Page 165 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

Example 5.3
The equivalent resistance R T of three resistors R 1 , R 2 , and R 3 in parallel is obtained from

1 1 1 1
------ = ------ + ------ + ------
R T R 1 R 2 R 3
Given that initially R = 5 Ω , R = 20 Ω , and R = 4 Ω , compute the change in R T if R 2 is
3
2
1
increased by 10 % and R 3 is decreased by 5% while R 1 does not change.
Solution:
||
||
The initial value of the equivalent resistance is R = 5204 = 2 Ω .
T
Now, we treat R 2 and R 3 as constants and differentiating R T with respect to R 1 we obtain

∂R
∂R
1
1
T
T
------
– ---------------- = – ------ or ---------- = ⎛ ⎝ R T ⎞ 2
R ⎠
R 2 ∂R 1 R 2 1 ∂R 1 1
T
Similarly,
∂R ⎛ R T ⎞ 2 ∂R ⎛ R T ⎞ 2
T
T
---------- = ------ and ---------- = ------
∂R 2 ⎝ R ⎠ 2 ∂R 3 ⎝ R ⎠ 3
and the total differential dR T is
∂R ∂R ∂R R T ⎞ 2 R T ⎞ 2 R T ⎞ 2
T
T
T
R = ----------dR + ----------dR + ----------dR = ⎛ ⎝ ------ dR + ⎛ ⎝ ------ dR + ⎛ ⎝ ------ 3 dR
R ⎠
R ⎠
R ⎠
2
T
1
3
1
2
∂R
∂R
∂R
2
3
1
2
1
By substitution of the given numerical values we obtain
2
--
--
)
–
dR = ⎛ ⎝ 2 - ⎞ 2 0 () + ⎛ ⎝ ------ ⎞ 2 2 () + ⎛ ⎝ 2 - ⎞ 2 – ( 0.2 = 0.02 0.05 = – 0.03
20 ⎠
5 ⎠
4 ⎠
T
Therefore, the eequivalent resistance decreases by 3% .
Example 5.4
In a series RC electric circuit that is excited by a sinusoidal voltage, the magnitude of the imped-
2
ance is computed from Z = R + X C 2 . Initially, R = 4 Ω and X C = 3 Ω . Find the change
Z
R
Z
in the impedance if the resistance is increased by 0.25 Ω (6.25 % ) and the capacitive reac-
–
tance X C is decreased by 0.125 Ω ( 4.167% ).
5−4 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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