Download ETEC 3501 CH 3 - Wheatstone - Delta - Y File

ETEC 3501
Chapter 3
Simple Resistive Circuits
3.5 Measuring Voltage and Current
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Ammeter – an instrument designed to measure current. It is
placed in series with the circuit elements whose current is
being measured.
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Voltmeter – an instrument designed to measure voltage. It is
placed in parallel with the with the circuit elements whose
voltage is being measured.
•
Ideal ammeters and voltmeters have no effect on the circuit
variables they are designed to measure.
•
That is an ideal ammeter has a resistance of zero Ohms and
an ideal voltmeter has a resistance of infinite Ohms.
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An ammeter and voltmeter connected in a circuit are shown in
Figure 3.21.
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The effects of an ideal ammeter and voltmeter are illustrated in
Figure 3.22.
Figure 3.21 An ammeter connected to measure the current in R1 and a voltmeter connected to measure the voltage across R2.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.22 A short-circuit model for the ideal ammeter, and an open-circuit model for the ideal voltmeter.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
•
Digital meters – measure analog values using an analog-todigital (A/D) converter. Are typically accurate over a broad
range of values.
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Analog meters – based on d’ Arsonval movement (Figure 3.23).
Accuracy is dependent on value of the elements being
measured.
•
An analog ammeter consists of a d’ Arsonval movement in
parallel with a resistor (Figure 3.24).
•
An analog voltmeter consists of a d’ Arsonval movement in
series with a resistor (Figure 3.25).
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A typical d’ Arsonval movement would be rated at 50 mV and
1 mA. So at those values the needle would be at full scale.
Figure 3.23 A schematic diagram of a d’Arsonval meter movement.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.24 A dc ammeter circuit.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.25 A dc voltmeter circuit.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
•
To measure a specific range of current or voltage, the value of
parallel or series resistors must be sized accordingly.
•
These resistances as well as the resistance of the d’Arsonval
movement affect the circuit being measured.
•
These effects are demonstrated in Example 3.5 and 3.6
Example 3.5 – Using a d’Arsonval Ammeter
Example 3.6 – Using a d’Arsonval Voltmeter
Assessment Problems
(a) i = 1V / 100 Ohm = 10 mA
(b) R m = 50 mV / 10 mA = 5 Ohm
i = 1V/ 105 Ohm = 9.524 mA
(a) v = 60V (75k/ (15k+75k)) = 50V
(b) R m = 150k Ohm
R’ = 75k || 150k = 50k Ohm
v = 60V(50k/(15k + 50k)) = 46.15 V
3.6 Measuring Resistance – The Wheatstone
Bridge
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The Wheatstone bridge is used to precisely measure
resistances of medium value (1 Ohm to 1M Ohm).
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Accuracies of + 0.1 % are possible.
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The bridge circuit is shown in Figure 3.26.
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R1, R2 , R3 are known resistors and RX is the unknown resistor.
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A d’Arsonval movement in the microampere range
(galvanometer) is used for current detection.
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The variable resistor (R3) is adjusted until zero current is
indicated by the galvanometer.
Figure 3.26 The Wheatstone bridge circuit.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
•
RX can them be determined using Equation 3.33.
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For the bridge to be balanced what must the voltage difference
be between point a and point b of Figure 3.27 ?
Figure 3.27 A balanced Wheatstone bridge (ig = 0).
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
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Looking further at the bridge circuit of Figure 3.26, if the
galvanometer is replaced with an equivalent resistance Rm , it
results in the circuit shown in Figure 3.28
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To reduce this bridge circuit to a single resistance across the
battery, methods beyond simple series and parallel reduction
must be employed.
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The interconnecting resistors can be reduced to a single
equivalent resistance by means of a delta-to-wye equivalent
circuit.
Figure 3.28 A resistive network generated by a Wheatstone bridge circuit.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.29 A ∆ configuration viewed as a π configuration.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.30 A Y structure viewed as a T structure.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.31 The ∆-to-Y transformation.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
•
The terminals of a, b, and c shown in Figure 3.31 must have
equivalent behavior in the delta configuration as well as the
wye configuration to be equivalent circuits.
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Equations 3.41 through 3.43 are derived using series and
parallel circuit simplification.
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Manipulation of Equations 3.41 through 3.43 yields the Yconnected resistors in terms of the delta-connected resistors.
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Expressions for the three delta resistors in terms of Yconnected resistors are given in Equations 3.47 through 3.49
Example 3.7 – Applying a Delta-to-Wye Transform
Assessment Problem
Figure 3.36 Model of a defroster grid.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Figure 3.37 A simplified model of the defroster grid.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Electric Circuits, Ninth Edition
James W. Nilsson • Susan A. Riedel
Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.