Coupling Element and Coupled circuits. Coupled inductor Ideal transformer Controlled sources

Transcription

1 Couplng Element and Coupled crcuts Coupled nductor Ideal transformer Controlled sources

2 Couplng Element and Coupled crcuts Coupled elements hae more that one branch and branch oltages or branch currents depend on other branches. The characterstcs and propertes of couplng element wll be consdered. Coupled nductor Two cols n a close proxmty s shown n Fg. Fg. Coupled col and reference drectons

3 Coupled nductor Magnetc flux s produced by each col by the functons φ f, ) φ f (, ) ( Where f and f are nonlnear functon of and By Faraday s law dφ f d f d + dφ f d f d +

4 Coupled nductor near tme-narant coupled nductor If the flux s a lnear functon of currents and φ ( t) ( t) + M ( t) d + M In snusod steady-state d φ ( t) M ( t) + ( t) d M d + V jω I + jωm I V jωmi+ jω I Note that the sgns of and are poste but the sgn for M can be + or

5 Coupled nductor Dots are often used n the crcut to ndcate the sgn of M Fg. Poste alue of M

6 P At some pont P µ + d µ + µ + H H µ H H H H The energy stored at tme t d [ ] Ε ( t), ( t) ( t) + M ( t), ( t) + ( t)

7 Coupled nductor Coeffcent of couplng The couplng coeffcent s k M If the cols are dstance away k s ery small and close to zero and equal to for a ery tght couplng such for a transformer.

8 Coupled nductor Mult-wndng Inductors and nductance Matrx For more wndngs the flux n each col are φ I+ I+ 3 I3 +.. φ 3 3I + 3 I+ 33 I3 +.. φ I+ I+ 3I3 +..,, 33 are self nductances and,, are mutual nductances In matrx form φ

9 Coupled nductor φ φ φ φ dφ dφ 3 Fg 3 Three-wndng nductor + - dφ3 3

10 Coupled nductor Induced oltage The nduced oltage n term current ector and the nductance matrx s Example d Fg. 4 shows 3 cols wound on a common core. The reference drecton of current and oltage are as shown n the fgure. Snce Hand H has the same drecton but are not therefore s poste whle and H 3 3 are negate. H 3 Fg. 4 H 3 H 3 3

11 Coupled nductor It s useful to defne a recprocal nductance matrx whch makes Γ Γφ Γ φ +Γφ Γ φ +Γ φ where Γ, Γ and Γ Γ det det det Thus the currents are t Γ +Γ + ( t) ( t ') ' ( t ') ' (0) t0 0t Γ +Γ + ( t) ( t ') ' ( t ') ' (0) 0 0 t

12 Coupled nductor In snusod steady-state Γ Γ I V V jω jω Γ Γ I V + V jω jω + Seres and parallel connectons of coupled nductors Equalent nductance of seres and parallel connectons of coupled nductors can be determned as shown n the example.

13 Coupled nductor Example Fg. 5 shows two coupled nductors connected n seres. Determne the Equalent nductance between the nput termnals M 3 φ + M φ M + 3 +, + dφ dφ dφ + φ( 0) 0 φ φ + φ Fg. 5 φ 3 H

14 Coupled nductor Example 3 Fg. 6 shows two coupled nductors connected n seres. Determne the Equalent nductance between the nput termnals. 5 M 3 φ + M φ M + 3 +, dφ dφ dφ φ( 0) 0 φ φ φ + φ Fg. 6 Note H + ± M for seres nductors

15 Coupled nductor Example 4 Two coupled nductors are connected n parallel n Fg 6. Determne the Equalent nductance. Γ det 5 3 det Fg 6 Γ 5 M 3 det 5 3 det 3 3 Γ 3 det 5 3 det 3

16 Coupled nductor The currents are KV By ntegraton of oltage Therefore Γ φ +Γ φ φ 3φ Γ φ +Γ φ 3φ + 5φ ( t) ( t) and φ (0) φ (0) 0 φ ( t) φ ( t) + φ + φ φ φ H Note ΓΓ +Γ ± Γ for parallel nductors

17 Ideal transformer Ideal transformer s ery useful for crcut calculaton. Ideal transformer Is a coupled nductor wth the propertes dsspate no energy No leakage flux and the couplng coeffcent s unty Infnte self nductances Two-wndng deal transformer Fg. 7

18 Ideal transformer Fgure 7 shows an deal two-wndng transformer. Cols are wound on deal Magnetc core to produce flux. Voltages s Induced on each wndng. φ If s the flux of a one-turn col then φ nφ, and φ nφ dφ dφ ( t) n ( t) n Snce and we hae () In terms of magnetomote force (mmf) and magnetc reluctance mmf n Rφ + n Rφ R

19 Ideal transformer µ If the permeablty s nfnte becomes zero then n + n R 0 and ( t) n ( t) n () From () and () ( t) ( t) + ( t) ( t) 0 (3) The oltage does not depend on or but t depends only on

20 Ideal transformer For multple wndngs + Ideal n n + n+ n33 3 (equal olt/ turn) n n n n n Fg. 8

21 Ideal transformer Impedance transformaton n n R n R ( ) ) ( ) ( R n n n n n n n R ( ) n n n R R

22 Impedance transformaton In snusod stead state Z n n : n Z Fg. 9 V ( ) ( ) n V n Zn ( jω) Z ( j ) n ω n I I

23 Controlled sources Controlled sources are used n electronc dece modelng. There four knds of controlled source. Current controlled current source Voltage controlled current source Voltage controlled oltage source Current controlled oltage source 0 0 α g m 0 0 µ 0 m r Fg. 0

24 Controlled sources Current controlled current source : Voltage controlled current source : Current rato Transconductance α g m Voltage controlled oltage source : Voltage rato µ Current controlled oltage source : Transresstance r m

25 Controlled sources Example Determne the output oltage from the crcut of Fg. Mesh ( R s + R) s R R R + R s s R s s R µ R R Mesh R R R µ R + R R + R R R µ R + R R + R s s Fg.

26 Controlled sources Example Determne the node oltage from the crcut of Fg. s G C C G g m Fg. KC d d( ) G + C + C d( ) C G + s ()

27 Controlled sources ( ) + () Dff. (3) d( ) C + G + gm 0 () d ( G + gm ) + C s G (3) d d ds d + m + ( G g ) C G (4) then d d C + + from () ( C C) G s G + gm+ G ds C C C CC C CC d G G d G G G s (5)

28 Controlled sources The ntal conons ( V 0) V, (0) From (3) d (0) [ s (0) GV ( gm+ G ) V] (6) C ( t) From (5) and (6) and ( t) can be soled

29 Controlled sources Other propertes The nstantaneous power enterng the two port s ( ) ( t) p ( t) ( t) ( t) + ( t) ( t) Snce ether or s zero thus t If s connected at port p( t) ( t) ( t) R R Therefore p( t) R Power enterng a two port s always negate

30 Controlled sources Example 3 Consder the crcut of Fg. 3 n snusod steady-state. Fnd the nput mpedance of the crcut. I I I s I α I V Z Z n Fg. 3

31 Controlled sources α α + s I I I I I I I s n Z I I Z I V Z ) ( α Note f the nput mpedance can be negate and ths two port Network becomes a negate mpedance conerter. > α