Errors in Electrical Measurements
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1 1 Errors in Electrical Measurements Systematic error every times you measure e.g. loading or insertion of the measurement instrument Meter error scaling (inaccurate marking), pointer bending, friction, no calibration Random error temperature effect, noises (unwanted signals) Reading error parallax, read a wrong scale Recording error for many measured values 2 1
2 Background Electric charge, e = Coulombs Current, I = dq/dt Coulomb/sec = Amps Voltage, the difference in electrical potential between two points (Joules/Coulomb) Power, P = IV Watts Energy, E = Pt Units (kw hr) Ohm s Law, V = IR 3 V 1 Background (Cont d) Kirchhoff s Current Law (KCL) I 1 I 2 I 3 = 0 I A loop is any closed I 2 I I 3 1 Nodal Kirchhoff s Voltage Law (KVL) V 1 I 1 R 1 V 2 I 2 R 2 = 0 I 1 R 1 V 2 Mesh or Loop R 2 A loop is any closed path in a circuit, in which no node is encountered more than once. A mesh is a loop that has no other loops inside of it. A supermesh occurs when a current source is contained between two 4 essential meshes. 2
3 Background (Cont d) Thévenin s Theorem Linear Circuit - Source - Load A B R L R th Vth Equivalent Circuit A B R L Open circuit to find V th Norton s Theorem Short circuit to find I N A R I N N B Equivalent Circuit R L 5 Background (Cont d) Example No Load Equivalent Source R 1 R 1 V 1 V 1 R 2 R 2 No Source Equivalent Load V eq = R 2 V 1 R 1 R 2 Shorted R 1 R 2 R eq = R 1 R 2 = R 1 R 2 R 1 R 2 6 3
4 Background (Cont d) Ideal Voltage Source Vs. Ideal Current Source I I = V fixed / R 0 V 0 I V = I fixed R V Superposition Theorem (for linear resistive network containing several sources) Voltage source Short Circuit 0 V Current source Open Circuit 0 A Vector summation of the individual voltage or current caused by each separate source 7 Example Background (Cont d) 3V 10k 3V 10k 6V 5k V out V out1 = 3V/(10k 5k ) 5k = 1V i 1 5k 10k V out = V out1 V out2 = 1V 4V = 3V 6V i 2 5k V out2 = 6V/(10k 5k ) 10k = 4V 8 4
5 Background (Cont d) The Maximum Power Transfer Theorem R s A V s I L R B L V L If V s and R s are fixed, when I L then V L Maximum power is at R L = R s Therefore V L = V s / 2 and I L = V s / 2R L P L = (I L ) 2 R L = (V s / 2R L ) 2 R L = V s 2 / 4R L 9 Ammeter Loading For current measurement, open circuit to put an ammeter. V DC Circuit R I I = V/R V R I A A R A I A = V/(RR A ) 10 5
6 Ammeter Loading (Cont d) Loading error, I A I = V/(RR A ) V/R = VR V(RR A A) R(RR A ) = VR A R(RR A ) % Loading Error = (I A I)/I 100% % = R A 100% RR A It means if R A then error 11 Ammeter Loading (Cont d) The ideal ammeter has zero internal resistance, so as to drop as little voltage as possible as electrons flow through it. Example without the ammeter, 12 6
7 Ammeter Loading (Cont d) with the ammeter, 13 Ammeter Loading (Cont d) One ingenious way to reduce the impact that a currentmeasuring device has on a circuit is to use the circuit wire as part of the ammeter movement itself. All current-carrying wires produce a magnetic field, the strength of which is in direct proportion to the strength of the current. 14 7
8 Ammeter Loading (Cont d) By building an instrument that measures the strength of that magnetic field, a no-contact ammeter can be produced. Such a meter is able to measure the current through h a conductor without even having to make physical contact with the circuit, much less break continuity or insert additional resistance. 15 Ammeter Loading (Cont d) More modern designs of clamp-on ammeters utilize a small magnetic field detector device called a Hall-effect sensor to accurately determine field strength. 16 8
9 Ammeter Loading (Cont d) Some clamp-on meters contain electronic amplifier circuitry to generate a small voltage proportional to the current in the wire between the jaws, that small voltage connected to a voltmeter for convenient readout by a technician. 17 Voltmeter Loading For voltage measurement, voltmeter is placed in parallel with the circuit element. V DC Circuit R V I V V R V V R V Thévenin s R V th R th I th 18 9
10 Voltmeter Loading (Cont d) I th = V th / ( R th R V ) V V = I th R V = V th R V / ( R th R V ) Error = V V V th = R V 1 V th R th R V % Error = (V V V th )/V th 100% = R th 100% R th R V It means if R V then error 19 Voltmeter Loading (Cont d) The ideal voltmeter has infinite resistance, so that it draws no current from the circuit under test. Example without the voltmeter, 20 10
11 Voltmeter Loading (Cont d) with the voltmeter, 21 Ohmmeter Loading Using a circuit within the instrument 22 11
12 Analog Ammeters & Voltmeters The users have to read the meter manual to find the basic accuracy specifications. Moving-coil-based analog meters are characterized by their full-scale deflection (f.s.d.) and effective resistance of the meters (few -k ) Typical meters produce a f.s.d. current of 50 A - 1 ma (this is not the ranges of the meters) 23 Percentage of FSD The accuracy of analog ammeters and voltmeters are quoted as a percentage of the full-scale deflection of pointer on the linear scale for all reading on the selected range. e.g. an ammeter having accuracy 2% f.s.d. within the range 0-10 A, Relative error 2% of 10 A Absolute error 0.2 A When a measured value is close to full scale, or at least above 2/3 of full scale, the published accuracy is meaningful
13 Percentage of Midscale Moving-coil-based ohmmeters is characterized by their midscale deflection of the pointer on nonlinear scale. What figure lies exactly between infinity and zero? eg e.g. 3% of midscale Relative error 3% of 9 k Absolute error 0.27 k 25 Percentage of DMM Reading In the case of digital instruments, the accuracy is generally quoted as the percentage of the reading 1 digit. e.g. 0.5% Reading 1 Digit when you read a voltage 1.2 V with a digit of 0.03 V (depending on the range selected) error = 0.5% of 1.2 V 0.03 V = V absolute error = 3% of reading relative error 26 13
14 Potential Meter Before the moving coil meter, potentiometer was used in measuring the voltage in a circuit Calculated from a fraction of a known voltage (on resistive slide wire) Based on voltage divider (linearly) By means of a galvanometer (no deflection or null-balance) 27 Calibration of Potentiometer galvanometer V s x 1 x 2 V std_cell (known) The end of a uniform resistance wire is at x 1 The sliding contact or wiper is then adjusted to x 2 The standard electrochemical cell whose E.M.F. is known (e.g volts for Weston cell) 28 14
15 Calibration of Potentiometer (Cont d) Adjust supply voltage V s until the galvanometer shows zero or detect t null (voltage on R 2 equals to the standard cell voltage or no current flows through the galvanometer) x 2 /x 1 V s = V std_cell V s = x 1 /x 2 V std_cell 29 Calibration of Potentiometer (Cont d) An extremely simple type of null detector, instead of the galvanometer, is a set of audio headphones, the speakers within acting as a kind of meter movement. When a DC voltage is initially applied to a speaker, the resulting current through it will move the speaker cone and produce an audible click. Another click sound will be heard when the DC source is disconnected. The technician would repeatedly press and release the pushbutton switch, listening for silence to indicate that the circuit was balanced
16 Calibration of Potentiometer (Cont d) The headphone s sensitivity may be greatly increased by connecting it to a device called a transformer. Astep-down transformer converts low-current pulses, created by closing and opening the pushbutton switch while connected to a small voltage source, into highercurrent pulses to more efficiently drive the speaker cones inside the headphones (N p /N s = I s /I p ). 31 Measuring of Potentiometer galvanometer V s x 1 x 3 V unknown (to be measured) Slide the wiper (change x 3 3) until the galvanometer shows zero again V unknown = x 3 /x 1 V s = x 3 /x 2 V std_cell 32 16
17 Rheostat Variable resistor with two terminals 33 Loading for a Potentiometer Resistor Fixed Adjustable Potentiometer
18 Loading of a Potentiometer (Cont d) V s p p x x R L V th R th I R L Voltage Divider V th = ( x / p ) V s Thévenin s p x x Superposition R L 1 = 1 1 R th (x/p)r p ((p-x)/p)r p R th = (x/p)r p (1 x/p) 35 Loading of a Potentiometer (Cont d) V th = I (R th R L ) (x/p)v s = I [ (x/p)(1 x/p)r p R L ] I = (x/p)v s (x/p)(1 x/p)r p R L V L = IR L = (x/p)v s R L (x/p)(1 x/p)r p R L = (x/p)v s (R p /R L )(x/p)(1 x/p) 1 It means the relationship between V L and x is nonlinear! 36 18
19 Loading of a Potentiometer (Cont d) Error, V th V L = (x/p)v s 1 1 (R p /R L )(x/p)(1 x/p) 1 = (x/p)v s (R p /R L )(x/p)(1 x/p) (R p /R L )(x/p)(1 x/p) 1 If R L >> R p then R p /R L 0 Error (x/p)v s (R p /R L )(x/p)(1 x/p) V s (R p /R L )[(x/p) 2 (x/p) 3 ], R L error % Error (R p /R L )[(x/p) (x/p) 2 ] 100% 37 Loading of a Potentiometer (Cont d) Let d(error)/dx = V s (R p /R L )[2x/p 2 3x 2 /p 3 ] = 0 We get 2x/p 2 = 3x 2 /p 3 x/p = 2/3 for the maximum Therefore, the max error = V s (R p /R L )[(2/3) 2 (2/3) 3 ] V s (R p /R L ) and the max % error 22.2 (R p /R L ) % 38 19
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