Engineering 43
Chp 2 Tutorial
Problem 2-24
Solution
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
ENGR/MTH/PHYS25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Amplifier Driving Speaker
 Consider an Amplifier Circuit
connected to a Speaker
Speaker
a.k.a. the
“LOAD”
Driving
Circuit
a.k.a. the
“SOURCE”
ENGR/MTH/PHYS25: Computational Methods
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Circuit Simplification
 Thévenin’s Equivalent Circuit Theorem (c.f. ENGR43)
Allows Tremendous Simplification of the Amp Ckt
RS
Thevenin
ENGR/MTH/PHYS25: Computational Methods
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VS
+
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Maximum Power Transfer
 The Simplest Model
for a Speaker is to
Consider it as a
RESISTOR only
 Since the “Load”
Does the “Work” We
Would like to
Transfer the
Maximum Amount of
Power from the
“Source” to the
“Load”
ENGR/MTH/PHYS25: Computational Methods
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RS
VS
+
─
RL
SPEAKER
MODEL
BASIC MODEL FOR THE
ANALYSIS OF POWER
TRANSFER
 Anything Less
Results in Lost
Energy in the
Driving Ckt in the
form of Heat
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
The Final Ckt Model
Source
 Driving Circuit
ENGR/MTH/PHYS25: Computational Methods
5
Load
 The Speaker
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Electrical Power Physics
 For ANY Electrical
Device with a:
• Potential, V, across it
• A current, I, thru it
+ V
-
P  VI
 Now OHM’s Law
Relates the Voltageacross and CurrentThru a resistor
i
I
+
 Then the Power
Used by the Device:
v
ENGR/MTH/PHYS25: Computational Methods
6
R
V  RI
-
Circuit Represent ation
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Voltage Division
 Recall the Reduced
Ckt Model
Source
Load
 This SINGLE LOOP
Ckt effectively
divides VS across
RS and RL
ENGR/MTH/PHYS25: Computational Methods
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 Analysis of this
“Voltage Divider” Ckt
produces a
Relationship
between VS & VL
RL
VL  VS
RS + RL
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Summary to This Point
 What we KNOW
• By Thévenin
Analysis of the
Driving Ckt we
determined VS & RS
 Note that VS & RS
are FIXED and
beyond our Control
as Speaker
Designers
ENGR/MTH/PHYS25: Computational Methods
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 The Speaker
Designer CAN,
however control the
Load Resistance, RL
 Thus Our Goal
Find RL such the Driving
Ckt Operates at the
Highest Efficiency; i.e.,
we seek RL that will
MAXIMIZE Driver→Load
Power Transfer
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
Analytical Game Plan
 Goal Find RL to Maximize PL(RL)
 From the Physics we Know
PL  VL I L
VL  RL I L
RL
VL  VS
RS + RL
ENGR/MTH/PHYS25: Computational Methods
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1
2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
The MATLAB Problem
Source
Load
RL
2
V
 And PL 
2 S
RS + RL 
 Define Transfer
RL
Ratio, r r 
RS + RL 2
 RS = 10Ω, 15Ω,
20Ω, 25Ω
 Then
 RL = 10Ω, 15Ω,
20Ω, 25Ω, 30Ω
 So to Maximize PL
need to Maximize r
ENGR/MTH/PHYS25: Computational Methods
10
PL  rVS2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
MATLAB Game Plan
 Concept  Test ALL possible Resistor
Combinations then Check for Best
 Because we have a small number of
allowable values for RS and RL, the
most direct way to choose RL is to
compute the values of r for each
combination of RS and RL.
• Since there are four possible values of RS
and five values of RL, there are
4(5) = 20 combinations.
ENGR/MTH/PHYS25: Computational Methods
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
MATLAB Plan (2)
 We can use an array operation in
MATLAB to compute r for each of these
combinations by defining two 5 × 4
2D-Arrays R_L and R_S.
• The five rows of R_L contain the five
values of RL, and its four columns are
identical.
• The four columns of R_S contain the four
values of RS, and its five rows are
identical.
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
RL
r
2
RS + RL 
MATLAB Plan (3)
 The Arrays we Need
10
15

R L  20

 25
30
10
15
20
25
30
10
15
20
25
30
10 
15 
20

25
30 
10
10

R S  10

10
10
15
15
15
15
15
20
20
20
20
20
25
25
25

25
25
• These Arrays MUST have the same size so
that we can perform element-by- element
operations with them.
ENGR/MTH/PHYS25: Computational Methods
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
The MATLAB Code
% Bruce Mayer, PE
% ENGR22 * 20Jan07 * Rev. 13Sep08
% Prob 2.24 * file Demo_Prob2_24_0809.m
%
% Since all COLUMNS in RL are the same, Define one Col and
Replicate in Row Vector
% Define RL col
a = [10;15;20;25;30];
% Make Array R_L by using a in 4-element Row Vector
R_L = [a,a,a,a]
%
% Since all ROWS in RS are the same, Define one Row and
Replicate in Col Vector
% Define RS row
b = [10,15,20,25];
% Make Array R_S by using a in 5-element Col Vector
R_S=[b;b;b;b;b]
%
% Use Element-by-Element Operations to Calc r
%% First Sum RS & RL for the 20 combos
Rsum = R_S+R_L
%% Now sq the 20 sums
RsumSq = Rsum.^2
% need "dot" as this is element-by-element
%% Finally Divide RL by SQd sums
r = R_L./RsumSq
%
% Use the max(A) command to find the max value in each COL,
and the ROW in in Which the max Values Occurs
[max_ratio, row] = max(r)
The .m-File OutPut
R_L =
10
15
20
25
30
10
15
20
25
30
10
15
20
25
30
10
15
20
25
30
15
15
15
15
15
20
20
20
20
20
25
25
25
25
25
R_S =
10
10
10
10
10
25
30
35
40
45
30
35
40
45
50
35
40
45
50
55
ENGR/MTH/PHYS25: Computational Methods
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RL =
RL =
RL =
RL =
RL =
10
15
20
25
30
RS = 10
0.0250
0.0240
0.0222
0.0204
0.0187
RS = 15
0.0160
0.0167
0.0163
0.0156
0.0148
RS = 20
0.0111
0.0122
0.0125
0.0123
0.0120
RS = 25
0.0082
0.0094
0.0099
0.0100
0.0099
0.0167
0.0125
0.0100
max_ratio =
0.0250
row =
Rsum =
20
25
30
35
40
r =
1
2
3
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
3
3
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt
ENGR/MTH/PHYS25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_2-24_Solution.ppt