Ngā Tauira Pātai e Matapaki ana i ngā Āhuatanga o ngā Ara RLC
Pendahuluan
Ko te ara iahiko RLC tētahi o ngā ara iahiko hiko e kitea whānuitia ana i roto i ngā mahi hikohiko me te hangarau hiko. Kei roto i tēnei ara iahiko he ātete (R), he inductor (L), me te capacitor (C) kua whakaritea i roto i tētahi huarahi motuhake. He mea nui ngā ara iahiko RLC nā te mea e whakaatu ana i ngā āhuatanga nui pērā i te haruru, te whakaiti, me ētahi atu. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru e aro ana ki ngā āhuatanga o ngā ara iahiko RLC, me ō rātou whakamārama.
Te Mārama ki ngā Porowhita RLC
He ara iahiko RLC he ara iahiko hiko kei roto he ātete (R), he inductor (L), me he capacitor (C) kua whakaritea kia raupapa, kia whakarara rānei. E whakaatu ana tēnei ara iahiko i ētahi āhuatanga e ahu mai ana i ngā ture hiko pēnei i te Ture a Kirchhoff me te Ture a Ohm. Ko ngā āhuatanga matua e tātarihia ana i roto i ngā ara iahiko RLC ko te aukati, te haruru, te tauwehenga whakamātao, me te auautanga tūturu o te ara iahiko.
1. Ātete (R): He wāhanga e ātete ana i te rere o te iahiko, ā, e huri ana i te pūngao hiko hei wera.
2. Inductor (L): He wāhanga e penapena ana i te pūngao i roto i te āhua o te papa aukume, ā, e ātete ana ki ngā huringa o te iahiko hiko.
3. Pūngao (C): He wāhanga e penapena ana i te pūngao i roto i te āhua o te papa hiko, ā, e ātete ana ki ngā huringa o te ngaohiko.
Te Tātai Taketake o te Ara RLC
I mua i tā tātou urunga atu ki ngā tauira pātai, he maha ngā tauira taketake me mōhio tātou:
1. Āraitanga Katoa (Z): I roto i te ara iahiko raupapa, ko te āraitanga katoa ko te tapeke o ngā ātete, ngā ine hiko, me ngā pūnga hiko:
\[
Z = \sqrt{R^2 + \left(\omega L – \frac{1}{\omega C}\right)^2}
\]
ko \(\omega\) te auau koki (ngā rātiana ia hekona).
2. Auautanga Āhuarangi (f_0): Ko te auau e iti rawa ai te aukati o te ara iahiko (i roto i te ara iahiko raupapa) e mōrahi rānei (i roto i te ara iahiko whakarara). E mōhiotia ana tēnei e te tātai:
\[
f_0 = \frac{1}{2\pi\sqrt{LC}}
\]
3. Tauwehe Whakamarumaru (\(\zeta\)) : He uara e tohu ana mēnā kei te pāngia te ara iahiko e te whakamarumaru nui, te whakamarumaru rawa, ngā wiriwiri kāore i te whakamarumaruhia rānei:
\[
\zeta = \frac{R}{2\sqrt{\frac{L}{C}}}
\]
4. Auautanga Taiao (\(\omega_0\)) : Auautanga taiao o te ara iahiko RLC kāore he whakamātao:
\[
\omega_0 = \frac{1}{\sqrt{LC}}
\]
5. Urupare Ā-Wā: Mō te tātaritanga wā, he mea nui anō hoki te urupare ā-wā. Kei roto ko ngā otinga rerekētanga mō te iahiko me te ngaohiko i ngā huringa ohorere.
Ngā Tauira Pātai me ngā Kōrerorero
Tauira Raru 1: Ara iahiko raupapa RLC
Hoatu he ara iahiko RLC raupapa me ngā uara e whai ake nei:
– Ātete, \(R = 100 \Omega\)
– Pūwhakakorikori, \(L = 0.5 H\)
– Pūngao, \(C = 10 \mu F\)
Pātai:
1. Tātaihia te aukati katoa o te ara iahiko i te auau o te 50 Hz.
2. Whakatauhia te auau haruru o te ara iahiko.
3. Tātaihia te tauwehenga whakamākū.
Ngā Whakautu me te Kōrero:
1. Āraitanga Katoa i te Auautanga 50 Hz:
\(\omega = 2\pi f = 2\pi \times 50 = 100\pi \approx 314 \, rad/s\)
Tātaihia te tauhohenga ā-whakauru (\(X_L\)):
\[
X_L = \omega L = 314 \whakareatia ki te 0.5 = 157 \, \omega
\]
Tātaihia te tauhohenga capacitive (\(X_C\)):
\[
X_C = frac{1}{omega
\]
Te aukati katoa (Z):
\[
Z = \sqrt{R^2 + (X_L –
\]
2. Auautanga o te Whakaohooho:
\[
f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.5 \times 10 \times 10^{-6}}} = \frac{1}{2\pi\sqrt{5 \times 10^{-6}}} = \frac{1}{2\pi \times 0.00224} \approx 71.1 \, Hz
\]
3. Tauwehe Whakamarumaru (\(\zeta\)) :
\[
\zeta = \frac{R}{2\sqrt{\frac{L}{C}}} = \frac{100}{2\sqrt{\frac{0.5}{10 \times 10^{-6}}}} = \frac{100}{2\sqrt{50000}} = \frac{100}{2 \times 224.6} \approx 0.223
\]
Tauira Pātai 2: Porowhita RLC Whakarara
I hoatu he ara iahiko RLC whakarara me ngā uara e whai ake nei:
– Ātete, \(R = 200 \Omega\)
– Pūwhakakorikori, \(L = 1 H\)
– Pūngao, \(C = 50 \mu F\)
Pātai:
1. Tātaihia te aukati katoa o te ara iahiko i te auau o te 60 Hz.
2. Whakatauhia te auau haruru o te ara iahiko.
3. Tātaihia te tauwehenga whakamākū.
Ngā Whakautu me te Kōrero:
1. Āraitanga Katoa i te Auautanga 60 Hz:
\(\omega = 2\pi f = 2\pi \times 60 = 120\pi \approx 377 \, rad/s\)
Tātaihia te tauhohenga ā-whakauru (\(X_L\)):
\[
X_L = \omega L = 377 \whakareatia ki te 1 = 377 \, \omega
\]
Tātaihia te tauhohenga capacitive (\(X_C\)):
\[
X_C = frac{1}{omega
\]
Te aukati katoa (Z) i te āhua whakarara:
\[
\frac{1}{Z} = \sqrt{\frac{1}{R^2} + \left(\frac{1}{X_L} – \frac{1}{X_C}\right)^2}
\]
Tātaihia ia wāhanga ārai:
\[
\frac{1}{R} = \frac{1}{200} = 0.005
\]
\[
\frac{1}{X_L} = \frac{1}{377} = 0.00265
\]
\[
\frac{1}{X_C} = \frac{1}{53} = 0.01887
\]
Nā, ko te aukati katoa ko:
\[
\frac{1}{Z} = \sqrt{0.005^2 + (0.00265 – 0.01887)^2} = \sqrt{0.000025 + (-0.01622)^2} = \sqrt{0.000025 + 0.000263} = \sqrt{0.000288} \approx 0.017
\]
\[
Z \approx \frac{1}{0.017} \approx 58.8 \, \Omega
\]
2. Auautanga o te Whakaohooho:
\[
f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{1 \times 50 \times 10^{-6}}} = \frac{1}{2\pi\sqrt{0.00005}} = \frac{1}{2\pi \times 0.00707} \approx 22.5 \, Hz
\]
3. Tauwehe Whakamarumaru (\(\zeta\)) :
\[
\zeta = \frac{R}{2\sqrt{\frac{L}{C}}} = \frac{200}{2\sqrt{\frac{1}{50 \times 10^{-6}}}} = \frac{200}{2\sqrt{20000}} = \frac{200}{2 \times 141.42} \approx 0.707
\]
Whakamutunga
Mā te matapaki i ngā tauira raruraru i runga ake nei, ka taea e tātou te mārama ki ngā āhuatanga rerekē o ngā ara iahiko RLC i ngā whirihoranga raupapa me ngā whirihoranga whakarara. He mea nui te mārama ki te tatau i te aukati, te auau orooro, me te tauwehenga whakamātao hei tātari i te mahi a ngā ara iahiko RLC i roto i ngā tono hangarau hiko. Mā te whakaharatau tonu i roto i ngā momo raruraru ka whakapakari ake i tō tātou māramatanga me ngā pūkenga tātari mō ēnei ara iahiko.
