Module 4: General Formulation of Electric Circuit Theory

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1 Mdule 4: General Frmulatin f Electric Circuit Thery

2 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated auxiliary relatinships. Fr certain classes f prblems, such as representing the behavir f electric circuits driven at lw frequency, applicatin f these relatinships may be cumbersme. As a result, apprximate techniques fr the analysis f lw-frequency circuits have been develped. These specializatins are used t describe the "ideal" behavir f cmmn circuit elements such as wires, resistrs, capacitrs, and inductrs. Hwever, when devices are perated in a regime, r an envirnment, which lies utside the range f validity f such apprximatins, a mre fundamental descriptin f electrical systems is required. When viewed in this mre general cntext, what may initially appear t be unexpected behavir f a circuit element ften reveals itself t be nrmal peratin under a mre cmplex set f rules. An understanding f this lies at the cre f electrmagnetically cmpatible designs. In this sectin, electric circuit thery will be presented in a general frm, and the relatinship between circuit thery and electrmagnetic principles will be examined. The apprximatins assciated with circuit thery and a discussin f the range f validity f these apprximatins will be included. It will be seen that effects due t radiatin and inductin are always present in systems immersed in time-varying fields, althugh under certain cnditins these effects may be ignred. 4. Limitatins f Kirchff's laws The behavir f electric circuits is typically described thrugh Kirchff's vltage and current laws. Kirchff's vltage law states that the sum f the vltages arund any clsed circuit path is zer M n V n 0 and Kirchff's current law states that the sum f the currents flwing ut f a circuit nde is zer N M n I n 0 It is thrugh applicatin f these relatinships that mst descriptins f electric circuits prceed. Hwever, bth f these relatinships are nly valid under certain cnditins: - The structures under cnsideratin must be electrically small. At 60 Hz, the wavelength f a wave prpagating thrugh free space is 5 millin meters, while at 300 MHz a wavelength in free space is m lng. Radiatin and inductin effects arise when the current amplitude and phase vary at pints alng the cnductr. - N variatin exists alng uninterrupted cnductrs. 4-

3 - N delay time exists between surces and the rest f the circuit. Als all cnductrs are equiptential surfaces. - The lss f energy frm the circuit, ther than dissipatin, is neglected. In reality, lsses due t radiatin may becme significant at high frequency. In chapter, a time-dependent generalizatin f KVL was presented v(t) R surce R i(t) L di(t) dt. Althugh this expressin is valid fr time-changing fields, it is assumed that the circuit elements are lumped, i.e, the resistance and inductance are cncentrated in relatively small regins. This assumptin begins t break dwn at frequencies where the circuit elements are a significant fractin f a wavelength lng. In this regime, circuits must be described in terms f distributed parameters. Every part f the circuit has a certain impedance per unit length assciated with it. This impedance may be bth real (resistive) and imaginary (reactive). Als, interactins which ccur in ne part f the circuit may affect interactins which ccur everywhere else in the circuit. In additin, the presence f ther external circuits will affect interactins within a different circuit. Thus at high frequency, a circuit must be viewed as a single entity, nt a cllectin f individual cmpnents, and multiple circuits must be viewed as cmpsing a single, cupled system. 4. General frmulatin fr a single RLC circuit The general frmulatin f electric circuit thery will begin with an analysis f a single circuit cnstructed with a cnducting wire f radius a that may include a cil (inductr), a capacitr, and a resistr. It will be assumed that the radius f the wire is much smaller than a wavelength at the frequency f peratin a/ << r a << where Œ/. A current flws arund the circuit. The tangential cmpnent f electric current 4-3

4 J s ŝ 3J ŝ( 3E) E s is driven by the tangential cmpnent f electric field E s alng the wire, which is prduced by charge and current in the circuit. Currents in the circuit are supprted by a generatr. The generatr is a surce regin where a nn-cnservative (meaning that the ptential rises in the directin f current flw) impressed electric field 3E e maintains a charge separatin. This impressed field is due t an electrchemical r ther type f frce, and is assumed t be independent f current and charge in the circuit. The charge separatin supprts an electric field 3E (Culmb field) within and external t the surce regin which gives rise t the current flwing in the circuit. Figure. Generalized electric circuit. In the regins external t the surce, the impressed field 3E e des nt exist, hwever within the surce bth fields exist, therefre by Ohm's law the current density present at any pint in the circuit is given by 3J ( 3E 3E e ) where varies frm pint t pint. Frm this it is apparent that in rder t drive the current density 3J against the electric field 3E which ppses the charge separatin in the surce regin, the impressed electric field must be such that 3E e > 3E. 4-4

5 Psitins alng the circuit are measured using a displacement variable s, having it's rigin at the center f the generatr. The unit vectr parallel t the wire axis at any pint alng the circuit is ŝ. The tangential cmpnent f electric field alng the cnductr is therefre E s ŝ 3E and the assciated axial cmpnent f current density is J s ŝ 3J ŝ( 3E) E s. The ttal current flwing thrugh the cnductr crss-sectin is then I s Pc.s. J s ds. bundary cnditins at the surface f the wire circuit Accrding t the bundary cnditin ˆ t ( 3E 3E ) 0 the tangential cmpnent f electric field is cntinuus acrss an interface between materials. Applicatin f this bundary cnditin at the surface f the wire cnductr leads t E s (r a ) E s (r a ), where E inside s (s) E s (ra ) is the field just inside the cnductr at psitin s alng the circuit, and E utside s (s) E s (ra ) represents the field maintained at psitin s just utside the surface f the cnductr by the current and charge in the circuit. Therefre the fundamental bundary cnditin emplyed in an 4-5

6 electrmagnetic descriptin f circuit thery is E inside s (ra ) E utside s (ra ). determinatin f E inside s (s) In rder t apply the bundary cnditin abve, the tangential electric field that exits at pints just inside the surface f the circuit must be determined. This is nt an easy task, because the impedance may differ in the varius regins f the circuit. At any pint alng the cnducting wire, including cils and resistrs, the electric field is in general E inside s (s) I s (s) where E inside s (s) I s (s) is the internal impedance per unit length f the regin. - surce regin In the surce regin, bth the impressed and induced electric fields exist, therefre the current density is J s e (E s E e s ) where e is the cnductivity f the material in the surce regin, E s is the tangential cmpnent f electric field in the surce regin maintained by charge and current in the circuit, and apparent that E e s is the tangential cmpnent f impressed electric field. Frm this, it is 4-6

7 E s J s E e e s J s S e E e e S e S I s E e e S e s where S e is the crss-sectinal area f the surce regin. In an ideal generatr, the material in the surce regin is perfectly cnducting ( e ), and has zer internal impedance. Thus E s E e s in a gd surce generatr. In general, in the surce regin E s (s) e I s (s) E e s (s) where e e S e is the internal impedance per unit length f the surce regin. - capacitr The tangential cmpnent f electric field at the edge f the capacitr flwing t the capacitr lie in the same directin. Therefre, within the capacitr E s and the current E s (s) c I s (s) where c is the internal impedance per unit thickness f the dielectric material cntained in the capacitr. The ttal ptential difference acrss the capacitr is the line integral f the nrmal cmpnent f electric field which exists between the capacitr plates r B A A V AB V B V a E P s ds E P s ds P A A V AB I s P B B c ds B c I s (s)ds if it is assumed that a cnstant current I s flws t the capacitr. 4-7

8 Figure. Capacitr. The time-harmnic cntinuity equatin states / 3J j!. Vlume integratin f bth sides f this expressin, and applicatin f the divergence therem yields Q S ( ˆn 3J )ds j P V!dv resulting in I s jq where Q is the ttal charge cntained n the psitive capacitr plate. The ttal ptential difference between the capacitr plates is then V AB jq P B A c ds 4-8

9 Figure 3. Single capacitr plate. but by the definitin f capacitance C Q V AB therefre B Q C jq P A c ds. Frm this cmes the expected expressin fr the impedance f a capacitr A P B c ds jc jx c. - arbitrary pint alng the surface f the circuit By cmbining the results fr the three cases abve, a general expressin fr the tangential cmpnent f electric field residing just inside the surface f the cnductr at any pint alng the circuit is determined 4-9

10 E inside s (s) (s)i s (s) E e s (s) where (s) is the internal impedance per unit length which is different fr the varius cmpnents f the circuit, and E e s (s) is the impressed electric field which is zer everywhere utside f the surce regin. determinatin f E utside s (s) In Chapter it was shwn that an electric field may be represented in terms f scalar and vectr ptentials. Thus at any pint in space utside the electric circuit the electric field is 3E /- j 3A where - is the scalar ptential maintained at the surface f the circuit by the charge present in the circuit, and 3A is the vectr ptential maintained at the surface f the circuit by the current flwing in the circuit. The well knwn Lrentz cnditin states that / 3A jk - 0 where k µ0 j 0. In the free space utside the circuit µ µ, 0, and 0 thus 0 k µ 0. Applying this t the Lrentz cnditin yields - j / 3A which, upn substitutin int the expressin fr electric field utside the circuit gives 4-0

11 3E j /(/ 3A) 3 A. The cmpnent f electric field tangent t the surface f the circuit is then given by E utside s (s) ŝ 3E ŝ/- j(ŝ 3A) j ŝ /(/ 3A) 3 A r E utside s (s) 0-0s ja s j 0 0s (/ 3A) A s. satisfactin f the fundamental bundary cnditin The bundary cnditin at the surface f the circuit states that the tangential cmpnent f electric field must be cntinuus, r E inside s (s) E utside (s) s therefre the basic equatin fr circuit thery is E e s (s) (s)i s (s) 0 0s -(s) ja s (s) j 0 0s / 3A(s) A s (s). pen and clsed circuit expressins Frm the develpment abve, the expressin fr the impressed electric field is E e s (s) 0 0s -(s) ja s (s) (s)i s (s). 4-

12 Figure 4. General circuit. Integrating alng a path C n the inner surface f the circuit frm a pint s t a pint s, which represent the ends f an pen circuit, gives s E e P s s s s S (s)ds 0 P 0s -(s)ds A j P s (s)ds (s)i P s (s)ds s s s but, because the impressed electric field exists nly in the surce regin s E e P s s (s)ds P B E e A s (s)ds where is the driving vltage. Nw it can be seen that s P 0s -(s)ds d- -(s P ) -(s ) s s 0 s and thus the equatin fr an pen circuit can be expressed 4-

13 s s 0 -(S ) -(s ) A j P s (s)ds (s)i P s (s)ds. s s If the circuit is clsed, then s =s and -(s )--(s )=0. In this case, the circuit equatin becmes 0 QC (s)i s (s)ds jq C A s (s)ds. 4.3 General equatins fr cupled circuits Nw the cncepts develped abve are extended t the case f tw cupled circuits, each cntaining a generatr, a resistr, a cil (inductr), and a capacitr. Circuit will be referred t as the primary circuit, and circuit will be referred t as the secndary circuit. This case is represented by a pair f cupled general circuit equatins 0 Q C (s )I s (s )ds j Q C 3A (s ) 3A (s ) 3 ds 0 Q C (s )I s (s )ds j Q C 3A (s ) 3A (s ) 3 ds. Here 3A and 3A are the magnetic vectr ptentials at the surface f the primary circuit maintained by the currents and in the primary and secndary circuits, given by I s I s and 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 and 3A and 3A are the vectr ptentials at the surface f the secndary circuit maintained by the currents and in the primary and secndary circuits, given by I s I s 4-3

14 Figure 5. Generalized cupled circuits. 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 and 3A (s ) µ C I s (s ) e j R R (s, s ). Substituting these expressins fr the varius magnetic vectr ptentials int the general circuit equatins abve leads t 0 (s )I s (s )ds jµ C C C I s (s ) e j R R (s ds ) 3 jµ C C I s (s ) e j R R (s ds ) 3 4-4

15 0 (s )I s (s )ds jµ C C C I s (s ) e j R R (s ds ) 3 jµ C C I s (s ) e j R R (s ). Nte that C and C, lie alng the inner periphery f the circuit while C and C lie alng the centerline. When the circuit dimensins and 0, 0 are specified, the equatins abve becme a pair f cupled simultaneus integral equatins fr the unknwn currents I s (s ) and I s (s ) in the primary and secndary circuits. These equatins are in general t cmplicated t be slved exactly. self and mutual impedances f electric circuits Reference currents I 0 and I 0 are chsen at the lcatins f the generatrs in the primary and secndary circuits, i.e., I 0 I s (s 0) at the center f the primary circuit generatr, and I 0 I s (s 0) at the center f the secndary circuit generatr. Nw let the currents be represented by I s (s ) I 0 f (s ) I s (s ) I 0 f (s ) where f (0) f (0).0, and f, f are cmplex distributin functins. The general circuit equatins frmulated abve may be expressed in terms f the reference currents as 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z 4-5

16 where Z = self-impedance f the primary circuit referenced t I 0 Z = self-impedance f the secndary circuit referenced t I 0 Z = mutual-impedance f the primary circuit referenced t I 0 Z = mutual-impedance f the secndary circuit referenced t I 0 and Z Z i Z e Z Z i Z e. Here Z i is referred t as the internal self-impedance f the primary and secndary circuits. This term depends primarily upn the internal impedance per unit length f the cnductrs present in the circuits, and includes effects due t capacitance and resistance. Z e is referred t as the external self-impedance f the primary and secndary circuits. This term depends entirely upn the interactin between currents in varius parts f the circuit, and includes effects due t inductance. The varius impedance terms are expressed as Z i Q C (s ) f (s ) ds Z i Q C (s ) f (s ) ds Z e jµ C C f (s ) e j R R (s ds ) 3 Z e jµ C C f (s ) e j R R (s ds ) 3 4-6

17 Z jµ C C f (s ) e j R R (s ds ) 3 Z jµ C C f (s ) e j R R (s ). It is nted that all f the circuit impedances depend in general n the current distributin functins f (s ) and f (s ). driving pint impedance, cupling cefficient, and induced vltage Cnsider the case where nly the primary circuit is driven by a generatr excitatin, i.e., 0 0 In this case the general circuit equatins becme 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z. This leads t a pair f equatins I 0 I 0 Z Z 0 I 0 Z Z Z Z which can be slved t yield 4-7

18 I 0 0 Z Z Z Z and I 0 0 Z Z Z Z which are the reference currents in the generatr regins. Frm these, it can be seen that the driving pint impedance f the primary circuit is Z in 0 I 0 Z Z Z Z Z Z Z Z Z. Fr the case f a lsely cupled electric circuit having Z Z Z Z «, Z and Z are bth negligibly small, and therefre (Z ) in Z. Fr this case the expressins 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z. becme 0 V i I 0 Z V i I 0 Z where V i I 0 Z is the vltage induced in the primary circuit by the current in the secndary circuit, and V i I 0 Z is the vltage induced in the secndary circuit by the current in the primary circuit. Because the circuits are cnsidered t be lsely cupled 0» V i 4-8

19 and therefre the general circuit equatins becme 0 x I 0 Z V i I 0 Z I 0 Z. near zne electric circuit Fr cases in which a circuit is perated such that the circuit dimensins are much less than a wavelength, then the phase f the current des nt change appreciably as it travels arund the circuit. The assciated EM fields are such that the circuit is cnfined t the near- r inductinznes. Mst electric circuits used at pwer and lw radi frequencies may be mdeled this way. In this type f cnventinal circuit R «, R «, R «, R «and therefre the fllwing apprximatin may be made e j R ij R ij! 4 R 4 ij 4!... j R ij R ij 3! 4 R 4 ij... x 5! fr i=, and j=,. Because the circuit dimensins are small cmpared t a wavelength f (s ) x f (s ) x and thus f i (s i ) e j R ij (s i j ) R ij (s i j ) x R ij (s i j ) fr i=, and j=,. Substituting the apprximatins stated abve int the expressins fr the varius circuit impedances leads t Z i Q C (s ) ds Z i Q C (s ) ds 4-9

20 Z e jx e jµ C C R (s ) Z e jx e jµ C C R (s ) Z jx jµ C C R (s ) Z jx jµ C C R (s. ) Since the integrals abve are frequency independent and functins nly f the gemetry f the circuit, the self and mutual inductances are defined as fllws: L e, where X e L e µ C C R (s ) is the external self-inductance f the primary circuit; L e, where X e L e µ C C R (s ) is the external self-inductance f the secndary circuit; 4-0

21 X L, where L µ C C 3 ds R (s ) is the mutual inductance between the primary and secndary circuit; X L, where L µ C C 3 ds R (s ) is the mutual inductance between the primary and secndary circuit. It is seen by inspectin that L L. radiating electric circuit At high, ultrahigh, and lw micrwave frequencies, the assumptins fr cnventinal nearzne electric circuits are nt valid since is nt usually satisfied. In a quasi-cnventinal R «r radiating circuit, the less restrictive dimensinal requirement f R «is assumed t be valid fr the frequencies f interest, therefre R «, R «, R «, R «. In this case e! j " R ij R ij! 4 R 4 ij 4! x j R ij R ij 3!... j R ij R ij 3! 4 R 4 ij... 5! fr i=, and j=,. In the apprximatin made abve, the higher rder term retained. Fr a quasi-cnventinal circuit R ij 6 is 4-

22 f (s ) x f (s ) x is bserved experimentally t remain apprximately valid. Thus f i (s i ) e j$ R ij (s i % j ) R ij (s i j ) x R ij (s i j ) j R ij (s i j ) 6 fr i=, and j=,. Frm this expressin it is apparent why the higher rder term R ij 6 in the expnential series was retained. The leading term in the imaginary part f the expnential series integrates t zer in the impedance expressins, i.e., Q j i j i 0 Q Ci Ci fr i=,. The impedance expressins fr the quasi-cnventinal circuit are thus given by Z e jµ C C % R (s ) j 6 C % R (s ) Z e jµ C C % R (s ) j 6 C % R (s ) Z jµ C C % R (s ) j 6 C % R (s ) Z jµ C C % R (s ) j 6 C % R (s ) and the expressins fr the internal impedances f the primary and secndary circuits remain unchanged. It is seen that the impedance expressins abve cnsist f bth real and imaginary parts 4-

23 Z e R e jx e Z e R e jx e and Z Z R jx. Nw, recalling that v p jµ 3 j 6 µ v p Œ therefre R e 4 5 C C ' R (s ( ) ( L e, where X e L e µ C C ' ( R (s ( ) R e 4 5 C C ' R (s ( ) ( X e L e, where 4-3

24 L e µ C C ) * R (s * ) 4 R 5 C C ) R (s * ) * and X L, where L µ C C ) * R (s *. ) It is nted that the inductances, and L fr the quasi-cnventinal circuit are the same as thse fr the cnventinal near-zne circuit. In the quasi-cnventinal circuit, hwever, impedances Z e, Z e and Z becme cmplex due t the presence f R e, R e and R. The resistive cmpnents R e and R e d nt represent dissipatin lsses in the circuit (dissipatin lsses are included in the internal impedance terms Z i and Z i ), but instead indicate a pwer lss frm the circuit due t radiatin f EM energy t space. R e and R e are therefre radiatin resistances which describe the pwer lss frm a quasi-cnventinal circuit due t EM radiatin. L e L e References. R.W.P. King, and S. Prasad, Fundamental Electrmagnetic Thery and Applicatins, Prentice- Hall, R.W.P. King, Fundamental Electrmagnetic Thery, Dver Publicatins,

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