chapter 3: analysis of simple resistive circuits series resistance
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Chapter 3 EGR 271 – Circuit Theory I. 1. Reading Assignment: Chapter 3 in Electric Circuits, 9 th Edition by Nilsson . Chapter 3: Analysis of simple resistive circuits Series Resistance - PowerPoint PPT PresentationTRANSCRIPT
Chapter 3: Analysis of simple resistive circuitsSeries ResistanceTwo or more resistors in series can be replaced by a single equivalent resistance, Req, as shown below.
Use KVL to show that: eq 1 2 NR = R + R + + R
RN
R1 R2 R3I
V
+
_
. . .
Req
I
V
+
_
Reading Assignment: Chapter 3 in Electric Circuits, 9th Edition by Nilsson
1Chapter 3 EGR 271 – Circuit Theory I
Example: Determine the current I in the circuit shown below.
8V
420V
7
3 5
2
6V
15V
I
2Chapter 3 EGR 271 – Circuit Theory I
Parallel ResistanceTwo or more resistors in parallel can be replaced by a single equivalent resistance, Req, as shown below.
R1 RNR3R2
. . .
. . .
Req
I
V
+
_
I
V
+
_
Use KCL to show that: eq
1 2 N
1R 1 1 1
R R R
3Chapter 3 EGR 271 – Circuit Theory I
Example: Find the equivalent resistance for the circuit shown below.
48 7236
Example: Find the voltage V in the circuit shown below.
120 60 240803A 8A7AV
+
_
4Chapter 3 EGR 271 – Circuit Theory I
Special Case - 2 Resistors in ParallelAn easy formula can be used to calculate the equivalent of two resistors in parallel.
eq
1 2 N
1R 1 1 1 R R R
R2R1
Show that the general formula for parallel resistance can be simplified as follows:
1 2eq
1 2
R R productR R R sum
(For 2 or more resistors) (For 2 resistors only)
5Chapter 3 EGR 271 – Circuit Theory I
Example: Find the equivalent resistance in each case using the formula for two resistors only.
Equal Value Resistors• Show the result of two equal value resistors in parallel.• Show the result of N equal value resistors in parallel.
7030 3080 60
6Chapter 3 EGR 271 – Circuit Theory I
Example: Find the current I in the circuit shown below.
Series/Parallel Combination of ResistorsCircuits can sometimes be reduced through repeated series or parallel combinations of resistors.
160
1.2k960
40 180
240100V
I
7Chapter 3 EGR 271 – Circuit Theory I
Voltage Division• Applies to series circuits only.• Voltage divides proportionally between series R’s with the largest R getting
the most voltage.• Show that
ii S
eq
RV VR
Two useful tools for analyzing resistive circuits• Voltage Division (or the Voltage Divider Law)• Current Division (or the Current Divider Law)
8Chapter 3 EGR 271 – Circuit Theory I
Example: Use voltage division to find the voltage V1 in the circuit shown below.
140
80
40
60
240V
V1
+
_
Example: Use repeated voltage division to find the voltage V2 below.
120100
8040V 600 V2
+
_
9Chapter 3 EGR 271 – Circuit Theory I
Example: Find the voltage V1 using repeated voltage division.
+
V1
_
600
40
480V 240 900
30080
10Chapter 3 EGR 271 – Circuit Theory I
Current Division • Applies to parallel circuits only.• Current divides between parallel R’s with the smallest R getting the most
current.• Show that: total
i Si
RI I (general form)R
11Chapter 3 EGR 271 – Circuit Theory I
150 6004020A
I1
Example: Find the current I1 using current division.
12Chapter 3 EGR 271 – Circuit Theory I
Current Division - Special Case: Two Resistors Only Show that for two resistors only the general form of current division can be expressed as follows.
totali S
i
RI I (general form)R
oppositei S
i 2
RI I (form for 2 resistors only)
R R
12mA 300100
I1
Example: Find the current I1 using current division (special form for two R’s only).
13Chapter 3 EGR 271 – Circuit Theory I
Example: Find the current I1 using repeated current division.
I1
600
40
200V 240 900
30080
14Chapter 3 EGR 271 – Circuit Theory I
Example: Find I1, V2, I3, I4, V5, and V6 in the circuit shown below.
200V
100
260
500
800
300160200
2000
420
I3
V2
I1+
_
V5
+
_
I4
V6
+
_
15Chapter 3 EGR 271 – Circuit Theory I
Bridge CircuitsOne type of resistive circuit that cannot be simplified through series and/or parallel combinations is the “bridge circuit.” A bridge circuit is shown below (drawn twice). Study the circuit to verify that there are no series resistors and no parallel resistors.
R5
R1
R3 R4
R0
R2
Vs
R5
R1
R3 R4
R0
R2
Vs
16Chapter 3 EGR 271 – Circuit Theory I
Delta-to-Wye (-Y) and Wye-to-Delta (Y-) TransformationsBridge circuits contain resistors that are connected in delta () and wye (Y) configurations. One way to analyze this circuit is to use a -Y or a Y- transformation.
Y and connections of resistors are shown below:
R 1 R 2
R 3
a b
c Wye Circuit
R c
R a R b
a b
c
Delta Circuit
17Chapter 3 EGR 271 – Circuit Theory I
If the wye and delta circuits to be equivalent, then they should provide the same resistance between each pair of terminals (a-b, b-c, and c-a).
Development: Determine the resistance seen at each set of terminals and equate them as follows:Ra-b (Delta) = Ra-b (Wye)
Rb-c (Delta) = Rb-c (Wye)Rc-a (Delta) = Rc-a (Wye)
R 1 R 2
R 3
a b
c Wye Circuit
R c
R a R b
a b
c
Delta Circuit
18Chapter 3 EGR 271 – Circuit Theory I
Solving the equations on the previous page yields the following relationships:
1 2 2 3 3 1a
1
1 2 2 3 3 1b
2
1 2 2 3 3 1c
3
Y- Conversion Equations:
R R R R R R R R
R R R R R R R R
R R R R R R R R
b c1
a b c
c a2
a b c
a b3
a b c
-Y Conversion Equations:
R R R R R R
R R R R R R
R R R R R R
R 1 R 2
R 3
a b
c Wye Circuit
R c
R a R b
a b
c
Delta Circuit
Note: The equations above must be used along with the circuit diagrams shown. The labeling of the resistors and nodes in the diagrams is critical.
19Chapter 3 EGR 271 – Circuit Theory I
Example: Determine I in the circuit shown below using: a) -Y conversion
+ _
30
10
I
100 V
20
50
40
25
20Chapter 3 EGR 271 – Circuit Theory I
Example: Determine I in the circuit shown below using: b) Y- conversion
+ _
30
10
I
100 V
20
50
40
25
21Chapter 3 EGR 271 – Circuit Theory I
The “balanced bridge”A bridge circuit is “balanced” when the following relationship exists:
R1R4 = R2R3
When the bridge is balanced, it can be shown that
I5 = 0
R5
R1
R3 R4
R0
R2
Vs I5
22Chapter 3 EGR 271 – Circuit Theory I
Applications of bridge circuits An unknown resistance can be determined using a bridge circuit as follows:• Replace R4 with an unknown resistor• Place an ammeter in series with R5
• Adjust R1 until the bridge is balanced (i.e., I5 = 0)• Solve for R4
+ _
R1 I5
R5 Runknown R3
R2 1 4 2 3
2 3unknown 4
1
R R = R R
R RR = R = R
or
A bridge circuit with resistive values, sometimes called a Wheatstone bridge, can be used to determine unknown resistor values.Other bridge circuits (such as the Scherring bridge) can be used to determine unknown values of capacitors and inductors in a similar manner.
23Chapter 3 EGR 271 – Circuit Theory I
Impedance Bridge A bridge circuits used to find unknown values of resistance, inductance, or capacitance are sometimes called an impedance bridge.An impedance bridge is a common piece of test equipment and will be used in lab classes such as EGR 262 and EGR 270.
An older style impedance bridge that involved adjusting knobs until the needle indicated that the bridge was balanced. Ref: http://www.testequipmentconnection.com/products/908
A newer microprocessor-controlled impedance bridge.
http://www.testequipmentconnection.com/products/914
24Chapter 3 EGR 271 – Circuit Theory I
Strain gaugesStrain gauges are often attached to surfaces to measure forces as the surface moves (such as in the deflection of an airplane wing or a beam). The force can be determined from the amount of stretch in the wire by measuring the resistance of the wire with a bridge circuit. As a wire stretches note that resistance increases since
A
l R
Reference: http://www.allaboutcircuits.com/vol_1/chpt_9/7.html
25Chapter 3 EGR 271 – Circuit Theory I
Strain gauge mounted on a vehicle suspension system component for fatigue and durability testing
Ref: http://blog.prosig.com/2006/05/17/fatigue-durability-testing/
26Chapter 3 EGR 271 – Circuit Theory I
Strain gauge mounted on a human bone to study walking and running impact Biomechanics lab at Iowa State University
Ref: http://www.kin.hs.iastate.edu/research/labs/biomechanics_lab.htm
27Chapter 3 EGR 271 – Circuit Theory I