Laʻana o nā pilikia e kūkākūkā ana i nā kaulike o kēia manawa
Pendahuluan
ʻO ke au uila ʻē aʻe (AC) kahi ʻano o ke au uila e hiki ke hoʻololi i ke kuhikuhi i kēlā me kēia manawa. ʻAʻole e like me ke au pololei (DC), e kahe ana i hoʻokahi kuhikuhi, he ʻano nalu sinusoidal ko AC a he mea nui ia i nā noi like ʻole o kēlā me kēia lā, mai ka hale a i ka ʻoihana. ʻO ka ʻike pehea e hana ai me nā hoʻohālikelike o ke au uila ʻē aʻe he mea nui ia no ka hoʻomaopopo ʻana i kekahi mau ʻano koʻikoʻi o ka uila a me ka uila. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a me nā hoʻonā e wehewehe i ke kumumanaʻo o nā hoʻohālikelike o ke au uila ʻē aʻe.
Ka Hoʻohālikelike Laulā o ke Au Alternating
Hoʻākāka ʻia ke au hoʻololi e ka hoʻohālikelike sinusoidal:
\[ i(t) = I_m \sin(\omega t + \phi) \]
Ma hea:
– ʻO \( i(t) \) ke au koke e like me ka hana o ka manawa.
– ʻO \( I_m \) ka waiwai kiʻekiʻe (kiʻekiʻe loa) o ke au.
– ʻO \( \omega \) ka wikiwiki angular, me nā anakahi o nā radians i kekona.
– ʻO ka manawa ka \( t \).
– ʻO \( \phi \) ka pae mua o ke au.
Laʻana Nīnau 1: Ke hoʻoholo nei i ke au i kahi manawa kikoʻī
Nīnau:
Hāʻawi ʻia i ka hoʻohālikelike o ke au uila alternating \( i(t) = 10 \sin(100\pi t + \pi/3) \). E hoʻoholo i ka nui o ke au i ka manawa \( t = 0.01 \) kekona.
Kūkākūkā:
Ua ʻike ʻia:
\[ ʻO I_m = 10 \, \kikokikona{A} \]
\[ \omega = 100\pi \, \text{rad/s} \]
\[ \phi = \pi/3 \]
\[ t = 0.01 \, \text{kekona} \]
E hoʻololi i kēia mau waiwai i loko o ka hoohalike o kēia manawa:
\[ i(0.01) = 10 \sin(100\pi \times 0.01 + \pi/3) \]
\[ i(0.01) = 10 \sin(\pi + \pi/3) \]
\[ i(0.01) = 10 \sin(4\pi/3) \]
Ua ʻike ʻia:
\[ \sin(4\pi/3) = -\sin(\pi/3) \]
\[ \sin(\pi/3) = \sqrt{3}/2 \]
No laila:
\[ \sin(4\pi/3) = -\sqrt{3}/2 \]
\[ i(0.01) = 10 \times -\sqrt{3}/2 \]
\[ i(0.01) = -5\sqrt{3} \]
\[ i(0.01) \approx -8.66 \, \text{A} \]
No laila, ʻo ka nui o ke au i ka manawa \(t = 0.01 \) kekona ma kahi o \(-8.66 \, \text{A} \).
Laʻana Nīnau 2: Ke Hoʻoholo nei i ka Velocity Angular a me ka Frequency
Nīnau:
No ke au hoʻololi i hōʻike ʻia e ka hoohalike \( i(t) = 5 \cos(200\pi t – \pi/4) \), e hoʻoholo i ka wikiwiki angular (ω) a me ke alapine (f) o ke au.
Kūkākūkā:
Hāʻawi ʻia:
\[ i(t) = 5 \cos(200\pi t – \pi/4) \]
ʻO ka wikiwiki angular (\( \omega \)) ke koina o \( t \) i loko o ka hoʻopaʻapaʻa cosine, ʻo ia hoʻi \( 200\pi \).
\[ \omega = 200\pi \, \text{rad/s} \]
Hiki ke loaʻa ka alapine (f) me ka hoʻohana ʻana i ka pilina:
\[ \omega = 2\pi f \]
\[ f = \frac{\omega}{2\pi} \]
E pani i ka waiwai o ka wikiwiki angular:
\[ f = \frac{200\pi}{2\pi} \]
\[ f = 100 \, \text{Hz} \]
No laila, ʻo ka wikiwiki angular o ke au he \( 200\pi \, \text{rad/s} \) a ʻo ke alapine (frequency) o ke au he \( 100 \, \text{Hz} \).
Laʻana Nīnau 3: Ke hoʻoholo nei i ka waiwai RMS
Nīnau:
No ke au hoʻololi i hāʻawi ʻia e \( i(t) = 7 \sin(50t) \), e hoʻoholo i ka waiwai RMS (root mean square).
Kūkākūkā:
Ua ʻike ʻia:
\[ i(t) = 7 \sin(50t) \]
ʻO ke kumukūʻai RMS no ke au sinusoidal:
\[ I_{\text{RMS}} = \frac{I_m}{\sqrt{2}} \]
kahi ʻo \( I_m \) ka waiwai o ke au kiʻekiʻe.
ʻO ka waiwai kiʻekiʻe o ke au \( I_m \) he 7 A.
No laila:
\[ I_{\text{RMS}} = \frac{7}{\sqrt{2}} \]
\[ I_{\text{RMS}} = \frac{7 \sqrt{2}}{2} \]
\[ I_{\text{RMS}} \approx 4.95 \, \text{A} \]
No laila, ʻo ka waiwai RMS no kēlā au he 4.95 A.
Laʻana Nīnau 4: Ke Helu ʻana i ka Mana Awelike
Nīnau:
Aia kahi kaapuni uila i kahi resistor 10 ohm a he au hoʻololi ko ia \( i(t) = 6 \sin(120\pi t) \). E helu i ka mana awelika i hoʻopau ʻia e ka resistor.
Kūkākūkā:
Hāʻawi ʻia i kēia manawa:
\[ i(t) = 6 \sin(120\pi t) \]
ʻO ke kumukūʻai o ke au kiʻekiʻe (\( I_m \)) he 6 A.
ʻO ke kumukūʻai RMS o kēia manawa:
\[ I_{\text{RMS}} = \frac{I_m}{\sqrt{2}} \]
\[ I_{\text{RMS}} = \frac{6}{\sqrt{2}} \]
\[ I_{\text{RMS}} = 3\sqrt{2} \]
Palekana (\( R \)) = 10 ohms.
Hiki ke helu ʻia ka mana awelika (\(P \)) i loko o kahi resistor ma o:
\[ P = I_{\text{RMS}}^2 \times R \]
\[ P = (3\sqrt{2})^2 \times 10 \]
\[ P = 18 \times 10 \]
\[ P = 180 \, \kikokikona{W} \]
No laila, ʻo ka mana awelika i hoʻopau ʻia e ka resistor he 180 W.
Pani
Ma kēia ʻatikala, ua uhi mākou i kekahi mau pilikia hoʻohālike a ua kūkākūkā i nā kaulike no ke au uila. ʻO ka ʻike pehea e helu ai i ke au i kēlā me kēia manawa, ka wikiwiki angular, ka alapine (frequency), ka waiwai RMS, a me ka mana awelika he mea nui ia no ka hoʻomaopopo ʻana i ke au uila a me kāna mau noi i ke ola o kēlā me kēia lā. ʻO ka hoʻomaopopo ʻana i kēia mau manaʻo e kōkua iā mākou e hoʻolālā a kālailai i nā kaapuni uila like ʻole e hoʻohana ana i ke au uila.
