Ngā Tauira Pātai mō ngā Ture a Kirchoff
He tūāpapa nui ngā ture a Kirchhoff i roto i te tātaritanga ara iahiko hiko, inā koa kāore e taea te whakaoti i te ara iahiko mā te whakamahi noa i te Ture a Ohm. I te ao tūturu, he maha ngā manga o ngā ara iahiko hiko, he maha ngā pūtake ngaohiko, me ētahi ātete e hono tahi ana. Koinei te wāhi e āwhina ai ngā ture a Kirchhoff i a tātou ki te tatau pūnaha i te iahiko, te ngaohiko, me te ahunga o te rere o te iahiko i roto i ia manga o te ara iahiko. E matapakihia ana e tēnei tuhinga he whakarāpopototanga o ngā ariā o ngā ture a Kirchhoff me ētahi tauira raruraru noa, me ngā mahi whakaoti hei ngāwari ki te mārama.
Te Mōhio ki te Ture a Kirchhoff
I te nuinga o te wā, e rua ngā Ture a Kirchhoff e whakamahia whānuitia ana:
1. Te Ture a Kirchhoff I (KCL – Te Ture o Nāianei a Kirchhoff)
I ngā kupu māmā noa iho: ko te tapeke o ngā ia e tomo ana ki tētahi pūnga he ōrite ki te tapeke o ngā ia e wehe atu ana i taua pūnga.
Pāngarau:
\[
\tapeke I_{in} = \tapeke I_{out}
\]
ka taea rānei te tuhi:
\[
\tapeke I = 0
\]
me te tohu pai mō te ia e tau mai ana, me te tohu kino mō te ia e puta atu ana (e ai ki te tikanga i whakamahia).
2. Te Ture Kirchhoff II (KVL – Te Ture Ngaohiko a Kirchhoff)
I ngā kupu māmā noa iho: ko te tapeke ā-pūtaiao o ngā ngaohiko i roto i tētahi porowhita kati he ōrite ki te kore.
Pāngarau:
\[
\tapeke V = 0
\]
Ko te tikanga o tēnei ko te tapeke o te pikinga ngaohiko (hei tauira, mai i te pākahiko) he ōrite ki te tapeke o te hekenga ngaohiko (puta noa i te ātete, i tētahi atu wāhanga rānei) i roto i te koropiko.
E whakamahia ngātahitia ana ēnei ture e rua: ko te KCL hei tātari i ngā pūnga, me te KVL hei tātari i ngā koropiko (ngā mata).
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Tauira Pātai 1 (KCL): Iahiko i tētahi Pūnga
Pātai:
I tētahi pūnga, e toru ngā iahiko e haere mai ana, arā, ko \(I_1 = 2A\), ko \(I_2 = 3A\), me \(I_3 = 1A\). Mai i te pūnga, e rua ngā iahiko e puta mai ana, arā, ko \(I_4\) me \(I_5 = 4A\). Whakatauhia te uara o \(I_4\).
Otinga:
Whakamahia te Ture Tuatahi a Kirchhoff:
\[
I_{in} = I_{out}
\]
Rerenga mai:
\[
I_1 + I_2 + I_3 = 2 + 3 + 1 = 6A
\]
Rerenga Putanga:
\[
I_4 + I_5 = I_4 + 4
\]
Nā reira:
\[
6 = I_4 + 4
\ArrowRightarrow I_4 = 2A
\]
Whakautu: \(I_4 = 2A\)
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Tauira 2 (KVL): Porowhita Māmā me ngā Ātete Raupapa
Pātai:
He ara iahiko porowhita kotahi te mea he pākahiko 12 V me ngā ātete raupapa e rua \(R_1 = 2\Omega\) me \(R_2 = 4\Omega\). Tātaihia te ia o te ara iahiko me te maturuturunga ngaohiko puta noa i ia ātete.
Otinga:
Nā te mea he ātete porowhita kotahi me he ātete raupapa, he ōrite te iahiko puta noa i ngā wāhanga katoa.
Ātete katoa:
\[
R_{katoa} = R_1 + R_2 = 2 + 4 = 6\Omega
\]
Iahiko porowhita:
\[
I = \frac{V}{R} = \frac{12}{6} = 2A
\]
Te hekenga o te ngaohiko puta noa i \(R_1\):
\[
V_1 = I \cdot R_1 = 2 \cdot 2 = 4V
\]
Te hekenga o te ngaohiko puta noa i \(R_2\):
\[
V_2 = I \cdot R_2 = 2 \cdot 4 = 8V
\]
Tirohia te KVL:
\[
12 – 4 – 8 = 0
\]
I runga anō i te tikanga.
Whakautu: \(I = 2A\), \(V_1 = 4V\), \(V_2 = 8V\)
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Tauira 3 (KVL): Ngā Porowhita e Rua (Tikanga Mata)
Pātai:
He ara iahiko porowhita-rua tēnei. Kei te porowhita maui he pūtake ngaohiko \(V_1 = 10V\) me he ātete \(R_1 = 2\Omega\). Kei te porowhita matau he pūtake \(V_2 = 5V\) me he ātete \(R_2 = 3\Omega\). E rua ngā porowhita e whakamahi ana i te ātete waenga \(R_3 = 4\Omega\). Whakatauhia ngā iahiko mata \(I_a\) (porowhita maui) me \(I_b\) (porowhita matau).
Otinga:
Me kī ko ngā iahiko mata \(I_a\) me \(I_b\) he taha matau. Ko te iahiko i roto i te ātete noa \(R_3\) ko \(I_a – I_b\) (i runga i te ahunga o te whakaaro).
Whārite KVL Porowhita Mauī:
\[
10 – (2I_a) – 4(I_a – I_b) = 0
\]
\[
10 – 2I_a – 4I_a + 4I_b = 0
\Rightarrow 10 – 6I_a + 4I_b = 0
\Rightarrow 6I_a – 4I_b = 10
\]
Whārite KVL Porowhita Matau:
\[
5 – (3I_b) – 4(I_b – I_a) = 0
\]
\[
5 – 3I_b – 4I_b + 4I_a = 0
\Rightarrow 5 + 4I_a – 7I_b = 0
\Rightarrow 4I_a – 7I_b = -5
\]
Whakaotia te pūnaha:
1) \(6I_a – 4I_b = 10\)
2) \(4I_a – 7I_b = -5\)
Whakareatia te whārite (1) ki te 2:
\[
12I_a – 8I_b = 20
\]
Whakareatia te whārite (2) ki te 3:
\[
12I_a – 21I_b = -15
\]
Tangohia:
\[
(12I_a – 8I_b) – (12I_a – 21I_b) = 20 – (-15)
\Rightarrow 13I_b = 35
\Rightarrow I_b = \frac{35}{13} \approx 2.69A
\]
Whakakapia ki te whārite (1):
\[
6I_a – 4(2.69) = 10
\Rightarrow 6I_a – 10.76 = 10
\Rightarrow 6I_a = 20.76
\Rightarrow I_a \tata ki te 3.46A
\]
Whakautu: \(I_a \tata ki te 3.46A\), \(I_b \tata ki te 2.69A\)
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Tauira Pātai 4 (KCL + KVL): Porowhita Peka Whakarara
Pātai:
E honoa ana tētahi pūtake 12 V ki ngā peka e rua e whakarara ana. Kei roto i te peka 1 te \(R_1 = 6\Omega\), kei roto i te peka 2 te \(R_2 = 3\Omega\). Tātaihia te iahiko i ia peka me te iahiko katoa.
Otinga:
Nā te mea he whakarara, he ōrite te ngaohiko o ia peka, arā, 12 V.
Te nāianei o te peka 1:
\[
I_1 = \frac{12}{6} = 2A
\]
Te nāianei o te peka 2:
\[
I_2 = \frac{12}{3} = 4A
\]
Me KCL i ngā pūnga:
\[
I_{katoa} = I_1 + I_2 = 2 + 4 = 6A
\]
Whakautu: \(I_1 = 2A\), \(I_2 = 4A\), \(I_{katoa} = 6A\)
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Ngā Tohutohu mō te Whakaoti i ngā Raru Ture a Kirchhoff
1. Whakatauhia te ahunga o te iahiko i te tuatahi. Mena he kino te hua o te iahiko, ko te tikanga he rerekē te ahunga tuturu ki te whakaaro.
2. Kia rite ki ngā tohu (+) me (-) i te wā e tuhi ana i te KVL. Ko te pikinga ngaohiko mai i tētahi pūtake he mea pai, ko te hekenga ngaohiko puta noa i tētahi ātete he mea kino (i runga i te ahunga o te porowhita).
3. Whakangāwaritia te ara iahiko mēnā ka taea, hei tauira, whakakotahihia ngā ātete i roto i te raupapa, i te whakarara rānei i mua i te whakamahi i a Kirchhoff.
4. Whakamahia ngā tikanga pūnaha: te tātari pūnga mō te KCL, te tātari mata rānei mō te KVL.
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Te Katinga
Mā ngā ture a Kirchhoff ka āwhina i te whakaoti rapanga hiko uaua i roto i te huarahi hanganga. Mā te mōhio ki te KCL me te KVL, ka taea e koe te whakatau i te iahiko i roto i ia peka, te hekenga ngaohiko puta noa i ngā wāhanga, me te mārama ki te whanonga whānui o te ara iahiko. E whakaatu ana ngā tauira i runga ake nei ko te mea nui ko te waihanga i ngā whārite tika me te whakaoti marie. Mā te mahi auau, ka māmā ake te kite i ngā tauira, ahakoa mō ngā ara iahiko uaua ake.
Ki te hiahia koe, ka taea e au te hanga i ētahi atu pātai whakaharatau 10 (kāore he matapakinga, he matapakinga katoa rānei), ka taea rānei e au te tuhi i tētahi putanga me ngā hoahoa ara iahiko me te whakaahuatanga taipitopito ake.
