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

13.2 Elementary integration 437
4.00
3.00
2.00
1.00
y 0
0
–1.00
–2.00
–3.00
–4.00
290 291 292 293 294 295 296 297
t (ms)
y 1
298 299 300
Figure13.5
Output froman operational
amplifier integrator circuit with
v 1 being asquare wave input
andv 2 =v 3 =0.
Assumingi A
isnegligible, then
i f
=i 1
+i 2
+i 3
(13.7)
For the capacitor,
i f
=−C dv o
dt
(13.8)
Thenegativesignisaresultofthedirectionchosenfori f
.CombiningEquations(13.6)--
(13.8) yields
v 1
+ v 2
+ v 3
= −C dv o
R 1
R 2
R 3
dt
∫
v
v o
=− 1
R 1
C + v 2
R 2
C + v 3
R 3
C dt
The circuit therefore acts as an integrator. The minus sign in the integration is a result
of the circuit design which isknown as aninverting circuit. Atypical output from this
circuitisgiven inFigure 13.5.
13.2.3 Integrationoftrigonometricfunctions
The trigonometric identities given in Table 3.1 together with Table 13.1 allow us to
integrate a number of trigonometric functions.
Example13.5 Evaluate
(a) ∫ cos 2 t dt
(b) ∫ sin 2 t dt