Phetoho ea Laplace ho Li-equation
Phetoho ea Laplace ke sesebelisoa sa bohlokoa sa lipalo bakeng sa ho sekaseka le ho rarolla li-equation tse fapaneng, haholo-holo li-equation tse fapaneng. E sebelisoa haholo boenjiniere, fisiks, litsamaisong tsa taolo, lipotolohong tsa motlakase, le ho etsa mohlala oa matla a sistimi hobane e fetola mathata a rarahaneng sebakeng sa nako hore e be a bonolo sebakeng se rarahaneng (\(s\)). Sena se lumella phapang le kopanyo ho "fetoleloa" ho ba mesebetsi ea algebra e laolehang haholoanyane.
Ho utloisisa Phetoho ea Laplace
Ka kakaretso, phetoho ea Laplace ea mosebetsi \(f(t)\) e hlalositsoeng bakeng sa \(t \ge 0\) ke:
\[
\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t)\, dt
\]
moo \(s\) e leng nomoro e rarahaneng \(s = \sigma + j\omega\). Phetoho ena e hlahisa mosebetsi o mocha \(F(s)\) o "emelang" boits'oaro ba \(f(t)\) lefapheng \(s\).
Molemo o moholo oa phetoho ea Laplace ke bokhoni ba eona ba ho sebetsana le maemo a pele ka mokhoa o hlophisehileng, ao hangata e leng karolo ea bohlokoa ea li-differential equation.
Hobaneng ha Phetoho ea Laplace e le ea Bohlokoa ho Li-equation?
Mekhoa e mengata ea lefats'e la 'nete e hlalosoa ka li-equation tse fapaneng. Mehlala e kenyelletsa motsamao oa selemo-boima, potoloho ea RLC, kapa mehlala e itseng ea kholo. Li-equation tse fapaneng hangata li thata ho li rarolla ka kotloloho, haholo-holo haeba li kenyelletsa matla a ho kenya a seng bonolo, joalo ka mesebetsi ea mehato, li-impulse (deltas), kapa li-input tse bitsitsoeng ka likotoana.
Phetoho ea Laplace e nolofatsa bothata ka litšobotsi tse 'maloa tsa bohlokoa:
1. Phapang ho ea ho algebra
Haeba \( \mathcal{L}\{f(t)\} = F(s) \), joale:
\[
\ lipalo{L}\{f'(t)\} = sF(s) – f(0)
\]
\[
\mathcal{L}\{f”(t)\} = s^2F(s) – sf(0) – f'(0)
\]
Sena se bolela hore di-derivative, tseo hangata di leng thata ho di sebetsana, di fetolwa ho ba dibopeho tse bonolo tsa algebra.
2. Convolution e fetoha ho ata
Ts'ebetso ea convolution ka nako e fetoha katoloso sebakeng sa domain \(s\), e thusa haholo tlhahlobong ea litsamaiso tse otlolohileng.
3. Kopanya maemo a pele
Maemo a pele a kena ka ho toba ho di-equation tse karolong ya domain (s) ntle le tlhoko ya mehato e meng.
Tšebeliso ea Li-equation tse Fapaneng
A re re re na le equation ea phapang ea mola oa pele:
\[
y'(t) + ay(t) = g(t), \quad y(0)=y_0
\]
Ka ho sebelisa phetoho ea Laplace mahlakoreng ka bobeli:
\[
\ lipalo{L}\{y'(t)\} + a\mathcal{L}\{y(t)\} = \mathcal{L}\{g(t)\}
\]
Sebelisa thepa e nkiloeng ho eona:
\[
(sY(s) – y(0)) + aY(s) = G(s)
\]
E le hore:
\[
(s+a)Y(s) = G(s) + y_0
\]
\[
Y(s) = \frac{G(s) + y_0}{s+a}
\]
Mohato o latelang ke ho fumana phetoho e fapaneng ya Laplace ho fumana \(y(t)\). Maemong a mangata, sena se ka etswa ho sebediswa tafole ya phetoho ya Laplace kapa ho sebediswa mekgwa ya karolo e sa fellang.
Mehlala ea Litekanyo tsa Phapang ea Sehlopha sa Bobeli
Nahana ka equation:
\[
y”(t) + 3y’(t) + 2y(t) = 0
\]
ka maemo a pele:
\[
y(0)=1, \quad y'(0)=0
\]
Phetoho ea sebaka:
\[
\ lipalo{L}\{y”\} + 3\mathcal{L}\{y'\} + 2\mathcal{L}\{y\} = 0
\]
Phetolo ea thepa ea Laplace:
\[
(s^2Y – sy(0) – y’(0)) + 3(sY – y(0)) + 2Y = 0
\]
Kenya maemo a pele:
\[
(s^2Y – s\cdot 1 – 0) + 3(sY – 1) + 2Y = 0
\]
\[
s^2Y – s + 3sY – 3 + 2Y = 0
\]
Kopanya:
\[
(s^2 + 3s + 2)Y = s + 3
\]
\[
Y(s) = \frac{s+3}{(s+1)(s+2)}
\]
Ebe u etsa likaroloana tse sa fellang:
\[
\frac{s+3}{(s+1)(s+2)} = \frac{A}{s+1} + \frac{B}{s+2}
\]
Re fumana \(A=2\), \(B=-1\), e le hore:
\[
Y(s)=\frac{2}{s+1}-\frac{1}{s+2}
\]
Laplace ka ho fapana:
\[
y(t) = 2e^{-t} – e^{-2t}
\]
Sena se bontša hore ts'ebetso ea ho rarolla li-differential e ba e hlophisehileng haholoanyane le ea algebra.
Phetoho ea Laplace ho Li-equation tse nang le Li-input tse Ikhethang
Phetoho ea Laplace e thusa haholo ha ho kenngoa tšebetsong e sa tloaelehang. Mohlala, mosebetsi oa mohato oa Heaviside \(u(ta)\) o emela lets'oao le "butsoeng" ka nako e itseng. Haeba ho kenngoa ha sistimi ho fetoha ho \(t=a\), tharollo e tobileng e sebelisang mekhoa e tloaelehileng e ka rarahana ka lebaka la tlhoko ea ho sebelisa mesebetsi e otlolohileng. Ka phetoho ea Laplace, mesebetsi e joalo e na le melao e tloaelehileng e etsang hore lintho li be bonolo.
Ka ho tšoanang, Dirac impulse \(\delta(t)\) hangata e sebelisoa tlhahlobong ea sistimi ho leka likarabelo tsa impulse. Phetoho ea Laplace ea \(\delta(t)\) e bonolo haholo, e leng 1, e leng se etsang hore ho be bonolo ho bala karabelo ea sistimi.
Karolo ho Mekhoa ea Boenjiniere le Taolo
Khopolo-taba ea taolo, phetoho ea Laplace ke motheo oa ho theha mosebetsi oa phetisetso ea sistimi. Mohlala, ho tsoa ho equation e fapaneng ea sistimi e fetohang, mosebetsi oa phetisetso o ka fumanoa:
\[
G(s) = \frac{Y(s)}{U(s)}
\]
Mosebetsi ona oa phetisetso o thusa ho sekaseka botsitso, karabelo ea maqhubu, le litšobotsi tsa nakoana tse kang ho feta tekano le nako ea ho lula. Ka har'a lisebelisoa tsa elektroniki, phetoho ea Laplace e boetse e sebelisoa ho sekaseka lipotoloho tsa RLC, kaha likamano tsa hona joale le motlakase tse fapaneng li ka fetoloa sebopeho sa algebra.
Melemo le Meeli
Phetoho ea Laplace e na le melemo e mengata:
– Nolofatsa di-differential equation hore e be di-algebraic equation.
– Kenya maemo a pele ka kotloloho.
– E loketse matshwao le di-input tse sa tsweleng pele kapa tse sa tsamayeng ka potlako.
– E sebetsa hantle haholo bakeng sa litsamaiso tsa linear time-invariant (LTI).
Leha ho le jwalo, ho na le mefokolo e meng:
– Hase mesebetsi yohle e nang le phetoho ya Laplace (ho itshetlehile ka kopano ya karolo e kopaneng).
– E loketse haholoanyane bakeng sa litsamaiso tse otlolohileng; bakeng sa litsamaiso tse seng tse otlolohileng hangata ho hlokahala mekhoa e meng.
– Ts'ebetso e fapaneng ea Laplace ka linako tse ling e ba thata haeba sebopeho sa \(Y(s)\) se rarahane 'me se se tafoleng e tloaelehileng.
Qetello
Phetoho ea Laplace ke mokhoa oa bohlokoa oa ho rarolla li-equation tse fapaneng, haholo-holo li-equation tse fapaneng, ka ho li fetola sebakeng sa \(s\), ho li etsa hore li laolehe haholoanyane. Mokhoa ona o nolofatsa ho kenyelletsoa ha maemo a pele, o sebetsana le lintho tse rarahaneng tse kenang, 'me o tšehetsa tlhahlobo ea litsamaiso mafapheng a fapaneng a boenjiniere le mahlale. Ka lebaka la thuso ea eona e kholo, phetoho ea Laplace e fetohile karolo ea bohlokoa lipalo le boenjiniere ba sejoale-joale bo sebelisitsoeng.
Haeba o lakatsa, nka boela ka eketsa bothata bo felletseng ba mohlala (ka likaroloana tse sa fellang le mehato e fapaneng ea Laplace) kapa ka etsa mofuta oa sengoloa se shebaneng haholo le ts'ebeliso e itseng joalo ka potoloho ea motlakase kapa sistimi ea taolo.
