Misali na Matsalolin Tattaunawa Kan Sauya Daidaito na Yanzu
Pendahuluan
Alternating current (AC) wani nau'in wutar lantarki ne wanda zai iya canza alkibla lokaci-lokaci. Ba kamar direct current (DC) ba, wanda ke gudana a hanya ɗaya, AC yana da siffar sinusoidal waveform kuma yana da mahimmanci a aikace-aikace daban-daban na yau da kullun, daga gida zuwa masana'antu. Sanin yadda ake aiki da alternating current equations yana da mahimmanci don fahimtar wasu muhimman fannoni na lantarki da wutar lantarki. Wannan labarin zai tattauna misalai da yawa na matsaloli da mafita don fayyace manufar alternating current equations.
Daidaito na Gabaɗaya na Canjin Wutar Lantarki
Ana bayyana canjin wutar lantarki ta hanyar lissafin sinusoidal:
\[ i(t) = Ni_m \sin(\omega t + \phi) \]
Ina:
– \(i(t) \) shine kwararar lantarki nan take a matsayin aikin lokaci.
– \( I_m \) shine ƙimar kololuwa (mafi girma) na halin yanzu.
– \( \omega \) shine saurin kusurwa, tare da raka'o'in radians a kowace daƙiƙa.
– \( t \) lokaci ne.
– \( \phi \) shine matakin farko na wutar lantarki.
Misali Tambaya ta 1: Ƙayyade Wutar Lantarki a Wani Lokaci
Tambaya:
Idan aka yi la'akari da daidaiton halin yanzu mai canzawa \( i(t) = 10 \sin(100\pi t + \pi/3) \). Ka ƙayyade girman halin yanzu a lokacin \(t = 0.01 \) daƙiƙa.
Tattaunawa:
An sani:
\[ Ni_m = 10 \, \rubutu{A} \]
\[ \omega = 100\pi \, \text{rad/s} \]
\[ \phi = \pi/3 \]
\[ t = 0.01 \, \rubutu{daƙiƙa} \]
Sauya waɗannan dabi'u cikin lissafin yanzu:
\[ i(0.01) = 10 \sin(100\pi \sau 0.01 + \pi/3) \]
\[ i (0.01) = 10 \ sin(\pi + \pi/3) \]
\[i (0.01) = 10 \sin(4\pi/3) \]
An lura cewa:
\[ \sin(4\pi/3) = -\sin(\pi/3) \]
\[ \sin(\pi/3) = \sqrt{3}/2 \]
Don haka:
\[ \sin(4\pi/3) = -\sqrt{3}/2 \]
\[ i(0.01) = sau 10 -\sqrt{3}/2 \]
\[ i(0.01) = -5\sqrt{3} \]
\[i(0.01) \approx -8.66 \, \text{A} \]
Don haka, girman wutar lantarki a lokacin \(t = 0.01 \) daƙiƙa yana kusan \(-8.66 \, \text{A} \).
Misali Tambaya ta 2: Ƙayyade Saurin Kusurwa da Mita
Tambaya:
Ga wani canjin wutar lantarki da aka bayyana ta hanyar lissafin \( i(t) = 5 \cos(200\pi t – \pi/4) \), ƙayyade saurin kusurwa (ω) da mita (f) na wutar lantarki.
Tattaunawa:
An bayar:
\[i(t) = 5 \cos(200\pi t – \pi/4) \]
Saurin kusurwa (\( \omega \)) shine ma'aunin \( t \) a cikin hujjar cosine, wanda shine \( 200 \pi \).
\[ \omega = 200\pi \, \text{rad/s} \]
Ana iya samun mita (f) ta amfani da alaƙar:
\[ \omega = 2\pi f \]
\[ f = \frac{\omega}{2\pi} \]
Maye gurbin ƙimar saurin kusurwa:
\[ f = \frac{200\pi}{2\pi} \]
\[ f = 100 \, \rubutu{Hz} \]
Don haka, saurin kusurwar wutar lantarki shine \( 200\pi \, \text{rad/s} \) kuma mitar wutar lantarki shine \( 100 \, \text{Hz} \).
Misali Tambaya ta 3: Tantance Darajar RMS
Tambaya:
Ga wata na'urar lantarki mai canzawa da aka bayar ta \( i(t) = 7 \sin(50t) \), ƙayyade ƙimar RMS (tushen matsakaicin murabba'i).
Tattaunawa:
An sani:
\[ i(t) = 7 \sin(50t) \]
Ƙimar RMS don kwararar sinusoidal shine:
\[I_{\text{RMS}} = \frac{I_m}{\sqrt{2}} \]
inda \(I_m \) shine ƙimar halin yanzu mafi girma.
Ƙimar mafi girman darajar yanzu \(I_m \) shine 7 A.
Don haka:
\[ I_{\text{RMS}} = \frac{7}{\sqrt{2}} \]
\[ I_{\text{RMS}} = \frac{7 \sqrt{2}}{2} \]
\[I_{\text{RMS}} \approx 4.95 \, \text{A} \]
Don haka, ƙimar RMS na wannan wutar lantarki kusan 4.95 A ne.
Misali Tambaya ta 4: Lissafin Matsakaicin Ƙarfi
Tambaya:
Da'irar lantarki ta ƙunshi resistor mai ƙarfin ohm 10 kuma tana da wutar lantarki mai canzawa \( i(t) = 6 \sin(120\pi t) \). Lissafa matsakaicin ƙarfin da resistor ke amfani da shi.
Tattaunawa:
Ganin halin yanzu:
\[i(t) = 6 \sin(120\pi t) \]
Ƙimar halin yanzu mafi girma (\( I_m \)) ita ce 6 A.
Ƙimar RMS na halin yanzu shine:
\[I_{\text{RMS}} = \frac{I_m}{\sqrt{2}} \]
\[ I_{\text{RMS}} = \frac{6}{\sqrt{2}} \]
\[ I_{\text{RMS}} = 3\sqrt{2} \]
Resistor (\( R \)) = 10 ohms.
Ana iya ƙididdige matsakaicin ƙarfin (\(P \)) a cikin resistor ta hanyar:
\[ P = I_{\text{RMS}}^2 \times R \]
\[ P = (3\sqrt{2})^2 \sau 10 \]
\[ P = 18 \sau 10 \]
\[ P = 180 \, \rubutu{W} \]
Saboda haka, matsakaicin ƙarfin da resistor ke amfani da shi shine 180 W.
Penutup
A cikin wannan labarin, mun tattauna misalai da dama na matsaloli kuma mun tattauna daidaiton wutar lantarki mai canzawa. Sanin yadda ake ƙididdige lokacin wutar lantarki a kowace naúra, saurin kusurwa, mita, ƙimar RMS, da matsakaicin ƙarfi yana da mahimmanci don fahimtar canjin wutar lantarki da aikace-aikacensa a rayuwar yau da kullun. Fahimtar waɗannan ra'ayoyi yana taimaka mana mu tsara da kuma nazarin da'irori daban-daban na wutar lantarki waɗanda ke amfani da canjin wutar lantarki.
