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S29 8 20/08/
TH E MANCH ESTER M ETR O PO LITAN UNIVER SITY
FACULTY O F SCIENCE AND ENGINEER ING
D EPA R TMENT O F ENGINEER ING AND TECH NO LO GY
SESSIO N 2001/
Exam ination for tBEng (H O NS) ELh eECTR ICA LAND ELECTR O NIC ENGINEER ING YEA R O NE
UNIT 64EE1107 : MATH EMATICS AND ELECTR O NICS
Th ursday 9 M ay 2002 2.00 pm to 4.00 pm
Instructions to Candidates A nsw er TW O questions from each section. Students are perm itted to use th e follow ing calculator m odels: Casio Fx Casio Fx83W A Casio FSh arp ELx85W A 531 A list of oth er perm itted m odels is h eld by th e invigilator.
20/08/02 continued
SECTIO N A
- (a) A com m on em itb ipolar transistor. Th e am plter transistor am plifier uses base potentiom eter and em itterifier h as b e e n designed using an NPN resistor b iasing in w h ich th e em itter resistor is NO T decoupled by a capacitor. From first principles and using h -param eters, derive an expression for th e sm allsignalvoltage gain of th e am plifier. A ssum e h (^) oe is so sm allth at it can b e neglected. Your answ er sh ould include a circuit diagram and a sm all signale q uivalent circuit. [14] (b ) D e sign a circuit of th e configuration describ e d in part (a) using th e follow ing inform ation: VCC = 10 V h (^) fe = 100 ICQ = 0.5m A VCQ = 5 V Use th e nearest preferred value (NPV)for resistors in your design. [11]
- (a) Th e output stage of a pow er am plifier dissipates 50 W. It com prises tw o identicaltransistors w ith θJC = 1. 5 °C/W. Th ey are both b olted onto one h eatsink. If th e m axim um perm issib le transistor junction tem perature is 150 °C and th e m axim um am b ient tem perature is 50 °C: (i) w h at m ust th e m axim um th erm alresistance of th e h eatsink , θCA , b e?
(ii) w h at is th e m axim um tem perature of th e h eatsink? [6][5] (b ) Th e th erm alresistance, θCA , of th e b e st availab le h eatsink for th e am plifier describ e d ab ove is 2 °C/W. Fan cooling is th erefore used w h ich effectively reduces θCA to 0. 7 °C/W. (i)(ii) W h atA pproxim atel w ould b e th e jy by w h atunction tem perature now? percentage can th e pow er output (^) b e [6] increased before th e m axim um junction tem perature is reach ed?[8]
20/08/02 Question 4. continued overleaf
SECTIO N B
- (a) D ifferentiate th e follow ing (i) y = 3x^2 – 5x + 3 (ii) y = sin 3x + cos 4x (iii) y = x + (^) x^32 − 21 x^ [8] (b ) State w h ich rule sh ould be used to differentiate th e follow ing (product, quotient or ch ain rule). [D o not com plete th e differentiations] (i) y = 1n(cos x) (ii) y = (^) cos^1 n^ xx (iii) y = 1n x. cos x (iv) y = cos^2 (1nx) [4] (c) D ifferentiate y = (^1) + 1 e 2 x [3]
Question 4. continued 4
20/08/02 continued
(d) Form ulae for a closed cylinder w ith b ase radius r (m etres) and h eigh t h (m etres). Surface area A = 2 πr^2 + 2 πrh (m 2 ) Volum e V = πr^2 h (m 3 ) A closed cylindricaltank is to b e m anufactured w ith a surface area of 50m 2.
(i) Sh ow th at th e volum e of th is tank can b e expressed as V = 25r - πr^3
h
r (ii) Calculate th e value of r th at gives a m axim um volum e. (iii) Calculate th e m axim um volum e. [10]
S29 8 20/08/
- (a) Evaluate th e follow ing definite integrals
(i) ∫^20 ( 6 x^2 − 2 x+ 5 )dx
(ii) ∫^ π 0 6 (cos 3 x − sin 2 x)dx
(iii) ∫ 14 ^ x^1 +^1 xdx [11]
(b ) O b tain ∫x cos2xdx [5]
(c) D e cide w h ich of th e follow ing differentialequations can b e solved by th e m eth od of separating th e variab les.
A : dxdy^ = e 3 x +^2 y B: dxdy^ = 3 x + 2 y C: dxdy^ =sin ( 3 x + 2 y )
[D o not attem pt to solve th e equations] [3] (d) Solve th e differentialequation.
dxdy^ =^ (^2 y −^1 )^ e^3 x
given th at y = 1 w h en x = 0. [You m ay leave your answ er as an im plicit relation b e tw een y and x]. [6]
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