

)( 3213 sssmFSESD RRRRIV

)( 43214 ssssmFSESD RRRRRIV

(5.24a)

)( 3214

4 sssmFS

EFSs RRRR

IV

R

(5.24b)

(5.24c)

(5.24d)

5.4.3 Multi-range VoltmeterExample 5.7: using PMMC or D’Arsonval meter characterized by the full scale

current, IFS = 1 mA and internal resistance, Rm = 500 Ω, design a multirangeDC Voltmeter to measure voltages in the following ranges:(a) 0 to 1 V (b) 0 to 10 V (c) 0 to 100 V (d) 0 to 1000 V

Solution: From (5.20), Vm, the voltage drop across, Rm is calculated as:


(b)

(c)

(d)



mVxxRIV mFSm 5.0500101 3

500500111

1 mAV

RI

VR m

FS

EFSs

kkmA

VRR

I

VR sm

FS

EFSs 91

110

)( 12

2

kkmA

VRs 9010

1100

3

kkmA

VRs 900100

11000

4

5.4.3 Multi-range VoltmeterExample 5.8: using PMMC or D’Arsonval meter characterized by the full scale

current, IFS = 100 µA and internal resistance, Rm = 100 Ω, design a multirangeDC Voltmeter to measure voltages in the following ranges:(a) 0 to 100 mV (b) 0 to 1 V (c) 0 to 10 V (d) 0 to 100 V

Solution: From (5.20), Vm, the voltage drop across, Rm is calculated as:


(b)

(c)

(d)




9001001001001

1 AmV

RI

VR m

FS

EFSs

kkA

VRR

IV

R smFS

EFSs 91

1001

)( 12

2

kkA

VRs 9010

10010

3

kkA

VRs 900100

100100

4

5.4.4 Ammeter and Voltmeter AccuracyThe accuracy of Ammeter and Voltmeter is only decided by the accuracy of thePMMC meter.

Example 5.9: Ammeter AccuracyAn ammeter shows the accuracy to ±1% of full scale deflection, if full scale isIFS = 400 A, find the absolute and relative errors as the pointer stays at :(a) 100 A (b) 200 A (c) 400 A.

Solution: The absolute error = IFS x Inaccuracy = 400 x ±1% = ± 4 A(a) The reading at 100 A = 100 ± 4 = 96 A~104 A = 100 A ±4%(b) The reading at 200 A = 200 ± 4 = 196 A~204 A = 200 A ±2%(c) The reading at 400 A = 400 ± 4 = 396 A~404 A = 400 A ±1%

Example 5.10: Voltmeter AccuracyA voltmeter shows the accuracy to ±0.1% of full scale deflection, if full scale isVFS = 10 V, find the absolute and relative errors as the pointer stays at:(a) 2 V (b) 6 V (c) 8 V.

Solution: The absolute error = VFS x Inaccuracy = 10 x ±0.1% = ± 0.01 V(a) The reading at 2 V = 2 ± 0.01 = 1.99V~2.01V = 2 V ± 0.5%(b) The reading at 6 V = 6 ± 0.01 = ) 5.99V~6.01V = 6 V ± 0.17%(c) The reading at 8 V = 8 ± 0.01 = ) 7.99V~8.01V = 8 V ± 0.125%



5.4.5 Voltmeter Sensitivity• It is amount of resistance per one volt.

• The sensitivity of a Voltmeter is always specified by the manufacturer and isfrequently printed on scale of the instrument.

Example 5.11using PMMC or D’Arsonval meter characterized by the full scale current,IFS = 100 A and internal resistance, Rm = 1 kΩ, Determine the requiredmultiplier resistance Rs if the Voltmeter is to measure 50 V as a full scale andcalculate its sensitivity.Solution:

From (5.22), Rs, is given by:



VRR ms /VV

RySensitivitVoltmeter EFSEFS

t

(5.25)

kkkkx

VR

IVR m

FS

EFSs 49915001

1010050

6

VkVk

/1050

500VRySensitivitVoltmeter FSD

t

• The simplest ohmmeter circuit consists of a battery; standard resistance (Rst)and PMMC meter are connected in series and Rx is the resistance to bemeasured as shown in Fig. 5.8(a).

• The current that flows through the resistance is inversely proportion with thevalue of the resistance. So the scale can be calibrated to indicate theresistance value.

• Since the basic meter indicates in response to current flowing through it, avoltage source is needed to supply current to a resistor under test.


5.5 Ohmmeter

Fig. 5.8 (a) Basic circuit of ohmmeter; (b) Ohmmeter scale(a)

(b)

5.5.1 Ohmmeter Construction

Notes:1. Clearly the ohmmeter scale is a nonlinear scale as shown in Fig. 5.8(b).2. Since the measured resistance depends on the voltage of the battery, a very

stable battery is required.3. Resistors cannot be measured when connected in a circuit, because different

voltage sources would interfere.• The coil current is given by:

• Its range is found as follows:• Under short circuit conditions, i.e. Rx = 0 Ω, Rst is selected to give IFS. So the IFS

position is marked by zero ohms.

• Under open circuit conditions, i.e. Rx = ∞ Ω, the current will be zero and theneedle points to the far left (zero current) and is marked by infinity ohms.

• From (5.26), the value of the resistance Rx is given by:


5.5 Ohmmeter (Continued)

mstx

sm RRR

EI

mst

sFS RR

EI

(5.26)

(5.27)

mstm

sx RR

IER (5.28)

5.5.2 Ohmmeter Design

Example 5.11: The ohmmeter in Fig. 5.8 (a) is made of a battery, Es = 1.5 V, aPMMC or D’Arsonval meter characterized by the full scale current, IFS = 100 Aand internal resistance, Rm = 1 kΩ.do the following:

(a) Calculate the standard resistance, Rst.(b) Determine the instrument indication when RX = 0.(c) Determine how the resistance scale should be marked at 0.25 IFS, 0.5 IFS, and

0.75 IFS.Solution: (a) From (5.27), standard resistance, Rst, is given by:

(b) When RX = 0, the instrument indication is calculated from (5.26) as:

(c) At 0.25 IFS (25 A), the resistance scale is determined from (5.28) as:

At 0.5 IFS,(50 A)

At 0.75 IFS,(75 A)Prof Fawzy IbrahimEEG383 Ch.5 B. Measurements36 of 81

5.5 Ohmmeter (Continued)

kkkkx

RIER

RREI m

FS

sst

mst

sFS 141151

101005.1

6

AkkRRR

EImstx

sm 100

11405.1

or Full Scale Deflection (FSD).

kkkkkx

RRIER mst

m

sx 451560)114(

10255.1

6

kkkkkx

RRIER mst

m

sx 151530)114(

10505.1

6

kkkkkx

RRIER mst

m

sx 51520)114(

10755.1

6

• The series resistance ohmmeter can be converted to a multirange ohmmeter byemploying several values of the standard resistance (Rst) and a rotary switch asshown in Fig. 5.9.

• The main disadvantage of such a circuit is the need for zero adjustment foreach time the scale is change.


5.5 Ohmmeter (Continued)5.5.1 Multirange Ohmmeter

Fig. 5.9 (a) Multirange ohmmeter

The accuracy of ohmmeter is dominated by following conditions:1. The battery aging.2. The accuracy of PMMC meter.3. The measurement technique.

Note: It is clear that the ohmmeter scale is nonlinear. The useful range of theohmmeter scale is seen to be approximately from 10% to 81% of FSD.


5.5 Ohmmeter (Continued)5.5.3 Multirange Accuracy

Simple voltmeter / ammeter• Different connections are

provided for voltmeter andammeter functions, onestructure is shown in Fig. 5.10.This prevents damage causedby connected the ammeteracross a large voltage. Anohmmeter function may beadded by including a battery anda suitable series resistor, Rst.


5.6 Multimeter

Fig. 5.10 Multirange multimeter

• Since voltmeters, ammeters and ohmmeters are all based on moving coilmeters with resistors (and a battery for ohmmeter) connected in differentconfigurations, one moving coil meter may be used to design a multi-purposemeter with appropriate switches.


5.7 Digital Measuring Instruments

Fig. 5.11 (a) Digital measuring instruments block diagram; (b) Seven-segment display

• Fig. 5.11(a) shows a general block diagram digital measuring instruments.• The Analog to Digital Converter (A/DC) coverts the input analog signal to the

corresponding digital one as explained in Chapter 4.• The binary to 7-segment display decoder is used to convert the binary to a form

that can be used by the 7-display. It has four inputs for the binary (0000 to1001) and seven outputs for the display as shown in Fig. 5.11(b) and explainedDigital Logic Design Course.

• The Seven Segment Display consists of seven segment (a, b, c, d, e, f and g).Each segment is a Light Emitting Diode (LED) or Liquid Crystal Display (LCD)as shown in Fig. 5.11(b) and explained Electronics I Course.

(a) (b)

5.7.1 Digital Measuring Instruments Block Diagram


5.7 Digital Measuring Instruments (Continued)

Fig. 5.12 (a) Multirange Digital Ammeter circuit diagram; (b) I/V Converter circuit.

• Fig. 5.12 (a) shows the circuit diagram of Multirange Digital Ammeter.

(a)

5.7.2 Multirange Digital Ammeter

(b)

• The circuit that coverts the current sourceto a voltage source is shown in Fig.5.12(b).

• The voltage vo does not depend on thevalue of ZL and is given by:

• The I/V converter will not affect the currentthrough the circuit

shioi iRv (5.29)

5.7.2 Multirange Digital AmmeterExample 5.12

For the Multirange Digital Ammeter circuit diagram shown in Fig. 5.12(a),design a multirange DC Ammeter to measure current in the following ranges:(a) 0 to 10 mA (b) 0 to 1 mA (c) 0 to 100 µA (d) 0 to 10 µA

Solution(a) For the range 0 to 10 mA, Rsh1 is calculated from (5.29) as:

(b) For the range 0 to 1 mA, Rsh2 is calculated from (5.29) as:

(c) For the range 0 to 100 µA, Rsh3 is calculated from (5.29) as:

(d) For the range 0 to 10 µA, Rsh1 is calculated from (5.29) as:



500

1010510105|| 311

31 x

VRxRxVRIV shshshFSo

k

xVRxRxVRIV shshshFSo 5

10151015|| 322

32

kVRxRxVRIV shshshFSo 50

105101005|| 433

63

kVRxRxVRIV shshshFSo 500

10510105|| 544

64



Fig. 5.13 Digital Voltmeter

• Fig. 5.13 Shows the circuit diagram Digital Voltmeter.5.7.3 Digital Voltmeter

• A digital voltmeter (DVM) essentiallyconsists of an A/D converter, latch circuit, aset of seven segment displays and theirdrivers.

• The latch is used to save the output of A/Deven when the A/D output goes to zero(reset) after a complete cycle of the rampgenerator.

t1 t2



Fig. 5.14 A multirange digital ohmmeter circuit.

• Fig. 5.14 shows a Multirange Digital Ohmmeter.• For n-bit A/DC the number of quantization level VR/2n. Therefore, the minimum

level of the A/DC corresponds to Vo = VR/2n and the maximum level of the A/DCcorresponds to Vo = VR - VR/2n.

• The output of the buffer, Vo is equal to the voltage V+ and is given by:

or

5.7.4 Multirange Digital Ohmmeter

1

1

stx

stRo RR

RVVV

2

2

stx

stRo RR

RVVV

(5.81a)

(5.81b)



• The minimum value of the resistance Rx gives the maximum value of thevoltage V+ or Vo. That is:

Simplify this equation, we get:

or

Therefore• The maximum value of the resistance Rx gives the minimum value of the

voltage V+ or Vo. That is:


n

n

RnRnR

Rstx

stRo VVVV

RRRVVV

212

211

21min

1maxmax

(5.31a)

n

n

xn

n

stst RRR2

122

12min11

nstn

n

stn

n

ststn

n

x RRRRR21

2121

212

212

1111min

121

min nst

xRR

nR

stx

stRo

VRR

RVVV21max

1minmin



Simplify this equation, we get:

Therefore• From (5.31a) and (5.31b), lower and upper limit values of the resistance Rx is:

• Similarly, if another standard resistance, Rst2 is used, lower and upper limitvalues of the resistance Rx is:


(5.31c)

(5.31b)

1212 1

1

nstxn

st RRR

nststxn

stx

st xRRRRR

R 221

11max1max

1

121max nstx RR

1212 2

2

nstxn

st RRR (5.32)

5.7.4 Multirange Digital OhmmeterExample 5.13

For the Multirange Digital Ohmmeter circuit diagram shown in Fig. 5.14, if VR =5 V and the number of bits of the A/DC, n = 7, design a multirange Ohmmeterto measure resistances in the following ranges:(a) 1 Ω to 10 kΩ (b) 1 kΩ to 10 MΩ

Solution: (a) For the range 1 Ω to 10 kΩ , Rst1 is calculated from (5.31) as:

• The resulting range is:

(b) similarly, For the range 1 kΩ to 1 MΩ , Rst2 is calculated from (5.31) as:

The resulting range is:



127

121

12 1711

min stst

nst

x RRRR

kRR nstx 1.161212712 7

1max

kRx 1.161

kRRkRR st

stn

stx 127

121

12 1711

min

MRR nstx 1.161212712 7

1max

MRk x 1.161

• Fig. 5.15 (a) shows another Multirange Digital Ohmmeter circuit and 5.15 (b)shows Current source and current mirrors.

• MOSFET in saturation operates as a currentsource (VCCS) and the drain current ID is:



Fig. 5.15 (a) Another multirange digital ohmmeter circuit; (b) Current source and mirrors

5.7.4 Multirange Digital Ohmmeter Alternative Circuit

(5.81b)

(a)

2` )(21

tnGSnD VVL

WkI

(b)

)/()/( 22

LWLW

II

REF

)/()/( 11

LWLW

II

REF

)/()/( 33

LWLW

II

REF



• The minimum value of the resistance Rx gives the minimum value of thevoltage V+ or Vo. That is:

or

• The maximum value of the resistance Rx gives the maximum value of thevoltage V+ or Vo. That is:

• From (5.33a) and (5.33b), lower and upper limit values of the resistance Rx is:

5.7.4 Multirange Digital Ohmmeter Alternative Circuit

nR

xioVRIVV2minminmin

(5.33a)

i

n

n

R

i

nRR

xn

RRxi I

V

IVVRVVRI

2

122/2/ maxmax

i

n

n

R

i

nRR

x I

V

IVVR

2

122/

max

i

nR

x IVR 2/

min

(5.33b)

in

n

Rxi

nR IVRI

V /2

122/

(5.33c)

5.7.4 Multirange Digital Ohmmeter Alternative CircuitExample 5.14

For the Multirange Digital Ohmmeter circuit diagram shown in Fig. 5.15 (a), ifVR = 5 V and the number of bits of the A/DC, n = 7, design a multirangeOhmmeter to measure resistances in the following ranges:(a) I1 = 1 µA (b) I2 = 10 µA (c) I3 = 100 µA

Solution:(a) For the range I1 = 1 µA, the values of Rx are calculated from (5.33) as:

• The resulting range is:

(b) For the range I1 = 10 µA, the values of Rx are calculated from (5.33) as:

(c) For the range I1 = 100 µA, the values of Rx are calculated from (5.33) as:



MRk x 96.439

kRk x 6.496.390

kxI

VRi

nR

x 06.39101128/52/

6min

MxI

VVRi

nRR

x 96.4101

128/552/6max

kRk x 4969.3

5.8.1 D’ Arsonval or PMMC meter on AC Measurements• For very low frequency (<0.1 Hz), the pointer of the PMMC Instrument tends to

follow the instantaneous level of the ac.• As the current grows positively, the pointer deflection increases to maximum at

the peak of the ac.• The as the instantaneous level falls, the pointer deflection decreases towards

zero. When the ac goes negative, the pointer is deflected (off the scale) to theleft of zero.

• For higher frequencies, the damping mechanism of the instrument and theinertia of the meter movement prevent the pointer from following the changinginstantaneous levels. Instead, the pointer settles at the average value of the accurrent.

•

• Since the average value of a pure sinusoidal ac signal is zero as shown in Fig.5.16 (a), then if the PMMC Instrument is connected directly to measure the 60Hz ac source it will indicate zero.

• In order to use a PMMC meter to measure the alternating current, a rectifiercircuit must be used to rectify the AC into DC. One of the good rectifier circuitsis the Full-wave rectifier as shown in Fig. 5.16 (b).


5.8 Analog AC Measurement

oTo

av dttxT

x )(1(5.35)

• The sine wave voltage signal, v(t) isdescribed by:

where Vmax denotes the peak valueor amplitude in volts and denotesthe angular frequency in radians persecond; that is, = 2nf rad/s, wheref is the frequency in hertz, f= l/T Hz,and T is the period in seconds


5.8 Analog AC Measurement (Continued)

Fig. 5.16 (a) Sine-wave voltage signal; (b) PMMC meter on AC Measurements

5.8.1 D’ Arsonval or PMMC meter on AC Measurements

tVtv sin)( max (5.36)

(a) (b)

5.8.2 Root Mean Square (RMS) versus Mean of AC Signal• The Root Mean Square (RMS), Vrms is the value of the DC voltage that if

connected to the circuit it will yield the same power as that of the AC one asillustrated in Fig.5.17 and described by:

• The average value of the rectified AC voltage, Vav is calculated by (5.35) as:

• The form factor is defined as Vrms / Vav and is calculated by (5.35) as:



)sin(707.02

)(1max

max

0

2 onlywavetheforVVdttvT

VT

rms

Fig. 5.17 (a) AC circuit; (b) Equivalent DC circuit; (c) Relation between Vrms and Vav

(a) (b) (c)

0max

maxmax

0

637.02sin1)(1 VVdVdttvT

VT

av

(5.37)

(5.38)

1.12

2/2

2/

max

max

VV

VVFactorForm

av

rms (5.39)

5.8.3 Full-wave rectifier (AC) Ammeters and Voltmeters• In order to use a PMMC meter to measure the alternating current, a rectifier

circuit must be used to rectify the AC into DC. One of the good rectifier circuitsis the Full-wave rectifier as shown in Fig. 5.16 (b).

• The PMMC pointer will indicate the average of the rectified signal which iso.637 for pure sinusoidal signal Equ.(5.38).

• Then to measure the rms, the scale must be calibrated to indicate the rms thatcorresponding to the average by multiplying the form factor Equ.(5.39).

• The circuit of the AC Ammeters is like the one of the AC voltmeter except thatinstead of connecting a series resistance, a shunt resistance is connected inparallel.

Disadvantage:• The 0.7 V drop across the diode limits the operation of this circuit; since we can

not measure voltage less than 1.4 V.Examples:

for Vin = 1.0 sin t gives no reading.for Vin = 2.0 sin t gives 0.6 V reading.for Vin = 200 sin t the reading is acceptable.



5.8.4 Precision Rectifier Circuits• Rectifier circuits were studied in Electronics I Course, where the emphasis was

on their application in power-supply design.• In such applications the voltages being rectified are usually much greater than

the diode voltage drop, rendering the exact value of the diode drop unimportantto the proper operation of the rectifier.

• Other applications exist, however, where this is no the case. For instance, ininstrumentation applications, the signal to be rectified can be of a very smallamplitude, say 0.1 V, making it impossible to employ the conventional rectifiercircuits. Also, in instrumentation applications the need arises for rectifier circuitswith very precise transfer characteristics.

• In this section we study circuits that combine diodes and op amps to implementa variety of rectifier circuits with precise characteristics.

• Precision rectifiers, which can be considered a special class of wave-shapingcircuits, find application in the design of instrumentation systems.



5.8.4 Precision Rectifier Circuits1. Precision Half-Wave Rectifier-The "Superdiode"• Fig. 5.18(a) shows a precision half-wave-rectifier circuit consisting of a diode

placed in the negative-feedback path of an op amp, with R being the rectifierload resistance.

• For vI > 0 yields that vO = vI and the diode is forward-biased and feedback loopis closed. Rectification is perfect even for small input voltages.

• For vI <0, the diode is cutoff and vO= 0. Primary sources of error are gain errorand offset error due to nonideal op amp.

• For negative input voltages, output voltage vA is saturated at negative limit.Large negative voltages across input can destroy unprotected op amps.Response time of circuit is slowed down due to slow recovery of internal circuitsfrom saturation.

• vO is rectified replica of vI without loss of voltage drop as in diode rectifiercircuit.

• The transfer characteristic of this circuit is shown in Fig. 18(b), which is almostidentical to the ideal characteristic of a half-wave rectifier.

Disadvantage:1. When vi goes negative and v0 = 0, the entire magnitude of vi, appears between

the two input terminals of the op amp. If this magnitude is greater than fewvolts, the op amp may be damaged unless it is equipped with what is called



5.8.4 Precision Rectifier Circuits1. Precision Half-Wave Rectifier-The "Superdiode"

"overvoltage protection" (a feature that most modern IC op amps have).2. when vi is negative, the op amp will be saturated. Although not harmful to the op

amp, saturation should usually be avoided, since getting the op amp out of thesaturation region and back into its linear region of operation requires sometime. This time delay will obviously slow down circuit operation and limit thefrequency of operation of the superdiode half-wave-rectifier circuit.



Fig. 5.18 (a) The “superdiode” precision half-wave rectifier and (b) its almost ideal transfer characteristic.

(a) (a)

5.8.4 Precision Rectifier Circuits2. Non-Saturating Precision Half-Wave Rectifier• An alternative precision rectifier circuit that does not suffer from the

disadvantages mentioned above is shown in Fig. 5.19.• For vI >0, vx is negative (one diode-drop below zero), D2 is forward biased,

current in R2 is zero, vO = 0, D1 is reverse biased. Feedback loop is closedthrough D2.

• For vI <0, vx is one diode-drop above output voltage, diode D1 turns on, D2 isoff. Circuit behaves as inverting amplifier with gain - R2 / R1. Feedback loop isclosed through D1 and R2.

• The major advantage of the improved half-wave-rectifier circuit is that thefeedback loop around the op amp remains closed at all times. Hence the opamp remains in its linear operating region, avoiding the possibility of saturation

• The associated time delay required to "get out" of saturation. Diode D2"catches" the op-amp output voltage as it goes negative and clamps it to onediode drop below ground; hence D2 is called a "catching diode."



5.8.4 Precision Rectifier Circuits2. Non-Saturating Precision Half-Wave Rectifier



Fig. 5.19 Non-Saturating Precision Half-Wave Rectifier: (a) The circuit; (b) For Vi > 0; (c) For Vi < 0; (d) its almost ideal transfer characteristic.

(c)

(b)(a)

(d)

5.8.4 Precision Rectifier Circuits3. Precision Full-Wave Rectifier• A Precision full-wave-rectifier circuit is shown in Fig.5.20(a).• For vI >0, positive input at A. The output of A2 will go positive, turning D2 on,

which will conduct through RL and thus close the feedback loop around A2 . Avirtual short circuit will thus be established between the two input terminals ofA2 , and the voltage at the negative-input terminal, which is the output voltage ofthe circuit, will become equal to the input. Thus no current will flow through R1and R2, and the voltage at the inverting input of A : will be equal to the input andhence positive. Therefore the output terminal (F) of A1 [ will go negative until A1saturates. This causes D1 to be turned off.

• For vI <0, when A goes negative. The tendency for a negative voltage at thenegative input of A, causes F to rise, making D1 conduct to supply RL andallowing the feedback loop around A1 to be closed. Thus a virtual groundappears at the negative input of A1 and the two equal resistances R1 and R2force the voltage at C, which is the output voltage, to be equal to the negativeof the input voltage at A and thus positive. The combination of positive voltageat C and negative voltage at A causes the output of A2 to saturate in thenegative direction, thus keeping D2 off.

• The overall result is perfect full-wave rectification, as represented by thetransfer characteristic in Fig. 5.20 (b).



5.8.4 Precision Rectifier Circuits3. Precision Full-Wave Rectifier



Fig. 5.20 (a) Precision full-wave rectifier; (b) Its transfer characteristic

5.8.4 Precision Rectifier Circuits4. A Precision Bridge Rectifier for Instrumentation Applications• The bridge rectifier circuit studied in Electronic I Course can be combined with

an op amp to provide useful precision circuits as shown in Fig. 5.21.• This circuit causes a current equal to |vA|/R to flow through the moving-coil

meter M.• Thus the meter Provides a reading that is proportional to the average of the

absolute value of the input voltage vA.• All the non-idealities of the meter and of the diodes are masked by placing the

bridge circuit in the negative-feedback loop of the op amp.• When vA is positive, current flows from the op-amp output through D1, M, D3,

and R.• When vA is negative, current flows into the op-amp output through R, D2, M,

and D4 . Thus the feedback loop remains closed for both polarities of vA. Theresulting virtual short circuit at the input terminals of the op amp causes areplica of vA to appear across R. The circuit of Fig. 5.21 provides a relativelyaccurate high-input-impedance ac voltmeter using an inexpensive moving-coilmeter.



5.8.4 Precision Rectifier Circuits4. A Precision Bridge Rectifier for Instrumentation Applications



Fig. 5.21 Use of the diode bridge in the design of an ac voltmeter.

SPRQ

5.9.1 Resistance Measurement using OhmmeterThis technique was already discussed in details in Section 5.5.

5.9.2 Resistance Measurement using Wheatstone Bridge•Wheatstone Bridge as shown in Fig. 5.22. consists of four arms, which have threeknown precision resistors and one unknown resistance.•The bridge balance is checked using a Galvanometer.•Very high sensitivity to resistance imbalance.•It is widely used in precision measurements of resistance from 1 to 106 .•Accuracy to ±0.2%.•To balance the bridge, change S till the galvanometer indicate zero. Then:

So: or


5.9 Resistance Measurement

dc VV

SIQI

RIPI

VV

VV

db

ad

cb

ac

3

2

4

1

QPSR

SQ

RP

dbcbadac VVVVIIII

&2

&1 3241

(5.40)

5.9.2 Resistance Measurement using Wheatstone Bridge


5.9 Resistance Measurement (Continued)

Fig. 5.22 Wheatstone Bridge (a) The circuit; (b) Precision resistor (P&Q); (c) Standard Resistor (S) and (d) Galvanometer

E+

-G

R S

P Q

I2

I3I4Standard

ArmUnknownArm

RatioArmsa

b

c d

I1

(a) (b)

(c)(d)

Example 5.15The Wheatstone Bridge as shown in Fig. 5.22 has the following arm values:P = 1.47k, Q = 500 and S = 2.0k. What value of unknown resistance Rbrings the bridge into the null conditions.

SolutionFrom (5.40), we have:


5.9 Resistance Measurement (Continued)

E+

-G

R S

P Q

I2

I3I4

StandardArmUnknown

Arm

RatioArmsa

b

c d

I1

1.47 2 5.880.5

PS k kR kQ k

5.9.2 Resistance Measurement using Wheatstone Bridge

Accuracy of Wheatstone BridgeSince

So, theoretically the relative error of R is summation of others, i.e.,

δR = δP + δQ + δS

Sensitivity of Wheatstone BridgeSensitivity of Wheatstone Bridge depends on :

1. The adjustable resistance (S)2. Galvanometer sensitivity

1. Sensitivity due to the adjustable resistance (S)Adjustable resistor and galvanometer determines the sensitivity of Wheatstone

Bridge. From equation

Adjustable resistor S affects the resolution of the measurement by


5.9 Resistance Measurement (Continued) 5.9.2 Resistance Measurement using Wheatstone Bridge

P SRQ

PSRQ

SQPR (5.41)

2. Sensitivity due to the Galvanometer sensitivity•The least detectable current of the galvanometer also determine the resolution ofthe measurement.•If the Galvanometer sensitivity (ΔIG) is defined as the maximum Galvanometercurrent that can not be detected.•The Sensitivity due to the Galvanometer sensitivity is given by:

where

and rg is the internal resistance of the Galvanometer.

Finally the sensitivity of Wheatstone Bridge is the maximum of:



(5.41)

GgTH IE

PRrRR

))((

PQPQ

RPRPSQRPRTH

//// E +

-G

R S

P Q

I2

I3I4Standard

ArmUnknownArm

RatioArmsa

b

c d

I1

SQPR

GgTH IE

PRrRR

))((

Sensitivity due ΔS Sensitivity due ΔIg

2. Sensitivity of Wheatstone BridgeExample 5.16

The parameters in a Wheatstone Bridge are: P = 2.5 k, Q = 1 k, S = 4k, E = 10V, minimum adjustable S = 0.1, rg = 1.2 k, and IG(min) = 2 A. Findthe value of R and the sensitivity or resolution of the Wheatstone Bridge.Solution

R = 2.5 × 4/1 = 10 k,R1 (due to S) = (2.5/1) (±0.1 ) = ±0.25 RTH= P//R + Q//S = 2 + 0.8 = 2.8 k

•Since R2 >> R1 , the sensitivity of the Wheatstone Bridge is R = R2 = 10.Usually determined by the sensitivity of the galvanometer,



(5.41)

10))((

)(2 GgTH

g IE

PRrRItodueR

• Due to the impurities of the coil material, the actual inductor modeled as:1. a pure inductor in series with a resistance (Rs and Ls) is used as a model of a

capacitor. This model is called series model as shown in Fig. 5.23(a).or

2. a pure inductor in parallel with a resistance (Rp and Lp). This model is calledparallel model as shown in Fig. 5.23(b).

• The series model can switch back to the parallel model or parallel model canswitch back to series model using the following formulas:


5.10 Inductance Measurement5.10.1 Inductor Equivalent Circuit

Fig. 5.23 Inductor equivalent circuit ); (a) series model and (b) parallel model

(a)(b)

2 2

2 2

s sp

s

s sp

s

R XRR

R XXX

2

2 2

2

2 2

p ps

p p

p ps

p p

R XR

X R

X RX

X R

s sX Lp pX L

(5.42)(5.43)

• The quality factor (Q) of an inductor is defined as :

• The larger Q mean best quality inductor (with the least leakage).

5.10.3 Measurement of Inductor Parameters Using Q Meter• The experiment setup to measure the coil parameters, Lx and Rx is shown in

Fig. 5.24. The signal source generates AC sine wave with amplitude veff = 1Vrms and known frequency f. The value of the variable capacitor is changed tillwe reach the resonance or the AC Voltmeter indicated maximum reading(vreading = 1 Vrms).


5.10 Inductance Measurement (Continued) 5.10.2 Inductor Quality Factor (Q)

(5.44)ps s

s s p

RX LQR R L

Fig. 5.24 Measurement of inductor parameters using Q meter

• The impedance of this circuit is given by:

• At resonance, Z = Rx because the reactive part is zero and we get:

• At resonance, vreading is maximum and we deduce the following:

• From (5.47), once Q is measured the inductance parameters are calculated as:


5.10 Inductance Measurement (Continued) 5.10.3 Measurement of Inductor Parameters Using Q Meter

(5.45)

vxx C

LjRZ

12

2 1||

vxx C

LRZ

or

vxvx CLC

L 11

QRL

RCCRCiv

x

x

xvvxvreading

111||

(5.46)

(5.47)

QCRand

CL

vx

vx

112 (5.48)

Example 5.17For the experiment setup shown in Fig. 5.24, if the input voltage source,Vrms = 1 V, its frequency, f =10 kHz, the voltmeter reading, Vreading = 1 Vrms andthe variable capacitance value, Cv = 10 µF. Calculate the coil parameters: Lxand Rx.

SolutionFrom (5.48) the coil parameters: Lx and Rx are calculated as follows:


5.10 Inductance Measurement (Continued) 5.10.3 Measurement of Inductor Parameters Using Q Meter

HxxxC

Lv

x

25)2(

101010)102(

112

3

6242

59.1210

1101010211

64 xxxxQCR

vx

• The experiment setup to measure the coil parameters, Lx and Rx is shown inFig. 5.25(a). The signal source generates AC sine wave with amplitude veff =Vmax and known frequency f.

• The reading of both AC Voltmers vR and vX are recorded. Knowing the value ofthe amplitude vin = Vmax . The coil parameters, Lx and Rx are determined usingthe phasor diagram as shown in Fig. 5.25(b) as.


5.10 Inductance Measurement (Continued) 5.10.4 Measurement of Inductor Parameters Using AC voltmeter

Fig. 5.25 (a) Measurement of inductor parameters using AC voltmeter; (b) The phasor diagram.

(a) (b)

xR

xRxRxR xvv

VvvxvvvvV2

coscos22

max22

222max

(5.49)

Rvi R

'sin XxLx vLiv 'cosXxRx viRv (5.50)

ivZ X

x 'cosxx ZR

'sin xx ZLj

Example 5.17For the experiment setup shown in Fig. 5.25, if the input voltage source,Vrms = 20 V, its frequency, f =10 kHz, R = 1 kΩ, the AC voltmeter reading acrossR, VR = 15 V and the AC voltmeter reading across the coil, VX = 10 V.Calculate the coil parameters: Lx and Rx.

SolutionFrom (5.49), (5.50) and Fig. 5.25, the coil parameters: Lx and Rx are calculatedas follows:

Anther solution


5.10 Inductance Measurement (Continued) 5.10.4 Measurement of Inductor Parameters Using AC voltmeter

oo

xR

xR

xxxvvVvv 76'&104

151022)20()15()10(

2cos

2222max

22

mHLLxxxvLiv xxXxLx 1076sin101021015'sin 33

16176cos101015'cos 3xxXxRx RxRxviRv

6601510

mAV

ivZ X

x

16076cos660'cosxx ZR

mHLLxZL xxxx 1076sin660102'sin 2

mAkV

RVi R 15

115

• The experiment setup to measure the coil parameters, Lx and Rx using ACbridge is shown in Fig. 5.26.

• The Maxwell Bridge is found to be suitable for measuring coils with lowQ (=ωLs/Rs). At balance, we have


5.10 Inductance Measurement (Continued) 5.10.5 Measurement of Inductor Parameters Using AC Bridges

Fig. 5.26 Measurement of inductor parameters using AC Bridges

(5.51)

(5.52)41

3S

RR RR

1 4 3SL R R C 3 3S

S

LQ R CR

11 3

3 4 4

S SR LR j R C jR R R

• Due to the impurities of the dielectric, the actual capacitor is modeled as:1. a pure capacitance in parallel with a resistance (Rp and Cp). This model is called

parallel model as shown in Fig. 5.27(a).or

2. a pure capacitance in series with a resistance (Rs and Cs) is used as a model ofa capacitor. This model is called series model as shown in Fig. 5.27(b).

• The series model can switch back to the parallel model or parallel model canswitch back to series model using the following formulas:


5.11 Capacitor Measurement 5.11.1 Capacitor Equivalent Circuit

Fig. 5.27 Capacitor equivalent circuit ); (a) parallel model and (b) series model.

(a) (b)

(5.44)(5.53)

2

2 2

2

2 2

p ps

p p

p ps

p p

R XR

X R

X RX

X R

2 2

2 2

s sp

s

s sp

s

R XRR

R XXX

where XP=1/(ωCP) and XS=1/(ωCS ).

• Dissipation factor (D) of capacitance is defined as :

• The smaller D mean best quality capacitor (with the least leakage).


5.11 Capacitor Measurement (Continued) 5.11.2 Capacitor Dissipation Factor (D)

(5.54)1p

s sp p p

XD R C

R C R

• The experiment setup to measure the capacitance parameters, Cx and Rx isshown in Fig. 5.28(a). The signal source generates AC sine wave withamplitude veff = Vmax and known frequency f.

• The reading of both AC Voltmers vR and vX are recorded. Knowing the value ofthe amplitude vin= Vmax. The capacitance parameters, Cx and Rx are determinedusing the phasor diagram as shown in Fig. 5.26(b) as.


5.11 Capacitor Measurement (Continued) 5.10.3 Measurement of Capacitance Parameters Using AC voltmeter

Fig. 5.28 (a) Measurement of capacitance parameters using AC voltmeter; (b) The phasor diagram.

(a) (b)

xR

xRxRxR xvv

VvvxvvvvV2

coscos22

max22

222max

(5.55)

Rvi R

'sin X

xCx v

Civ 'cosXxRx viRv (5.56)

ivZ X

x 'cosxx ZR

'sin xx ZLj

Example 5.17For the experiment setup shown in Fig. 5.26, if the input voltage source,Vrms = 20 V, its frequency, f =10 kHz, R = 2 kΩ, the AC voltmeter reading acrossR, VR = 10 V and the AC voltmeter reading across the capacitance, VX = 15 V.Calculate the capacitance parameters: Cx, Rx and Dissipation factor, D.

SolutionFrom (5.53), (5.54) and Fig. 5.26, the coil parameters: Lx and Rx are calculatedas follows:


5.11 Capacitor Measurement (Continued) 5.11.3 Measurement of Capacitance Parameters Using Q Meter

mAkV

RVi R 5

210

oo

xR

xR

xxxvvVvv 76'&104

15102)20()15()10(

2cos

2222max

22

kmAV

ivZ X

x 5515 72576cos3'cosxx ZR

FCkCx

ZC x

xx

x

4.592.276sin6601021'sin1

4

25.0105.572510102 93 xxxxxCRD ss

R3R4

C3

Rs

R1

E a b

Ls

• The experiment setup to measure the capacitance parameters, Cx and Rx usingAC bridge is shown in Fig. 5.29.

• The series capacitance bridge is found to be suitable for capacitors with highdielectric resistance (low D = ωCsRs). At balance, we have:


5.11 Capacitor Measurement (Continued) 5.11.4 Measurement of Capacitance Parameters Using AC Bridges

Fig. 5.29 Measurement of capacitance parameters using AC Bridges

(5.57)

(5.58)

1 2

3 4

Z ZZ Z

43

11 /1/1R

CjRR

CjR ss

3

41

RRRR s

4

31

RRCC s

11CRD

eeg383 measurement - chapter 5 - basic electric quantities measurement (new slides)

Documents