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 lwfrequency 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 timevarying 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 timedependent generalizatin f KVL was presented v(t) R surce R i(t) L di(t) dt. Althugh this expressin is valid fr timechanging 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 43
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 nncnservative (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. 44
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 crsssectin 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 45
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 46
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 crsssectinal 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. 47
8 Figure. Capacitr. The timeharmnic 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 48
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 49
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 40
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) 00s 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 43
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 44
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 45
16 where Z = selfimpedance f the primary circuit referenced t I 0 Z = selfimpedance f the secndary circuit referenced t I 0 Z = mutualimpedance f the primary circuit referenced t I 0 Z = mutualimpedance 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 selfimpedance 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 selfimpedance 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 46
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 47
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 48
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 49
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 selfinductance f the primary circuit; L e, where X e L e µ C C R (s ) is the external selfinductance f the secndary circuit; 40
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 quasicnventinal 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 quasicnventinal 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 quasicnventinal 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 43
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 quasicnventinal circuit are the same as thse fr the cnventinal nearzne circuit. In the quasicnventinal 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 quasicnventinal 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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