082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

32 Chapter 1 Review of algebraic techniques
Engineeringapplication1.7
Resistorsinparallel
When carrying out circuit analysis it is often helpful to reduce the complexity of a
circuit by calculating an equivalent single resistance for several resistors connected
togetherinparallel.Thissimplifiedversionoftheoriginalcircuitthenbecomesmuch
easier to understand. Figure 1.4 shows the simplest case of two resistors connected
together inparallel.
R 1
R 2
Figure1.4
Two resistorsin parallel.
The equivalent resistance,R E
, ofthis simple network isfound from the formula
1
R E
= 1 R 1
+ 1 R 2
Bycombining the fractions on the r.h.s.we see
1
R E
= R 2 +R 1
R 1
R 2
and hence
R E
= R 1 R 2
R 1
+R 2
Consider the case whenR 1
andR 2
are equal and have valueR. The equivalent resistance
then becomes
R E
=
RR
R +R = R2
2R = R 2
So
R E
= R 2 = 0.5R
Thereforetheeffectofputtingtwoequalresistorsinparallelistoproduceanoverall
equivalent resistance which ishalf thatof a singleresistor.
EXERCISES1.5
1 Classifyeachfraction aseitherproperorimproper.
(a)
(d)
x +2
x 2 +2
x 2 +2
x +2
(b)
(e)
2
x +2
x 2 +2
x 2 +1
(c)
(f)
2 +x
2
x 2 +1
x 2 +2
2 Classifyeachofthe following algebraic fractions as
properorimproper.
3t+1 10v 2 +4v−6 6−4t+t 3
(a)
t 2 (b)
−1 3v 2 (c)
+v−1 6t 2 +1
(d)
9t+1
t +1
(e)
100f 2 +1
f 3 −1
(f)
(x +1)(x +2)
(x +3) 3