Notes on Inductance and Circuit Transients Joe Wolfe, Physics UNSW. Circuits with R and C. τ = RC = time constant

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1 Nes n Inducance and cu Tansens Je Wlfe, Physcs UNSW cus wh and - Wha happens when yu clse he swch? (clse swch a 0) - uen flws ff capac, s d Acss capac: Acss ess: c d s d d ln + cns. 0, ln cns. ln ln ln ln a ln b ln a b e / /c e / () me cnsan Uns: ΩF A. s. s - d e - / e - / e - / /e /e /e /e

2 Example A eal uses delecc wh delecc cns. 4.0 and cnducvy σ 0-5 Ω-m-. I s chaged hen lef dscnneced f hu. Wha facn f chage emans? I A K, σ d - (σ /ρ) A d K A d ρ A d Aσ d K Aσ. A d K σ 4.0 x Fm-.78 x 04 s Ω-m- e - / Q Q e / e 3600/ 0.8 Example Elecnc applance eues 0 W a 40 D, wh max vaan f 0%. Wha capacance needed se enegy beween cycles? A + m 90%* m A has 50 cycles pe secnd, uns ff 00 mes/secnd, e evey 0 ms. m e / ms,?, wha s? s he effecve essance f he ccu. We knw and P, s use P /P 80 Ω. e / 0.90 ln (0.90) 0 ms 80 Ω * ln µf Example he membane f a "pacemake" neune f he hea has a capacance f 0 pf and a cnducance G 3.0 pω. Inally he penal dffeence acss hs membane s 80 m. The neune begns a pulse when he penal dffeence eaches 60 m. Hw lng befe begns a pulse? Neglecng he duan f he eleccal mpulse self, wha s he pulse ae? () ln ln ln s The "leakness" f he membane s nw suddenly nceased by a fac f. Hw lng beween eleccal mpulses? and wha s he appxmae pulse ae, agan neglecng he duan f he ndvdual mpulses. - ln 0.48 s - ln...

3 3 chagng Wha happens when yu clse he swch? (clse swch a 0, 0, 0) - uen flws n capac, s I + d Kchff + ake devaves, use + d 0 d + eaange: d - ln - + cns nal cnn ln cns e -/ e-/ () whee / and /e /e ( /e) ( /e ) Inducance Self-nducn B flux hugh each un f cl φ B d(nφ Faaday Β ) Inducance, NΦ B (3) ( s he al flux lnkage pe un cuen) S d (4) Ne: n cl ae f change n cf acss negal f Uns: heny l secnd/amp

4 4 enz's law: acs n decn manan cuen (and B) + f nceases + f deceases esss change n Example Aumble gnn cl (schemac) (n n syllabus) n cnac n ce spng spng mves n gh clse cnac. flws whch causes B B aacs n lef, whch pens ccu d 0 uckly s lage Ideal Slend A l n uns/m NΦ B n l BA bu B µ n NΦ B nlµ na µ nla (5) Ne: f nn-magnec maeals, µ µ ο 4π 0-7 Hm- magnec maeals, µ >> µ n ces f lage. Example l n kele has 60 Ω. I s 0 cm lng, has un/mm, and s 4 mm n damee. ) wha s? ) s cnneced 40 ms. Swch has beakdwn penal f 000 and s uned ff a peak f cycle ( 340 ). Esmae he duan f spak. (Neglec he ). Peak p A. (5) µ ο n l A 4π 0-7.(03 ). (0.). π( 0-3) SI 3. µh. - d ~ 000 dung spak d ~ cns. 000 d 5.7 x (In pacce s lage because f he 's.) 8 ns.

5 5 - Tansens ecall - d (enz's law: back emf) Ths pevens sudden change n, s cnnuus. a ) swch a a. Afe a lng me, d b 0, c/ ) swch b a 0 a 0, c / Kchff: - d d ln - + ln - e -/ c e-/ whee / (6) Uns: H Ω A/s. A s /e /e ) swch a b f lng me ( 0). Swch a a 0. c - d c slve ( - e- / ) whee / (7) ( /e) ( /e ) ' Example 6 64 H 3k ( 3 kω) 47k Swch clsed f lng me. Open a 0. Wha des he O shw? Swch clsed, afe a lng me d 0 Kchff's law: and T + O shws 6

6 6 Swch pened fbds abup change n a 0 (0) /, 0 al sees essance s + s - ( + )/ e hee + - s O - - O - e -(+)/ 6 - Enegy n B ~ f Pwe suppled.e. du B U B du B d. ' d' U B (8) c.f. U E Enegy densy U B vl U B Al Al (5) µ n la (5) (fm mag.) B µ n U B vl B µ (9) (c.f. U E ο E ) Muual Inducn N B N N φ G Muual Inducance M (f w... ) M N Φ c.f. N Φ s M N Φ M d N dφ - (Faaday's law) (lng pf) M M M s: M d, M d (0) (S.I. un heny s A - )

7 7 Example Tw lng cncenc slends. M? x x x x x x x x x x x x n / un lengh and n / un lengh M M N Φ n l Φ B µ n, and Φ b A M n l (µ ο n ) π M π l µ n n () scllans (assume deal : 0, deal : ) Kchff's law: - d d, s - d d d - - ω say slve: cns sn (ω + cns) m sn (ω + φ) whee ω () - d - ω m cs (ω + φ) - m cs (ω + φ) Enegy U E. [...cs...] U B [...sn...] U T U E + U B m [ ω cs ( ) + sn ( )] m [ cnsan] als m m ω s U T m ω m Example Fn end f a ad: max mn. pf max (3 n //) ο A 3 d 3 π d pf ) If we wan une he AM band (54-60 khz) wha values f and? Hw d we make he? f max π ω max π ( mn + ) f mn π ( max + ) f max f... max + mn mn + (f max - f mn ) max f mn - mn f max pf f π. 4π f max +. 4p f mn 490 mh (5) µ ο n la say n 000/m, A 5 cm, l? l µ na 00 m. Ops, y n ce: µ µ µ ~ 000

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