SOME USEFU MAHEMAICS
SOME USEFU MAHEMAICS I is esy o mesure n preic he behvior of n elecricl circui h conins only c volges n currens. However, mos useful elecricl signls h crry informion vry wih ime. Since he signl is consnly vrying wih respec o ime, new echniques mus be evelope since he shpe of hese signls my be ifferen n we my wish o compre circui behvior uner ifferen ypes of ime vrying inpus. he suen probbly knows he four ifferen ypes of c volges n currens h cn exis in circui. hey re pek o pek, pek, rms (or effecive), n verge. Pek n pek o pek re mesure wih n oscilloscope. hey my be re irecly. Equions for convering beween pek n rms, n pek n verge re well known. However, i is of ineres o see how we obine hese relionships. e us efine volge s h(). I hs mximum or pek vlue of IP. he equion o represen sinusoil curren woul be: h() IP sin f Since f, hen f. h() I P sin AVERAGE VAUE he verge vlue of ime vrying curren wveform over he perio is he vlue h c curren woul be require o eliver uring he sme perio. herefore he verge curren IAV is: IAV re uner he curve perio I AV IP sin I P I AV I AV sin I I - P AV [ cos ] I I - P AV [ cos ] IP IAV - [() - ()] his inices h he verge vlue of sinusoil wveform is zero. Roo Men Squre or Effecive Vlue We jus sw h he verge vlue of sinusoil wveform is zero. his oes no provie much useful informion. Anoher meho o efine he chrcerisics of wveform is he RMS vlue. his is imporn since his reles o he power elivering cpbiliy of he wveform. RMS is Pge
SOME USEFU MAHEMAICS lso clle he effecive vlue since c source woul eliver he sme power s he rms vlue of he ime vrying source. he rms vlue of wveform is foun by king squring he insnneous vlue for he wveform ech insn n king he verge (men) of hese vlues. Since he iniviul vlues re squre, here re no negive conribuions n he sum cnno be zero unless he inpu is zero for ll ime. Finlly, we ke he roo of his men vlue o obin he resul. h() IP sin f. Irms Irms Irms Irms P I sin I P sin I P sin 4 [ - ( )] 4 sin 4 I P ( - ) 4 Irms IP.77 IP he rms vlue is he mesuremen h we obin from sinusoil source when i is mesure wih meer. Unless oherwise inice, ll c volges n curren re rms vlues. Euler s Formul jθ e cos θ + j sin θ -jθ e cos θ - j sin θ < > < b > Now < > n < b > n solve for cos θ cos θ ( e + e ) cos θ e + e Subrc < b > from < > n solve for sin θ sin θ j ( e - e ) sin θ - j e - e Pge
SOME USEFU MAHEMAICS INEGRAION All consns of inegrion re ignore sin - cos sin ω sin ω - 4ω cos sin sin ω cos ω x + n - ln cos 4ω e co ln sin e ( - ) sin ω - cos ω e ω e ( - + 3 ) cos ω sin ω ω sin ω cos ω n+ n, n - ω ; > n+ sin ω ln ; ω ; ω < eu e e ; m + n even eu sin mx cos nx m ; m + n o ω - ω m + n sin - ω sin mx sin nx x cos mx cos nx x ω - ω n where m & n re inegers n m n + ω x sin x x ( sin x - x cos x ) x cos x x ( cos x + x cos x ) x sin x x 3 ( x sin x + cos x - x sin x ) x cos x x 3 ( x cos x - sin x + x sin x ) x x e e sin bx x ( sin bx - b cos bx) + b x x e e cos bx x ( cos bx + b sin bx) + b Pge 4
SOME USEFU MAHEMAICS DERIVAIVES x n nx n- (u + v) u + v (uv) vu + uv (e x ) e x (e x ) e x (ln u) (/u) sin u cos u sin (u) cos u cos u sin u cos (u) sin (u + c) (n u) sec u u (co u) csc u u (sec u) sec u n u u (csc u) csc u co u u u uv vu - v v - (sin u) u - u - (n u) u + u Pge 5
formul shee - EE Formuls for ENGI 4 -------------------------------------------------------------------------------------------------- v i i C v i v + i v i C + v α ω ο α <ω ο p v i i i p v i C v x( ) x x( ) x + e α + e α D + D ( + B sin( ω ) ) B cos ω ω ω o α --------------------------------------------------------------------------------------------------- v( ) V m cos ω + θ V V m e jθ v() {V e jω } Z R R Z C Z j ω Y j ω C v w i C w v τ R C τ R x( ) x Z R + j X Z + ( x x( ) ) e -------------------------------------------------------------------------------------------------- R α α ω o R C C s α α + ω o α >ω o x( ) x( ) A e s + A e s + s α α ω o τ
Formuls for ENGI 4 --------------------------------------------------------------------------------------------------- v( ) V m cos ω + θ V V m e jθ v() {V e jω } Z R R Z C Z j ω Y j ω C * Z h is complex conjuge Z R + j X Z V p( ) P + P cos( ω ) Q sin( ω ) m I m P cos θ v θ i pf cos θ v θ i V m V rms Z Z h V I* S P + jq I* is complex conjuge Q V m I m sin θ v θ i I m R P Q I m X M k w i + i ± Mi i v v N i N i N N N Z N Z ω o C prllel Q series Q Q ω o B ω ω B ω o R C R C B ω o B R R C RC R ω ω ω o + Q ω o ω o + Q ω o Q Q + + A B log H j ω (