Tauira o ngā Pātai Kōrero mō te Ara RLC
Ko te ara iahiko RLC he momo ara iahiko hiko kei roto ko te ātete (R), te inductor (L), me te capacitor (C) e honoa ana i roto i te raupapa, i te whakarara rānei. E whakamahia whānuitia ana ngā ara iahiko RLC i roto i ngā momo tono hiko, tae atu ki ngā tātari ngaru, ngā oscillator, ngā huinga tohu, me ētahi atu. He pūkenga nui te mārama ki te tātari me te whakaoti rapanga i roto i ēnei ara iahiko mō ngā ākonga hangarau hiko me ngā tohunga ngaio i roto i te mara.
I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru me ā rātou otinga e pā ana ki te tātaritanga ara iahiko RLC. Kua hangaia ēnei raruraru hei āwhina i a koe ki te mārama ki ngā mātāpono me ngā whakamahinga taketake o ngā ara iahiko RLC.
Tauira Raru 1: Ara iahiko raupapa RLC
Pātai:
Hoatu he ara iahiko raupapa kei roto ko te ātete me te ātete \( R = 10 \, \Omega \), he inductor me te inductance \( L = 0.1 \, H \), me te capacitor me te capacitance \( C = 100 \, \mu F \). Ka hoatu he ngaohiko sinusoidal ki tēnei ara iahiko \( V(t) = 100 \sin(1000 t) \V). Whakatauhia:
1. Te tauhohenga ā-whakauru \( X_L \) me te tauhohenga ā-pūmanawa \( X_C \).
2. Te aukati katoa o te ara iahiko.
3. Iahiko mōrahi \( I_{max} \).
4. Te ngaohiko mōrahi i runga i ia wāhanga.
Kōrero:
1. Te Tauhohenga Ārai \( X_L \) me te Tauhohenga Āwhina \( X_C \)
Ka tatauhia te tauhohenga ā-whakauru \( X_L \) mā te tātai:
\[ X_L = \omega L \]
me \( \omega = 1000 \, rad/s \) me \( L = 0.1 \, H \),
\[ X_L = 1000 \whakareatia ki te 0.1 = 100 \, \Omega \]
Ka tatauhia te tauhohenga capacitive \( X_C \) mā te tātai:
\[ X_C = \frac{1}{\omega C} \]
me \( C = 100 \, \mu F = 100 \times 10^{-6} \, F \),
\[
2. Te Āraitanga Katoa o te Arahiko
Ko te aukati katoa \( Z \) mō te ara iahiko raupapa RLC ko:
\[ Z = \sqrt{R^2 + (X_L – X_C)^2} \]
Whakakapia ngā uara:
\[ Z = \sqrt{10^2 + (100 – 10)^2} = \sqrt{10^2 + 90^2} = \sqrt{100 + 8100} = \sqrt{8200} = 90.55 \, \Omega \]
3. Iahiko Mōrahi \( I_{max} \)
Ka taea te tatau i te iahiko mōrahi mā te whakamahi i te ture a Ohm:
\[ I_{max} = \frac{V_{max}}{Z} \]
me \( V_{max} = 100 \, V \) me \( Z = 90.55 \, \Omega \):
\[ I_{max} = \frac{100}{90.55} = 1.104 \, A \]
4. Te Ngaohiko Mōrahi i Ia Wāhanga
Ngaohiko i runga i te ātete:
\[ V_R = I_{max} \times R = 1.104 \times 10 = 11.04 \, V \]
Ngaohiko i runga i te inductor:
\[ V_L = I_{max} \times X_L = 1.104 \times 100 = 110.4 \, V \]
Ngaohiko i runga i te pūnga:
\[ V_C = I_{max} \times X_C = 1.104 \times 10 = 11.04 \, V \]
Tauira Pātai 2: Porowhita RLC Whakarara
Pātai:
Homai he ara iahiko whakarara kei roto ko te ātete me te ātete \( R = 50 \, \Omega \), he inductor me te inductance \( L = 0.5 \, H \), me te capacitor me te capacitance \( C = 10 \, \mu F \). Ko te pūtake ngaohiko he sinusoidal me te ngaohiko \( V(t) = 200 \cos(2000 t) \, V \). Whakatauhia:
1. Te tauhohenga ā-whakauru \( X_L \) me te tauhohenga ā-pūmanawa \( X_C \).
2. Te urunga katoa o te ara iahiko.
3. Iahiko mōrahi \( I_{max} \).
Kōrero:
1. Te Tauhohenga Ārai \( X_L \) me te Tauhohenga Āwhina \( X_C \)
Ka tatauhia te tauhohenga ā-whakauru \( X_L \) mā te tātai:
\[ X_L = \omega L \]
me \( \omega = 2000 \, rad/s \) me \( L = 0.5 \, H \),
\[ X_L = 2000 \whakareatia ki te 0.5 = 1000 \, \Omega \]
Ka tatauhia te tauhohenga capacitive \( X_C \) mā te tātai:
\[ X_C = \frac{1}{\omega C} \]
me \( C = 10 \, \mu F = 10 \times 10^{-6} \, F \),
\[
2. Te Whakaurunga Katoa o te Porowhita
Ko te urunga \( Y \) te taurite o te aukati. Mō te ara iahiko RLC whakarara:
\[ Y = \sqrt{G^2 + (B_L – B_C)^2} \]
te wāhi \( G = \frac{1}{R} \),
\[ G = \frac{1}{50} = 0.02 \, S \]
Te whakaroa ā-whakauru \( B_L \):
\[ B_L = \frac{1}{X_L} = \frac{1}{1000} = 0.001 \, S \]
Te ātete ā-pūngao \( B_C \):
\[ B_C = \frac{1}{X_C} = \frac{1}{50} = 0.02 \, S \]
Whakakapia ngā uara katoa ki roto i te tātai:
\[ Y = \sqrt{0.02^2 + (0.001 – 0.02)^2} = \sqrt{0.0004 + 0.000361} = \sqrt{0.000761} = 0.0276 \, S \]
Ko te aukati katoa \( Z \) ko te taurite o te whakaurunga:
\[ Z = \frac{1}{Y} = \frac{1}{0.0276} = 36.23 \, \Omega \]
3. Iahiko Mōrahi \( I_{max} \)
\[ I_{max} = \frac{V_{max}}{Z} \]
me \( V_{max} = 200 \, V \) me \( Z = 36.23 \, \Omega \):
\[ I_{max} = \frac{200}{36.23} \approx 5.52 \, A \]
Whakamutunga
He mea nui te mārama ki te tatau i te tauhohenga, te urunga, te aukati, me te iahiko i roto i te ara iahiko RLC mō ngā tono i roto i te hangarau hiko. Mā te mahi tonu mā roto i ngā tauira maha pērā i ngā mea kua kōrerohia e tātou, ka hohonu ake tō māramatanga ki te tātari ara iahiko RLC. Mā te mahi tonu ka whakapai ake hoki i ō pūkenga whakaoti rapanga me te hoahoa ā muri ake nei o ngā ara iahiko hiko uaua.
