Heaviside :

Khi giai cac bai toan mach dien= pp toan tu,thu duoc nghiem toan tu co dang 1 phan thuc huu ty :

F(s) = H1(s)/H2(s) = (a0 + a1s + a2s2 + ... + ans^n)/(b0 + b1s + ... + bm^m)

Trong do a0 - an,b0 -bm la cac hang so.gia thiet tu va mau ko co nghiem chung va n<m.den day co the phan tich phan thuc F(s) thanh tong cac phan thuc toi gian,tu do dua vao bang ham goc - anh va tinh chat tuyen tinh cua phep bdoi Laplace de tim goc cua F(s)

Xe cac truong hop :

1. H2(s) chi co cac nghiem don,nghia la

H2(s) = bm(s-s1)(s-s2)...(s-sm)

luc do,co dang khai trien F(s)

F(s) = A1/(s-s1) + A2/(s-s2) + ... +Am/(s-sm) = ∑ Ak/(s-sk)

De xac dinh he so Ak,nhan ca 2 ve cua bthuc voi (s-sk) sau do cho s -> sk sao choc ac so hang o ve phai,tru Ak deu triet tieu va nhu vay viet dc :

Ak = lim[F9s)(s-sk)] = lim [H1(s)/H2(s) . (s-sk)]

Vi H2(sk) = 0 nen gioi han ben fai co dang 0/0.ap dung quy tac L'hospital :

Ak = lim[(H1(s) + (s-sk)H1(s))/H'2(s)] = H1(sk)/H'2(sk)

Ap dung bang anh Laplace cho moi so hang cua F(s),ta duoc ham goc cua F(s) :

f(t) = ∑ H1(sk)/H'2(sk) . exp(skt)

2. H2(s) co nghiem boi.vd 1 nghiem la sL boi r sao cho co the viet :

H2(s) = bm(s-s1)(s-sk)(s-sl)^r voi k + r = m

Luc do khai trine cua F(s) :

F(s) = A1/(s-s1) + A2/(s-s2) + ... + Ak/(s-sk) + (Al0 + Al1(s-s1) + ... + Alr(s-sl)^(r-1))/(s-s1)

Can phai xet so hang cuoi cung cua bieu thuc ung voi nghiem boi sl

Truoc het,de tinh Al0,nhan ca 2 ve voi (s-s1)^r roi cho s->s1 :

Al0 = lim[F(s)(s-ss1)^r] = [H1(s)/H2(s) . (s-s1)^r ]

Tiep tuc tinh Al1 sau khi nhan ca 2 ve voi (s-s1)^r,dao ham ca 2 ve theo s roi cho s->s1,se co :

d/ds[F(s)(s-s1)^r]s=s1 = Al1

tuong tu ta co :

½ . d2/ds2 [ F(s)(s-s1)^r ] s=sl = Al2

.........

1/(r-1)! . (d^(r-1)/ds(r-1)[F(s)(s-sl)^r] s=sl = All - 1

Sau khi xac dinh dc cac he so Ali ( I = 0,...,r-1),tim goc cua moi so hang tuong ung voi cac he so do. So hang co he so Ali la Ali/(s-sli)^(r-1) . ham goc tuong ung voi no la :

Fli(t) = Ali^(t^r-i-1)/(r-i-1)! . exp(slt)

Cuoi cung,duoc ham goc cua F(s) nhu sau :

k

f(t) = ∑H1(sj)/H2(sj) exp(sjt) + [Al0/(r-1)! . t^(r-1) + Al1/(r-2)!. T^(r-2) + ...+ Al(r-1)/0!] exp(slt)

r-1

= ∑ H1(s)/H'2(sj) exp(sjt) + ∑ Ali.t^(r-i-1) / (r-l-i) ! exp (slt)

3. truong hop cac nghiem cua H2(s) co 1 nghiem phuc . neu sk la nghiem don thi kem theo no co 1 nghiem phuc lien hiep s*k. co

H1(s*k)/H'2(s*k) = H*1(sk)/H'*2(sk)

Sao cho ung voi cac ham goc :

H1(sk)exp(skt)/H'2(sk) + H1*(sk)exp(s*k)/H'*2(sk) =

= 2Re[H19(k)exp(skt)/H2(sk)] = 2 | Ak | exp (δkt)cos(ωkt + ak)

Trong do dat : H1(sk)/H'2(sk) = Ak = | Ak | exp exp (jak)

Sk = δk + jωk

Cac dinh luat Kirchhoff trong mach dien :

+ dinh luat KC 1 : tong cac dong dien di vao 1 nut nao do bang tong cac dong dien tu nut do di ra

+ dinh luat KC 2 : tong dai so cac dien ap sut tren cac thong so thu dong cua mot vong kin bang tong dai so cac suc dien dong co trong vong kin do

∑Uk(t) = ∑ej(t)

K J

Phat bieu gon hon : tong dai so cac dien ap sut trong 1 vong kin bang 0

He phuong trinh tong quat cua mach dien :

Giai bai toan mach dien -> lap phuong trinh

So cac pt lap ko phai 1 -> lap he pt,chon an

Bang? Ham' - Anh Laplace :

Ham goc f(t) Anh Laplace F(s)

1. df(t)/dt sF(s) - f(0)

2. ∫ f(t)dt 1/s [ F(s) + ∫(- vo cung den 0 ) f(t)dt

3. -tf(t) dF(s)/ds

4. 1/t. f(t) 0-s∫F(z)dz

5. e^(-at) f(t) F(s + a)

6. f(t-a) l(t-a) e^(-as) F(s)

7. f(t/a) aF(as)

8. l(t) l/s

9. t^n n!/(s^(n+1)

10. e^-at 1/(s+a)

11. sinωt ω/(s^2 + ω^2)

12 cosωt s/( s^2 + ω^2)

13. δt 1