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DESIGN NOTE TRIUMF TRI-DN-96-2 March 1996 From Maxwell’s equations for a cavity resonator to a parallel LCR circuit using simple maths S. Koscielniak TRIUMF ABSTRACT The purpose of these calculations is to demonstrate that a resonant cavity behaves like a parallel LCR circuit in the neighbourhood of resonance. This property has been well known for a long time, but the rigorous development is very lengthy and mathematical. The argumenmts used here are rough; but nonetheless results are useful and easily comprehended. TRIUMF 4004 WESBROOK MALL, VANCOUVER, B.C. CANADA V6T 2A3 From Maxwell’s equations for a cavity resonator to a parallel LCR circuit using simple maths The purpose of these calculations is to demonstrate that a resonant cavity behaves like a parallel LCR circuit. This property has been well known for a long time, but the rigorous development[1] is very lengthy and mathematical. The arguments used here are very rough; but nontheless results are useful. We shall consider the TM010 mode in a pill-box cavity. In section 1 we ﬁnd the resonance frequency. In section 2 we ﬁnd the magnetic ﬁeld in terms of a driving current. In section 3 we include the eﬀect of losses and ﬁnd the quality factor. In section 4 we ﬁnd the shunt resistance. 1 Self-excited, lossless We postulate magnetic and electric ﬁelds of the form: H = H h(x)e+jωc t , ˜ E = E e(x)e+jωc t . ˜ (1) ˜ ˜ Here H, E are phasors (i.e. complex quantities) and the vectors h, e specify the ﬁeld shapes. Here x is a general coordinate vector. Take cylindrical polar coordinates r, φ, z and right-handed orthogonal unit vectors nr , nφ , nz . Pick ﬁeld shapes which satisfy the boundary conditions of a pill-box resonator, of length d and radius a. h = nφ J1 (kr) , e = nz J0 (kr) , (2) with J0 (ka) = 0 or ka = P01 ≈ 2.405. Applying Stokes’ theorem1 to Maxwell’s equation for the ‘curl’ of magnetic ﬁeld, we ﬁnd H · dl = jωc E · dS . (3) We take a circular path (see ﬁgure 1) close to the circumference of the cavity, that is just inside the dielectric. h · dl = 2πaJ1 (ka) (4) a e · dS = 2π J0 (kr)rdr = 2π(a/k)J1(ka) (5) 0 Hence ˜ ˜ H = j(ωc /k)E . (6) Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of electric ﬁeld, we ﬁnd −µjωc H · dS = E · dl . (7) 1 path z E H resonator wall Figure 1: integration path for magnetic ﬁeld integration path H z E Figure 2: integration path for electric ﬁeld 2 We take a rectangular path (see ﬁgure 2) along the centre line, and along the cavity walls. a h · dS = d J1 (kr)dr = d/k (8) 0 e · dl = d (9) Hence ˜ ˜ −µj(ωc /k)H = E . (10) We have two equations (6, 10) for two unknowns: the resonance frequency, ωc , and the ﬁeld ˜ ˜ ratio H/E . We obtain ωc = ±kc and ˜ ˜ E = ∓jη H , (11) where √ c = 1/ µ and η= µ/ = 376.7 ohm . (12) η is known as the free-space impedance. Let us ﬁnd the charges and current on the cavity walls, which can be considered the ﬁeld sources. 1.1 Surface current and charge The current density, Jc , ﬂows longitudinally over the cylindrical side wall at r = a. If we take an integration contour similar to that for equation (6) but just inside the conducting material, then we shall ﬁnd: H · dl = 0 = (Jc + jωc E) · dS . (13) ˜ Hence the total wall current, Ic , is E˜ ˜ Jc · dS = Ic = 2πaJ1 (P01 ) ampere . (14) jη Here we replaced ωc /k by c. The longitudinal electric ﬁeld starts and terminates at the surface charges on the end walls at z = 0 and z = d. If we apply Gauss’ theorem to an inﬁnitesimal cylindrical volume element (in contact with the conductor) we shall ﬁnd the surface charge density is ˜ σc (r) = EJ0 (kr) coulomb/(metre)2 . (15) 2 Driven, oﬀ resonance, lossless Suppose the cavity is driven by a charged particle beam current. For simplicity, we shall suppose the wavelength between bunches is long compared to the cavity gap-width, and so can ignore 1 The spatial relation of dl to dS is analogous to that of nr and nφ to nz ; if l is taken anti-clockwise in the plane of the page, then dS points out of the page. 3 that the beam component is a travelling wave. Jb = Jb j(x)e+jωt , ˜ where j = nz jb (r) . (16) We take the magnetic and electric ﬁelds to have the same time dependence as the beam current, that is e+jωt with ω = ωc . Applying Stokes theorem to Maxwell’s equation for the ‘curl’ of magnetic ﬁeld, we ﬁnd H · dl = (Jb + jωE) · dS . (17) ˜ The total beam current, Ib , is given by ˜ Jb · dS = Ib . (18) We take a circular path close to the circumference of the cavity, that is just inside the dielectric. ˜ Ib ω˜ ˜ H = + j E. (19) 2πaJ1 (ka) k Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of electric ﬁeld, we take a rectangular path along the centre line, and along the cavity walls. ˜ ˜ −µj(ω/k)H = E . (20) ˜ ˜ ˜ We have two equations (19, 20) for two unknowns: E and H in terms of the total current Ib . We eliminate E˜ by taking (19) − j(ω/k) × (20). The physical analogue of our ability to eliminate ˜ E by diﬀerencing the simultaneous equations is that the peak electrical and magnetic stored 2 energies are (almost) equal. We use the fact that µ/k 2 = 1/ωc to ﬁnd 2 ˜ ˜ ωc Ib H = (21) (ωc − ω 2) 2πaJ1 (P01 ) 2 ˜ ω ˜ E = −jη H . (22) ωc Note, the magnetic and electric stored energies are equal at resonance. Evidently, if ω > ωc then more energy is stored in the electric ﬁeld than in the magnetic ﬁeld; this is the analogue of the fact that more energy is stored in the capacitance of a parallel LCR circuit if driven above its resonance frequency. The excess energy is alternately passed back and forth between the beam and the cavity each quarter cycle of the ﬁeld oscillation. 2.1 Relation of voltage to current The peak, instantaneous gap voltage is d ωωc ˜ Ib ˜ V =− E · nz dz = +jηd . (23) 0 (ωc − ω 2 ) 2πaJ1 (P01 ) 2 Evidently, if ωc > ω then the gap voltage (induced by the beam current) leads the beam current by 90◦ ; whereas if ωc < ω then the gap voltage lags by 90◦ . 4 2.2 Wall current Again, we may try to ﬁnd the surface current that ﬂows in the cavity walls. We take an integration contour similar to that for equation (17) but just inside the conducting material, then we shall ﬁnd: ˜ ˜ ˜ 0 = Ic + Ib + j(ω/k)E × 2πaJ1 (P01 ) . (24) ˜ If we eliminate E (by using equations 21, 22) , we shall ﬁnd 2 ˜ ˜ ωc −Ic = Ib . (25) (ωc − ω 2 ) 2 ˜ Incidentally, in equation (24) we may associate the ﬁrst term, Ic , with the current through ˜ the lumped inductance, and the last term (∝ j(ω/k)E) with the current through the lumped capacitance of the equivalent LC circuit. 2.3 generator current From Ampere’s theorem (relating the line integral of H to current sources) it is evident that a current Ig ﬂowing in a coupling loop2 and moving parallel to the cavity axis will have exactly ˜ the same eﬀect as the beam current, provided that both currents ﬂow in the same sense and have the same magnitude. Consequently, one may use superposition to ﬁnd the eﬀect of cavity excitation by generator and beam together. 3 Self-excited, lossy We postulate ﬁelds of the form: H = H h(x)e(+jωc −α)t , ˜ E = E e(x)e(+jωc −α)t . ˜ (26) If the conductivity, σ, is large then the boundary conditions are hardly disturbed by their lossy nature; and so, in the interior of the cavity, we can use the same ﬁeld shapes h and e as before (equation 2). However, we must realize that there are now small electric ﬁelds, ∆Ee(+jωc −α)t , parallel to the bounding surfaces (and just inside the conducting material). Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of magnetic ﬁeld, we take a circular path close to the circumference of the cavity, that is just inside the dielectric. ˜ ˜ H = E(jωc − α)/k . (27) 2 We assume that the current carrying conductor is almost the length of the cavity. 5 Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of electric ﬁeld, we take a rectangular path along the centre line, and along the cavity walls. ˜ ˜ 1 −µH(jωc − α)/k = E + ∆E · dl . (28) d Evidently, we must ﬁrst ﬁnd an expression for the losses term appearing on the far right of equation (28). Now ∆E = Jc /σ and the conduction current density can be found by employing the relation ∇ ∧ H = Jc at the conducting surface. Let δ indicate the Dirac delta function. Let Jc = nr Jr + nz Jz and recall that h = nφ J1 (kr). One ﬁnds: Jr ˜ = −HJ1 (kr)δ(z) , Jr ˜ = +HJ1 (kr)δ(z − d) , Jz ˜ = −HJ1 (ka)δ(r − a) . (29) z=0 z=d r=a We substitute and ﬁnd ˜ H δ(z) δ(z − d) ∆E · dl = + + d J1 (ka)δ(r − a) . (30) σ k k This last equation (30) is not much use to us. However, we recall that most of the electric ﬁeld is contained within a thickness δs , the skin-depth, of the surface. Hence, let us integrate perpendicular to the path l to obtain: ˜ H 2 δs ∆E · dl = + d J1 (ka) . (31) σ k Hence, we can write (28) in the form ˜ −µH ˜ (jωc − α) = E + H 2 + J1 (ka) . ˜ (32) k σδs kd ˜ ˜ We have two equations (27, 32) for two unknowns: the decay rate, α, and the ﬁeld ratio H/E . It is usual to replace 1/(σδs ) by the surface resistivity Rs (ohms). We ﬁnd that µ ˜ 2 + J1 (P01 ) = 0 . 2 (jωc − α)2 E + Rs H 1+ ˜ (33) k kd ˜ We eliminate H by using equation (27), and divide through out by (jωc − α)/ωc so as to leave the eigenvalue equation: (jωc − α) ωc Rs 2 + + + J1 (P01 ) = 0 . (34) ωc (jωc − α) η kd 2α α Rs 2a or 1+ = + J1 (P01 ) . (35) ωc j2ωc η P01 d Now, we assume the losses are small, so that α/(2ωc) 1. Further, we note that 2α/ωc = 1/Q where Q is the quality factor. Hence 1 Rs 2a Rs a 2 P01 = + J1 (P01 ) ≈ + . (36) Q η P01 d η d P01 2 This may be compared with the exact value from text books[2], 1 Rs a 2 = +1 . (37) Q η d P01 Note that P01 ≈ 2.405, and that the agreement is remarkable. 6 4 Driven, lossy H, E and Jb have the time dependencies assumed in section 2 – except that we may remove the restriction ω = ωc . Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of magnetic ﬁeld, we take a circular path close to the circumference of the cavity, that is just inside the dielectric. ˜ Ib ω˜ ˜ H = + j E. (38) 2πaJ1 (ka) k Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of electric ﬁeld, we take a rectangular path along the centre line, and along the cavity walls. ω˜ ˜ η ˜ −µj H = E + H . (39) k Q The last term in equation (39) represents the losses due to ﬁnite conductivity. We have two ˜ ˜ ˜ equations (38, 39) for two unknowns: E and H in terms of the total current Ib . ˜ We eliminate E to ﬁnd: ω2 ˜ ˜ Ib ωη ˜ 1− µ 2 H = − j H. (40) k 2πaJ1 (ka) kQ Now ck = ωc and so 2 (ωc − ω 2 ) j ω ˜ ˜ Ib 2 + H = . (41) ωc Q ωc 2πaJ1 (ka) We now perform some mathematical manipulation. We multiply throughout by Qωc /(jω). We deﬁne the detuning angle by 2 Q(ωc − ω 2 ) tan Ψ = . (42) ωωc Hence, the magnetic ﬁeld phasor is given by: Q ωc ˜ Ib ˜ [1 − j tan Ψ] H = . (43) j ω 2πaJ1 (ka) This may be simpliﬁed, using trigonometric identities, to give: ˜ Ib H = −jQ cos Ψe+jΨ ˜ . (44) 2πaJ1 (ka) Notice how the ampliﬁcation factor Q cos Ψ replaces the resonant denominator appearing in equation (21). We may go on to solve for the electric ﬁeld phasor: ˜ 1 ω ˜ E = − +j ηH . (45) Q ωc 7 Now, ω ≈ ωc and Q 1, and so to good approximation: η˜ ˜ Ib ˜ E = H = −ηQ cos Ψe+jΨ . (46) j 2πaJ1 (ka) 4.1 gap voltage and shunt resistance Finally, we may ﬁnd the peak, instantaneous gap voltage ˜ d ηdQ cos Ψe+jΨ ˜ V =− E · nz dz = + Ib . (47) 0 2πaJ1 (P01 ) If we compare with the conventional representation V = Rshunt cos Ψe+jΨ Ib , ˜ ˜ (48) then it is clear, the parallel (or shunt) resistance is given by: Rshunt ηd = . (49) Q πa2J1 (P01 ) This may be compared with the value in text books[3]: Rshunt ηd ηd ηd = 2 ≈ ≈ . (50) Q πaP01 J1 (P01 ) πaP01 J1 (P01 ) × (P01 /2) πa2.9J1 (P01 ) Given the roughness of our derivation, the agreement is remarkable. 4.2 power ﬂow We may calculate the power W exchanged with the cavity: 2W = R[I V ∗ ] = R[I ∗ V ] ˜˜ ˜ ˜ (51) = +Rshunt cos2 Ψ|Ib |2 ˜ (watt) . (52) The electrical power is positive, indicating that energy ﬂows into the cavity and out of the beam. 4.3 Eﬀect of detuning Let us consider the voltage and current relations for a parallel LCR resonator. Let us assume e+jωt time dependence. ˜ 1 1 ˜ I= + j ωC − V . (53) R ωL 8 2 ˜ ˜ At resonance we ﬁnd ωc = 1/(LC), and that voltage and current are in phase: V = RI. Oﬀ (but close to) resonance we ﬁnd ˜ 1 1 ˜ I≈ + j ωc C + (ω − ωc ) V (54) R ωc L Clearly if ω > ωc then current leads and voltage lags. Conversely, if ω < ωc then voltage leads and current lags. Let us compare this behaviour with the pill-box resonator. From equation (42) we ﬁnd that Ψ < 0 when ω > ωc , and so from (48) voltage lags the current. Conversely, when ω < ωc then Ψ > 0, and so the voltage leads the current. Evidently, with respect to detuning, the resonator displays the same behaviour as the parallel LCR circuit. 5 Driven, on resonance, lossless This particular case is somewhat away from the main stream of our presentation, but is given here for completeness. The ﬁelds are anticipated to increase linearly with time. We postulate magnetic and electric ﬁelds of the form: H = [ωtH + h] h(x)e+jωt , ˜ ˜ E = [ωtE + e] e(x)e+jωt . ˜ ˜ (55) ˜ ˜ The ‘extra’ ﬁelds, h and e, account for the fact that the beam current will give rise to ﬁelds even at time t = 0. Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of magnetic ﬁeld, we take a circular path close to the circumference of the cavity, that is just inside the dielectric. ˜ ˜ ˜ ω ˜ [ωtH + h] = Ib + e [E(1 + jωt) + j˜] . (56) k Here, for brevity, we introduced the current per unit length of azimuth: ˜ Ib ˜ Ib = . (57) 2πaJ1 (P01 ) Applying Stokes’ theorem to Maxwell’s equation for the ‘curl’ of electric ﬁeld, we take a rect- angular path along the centre line, and along the cavity walls. ω ˜ ˜ ˜ −µ [H(1 + jωt) + j h] = [ωtE + e] . ˜ (58) k We compare the coeﬃcients of t to obtain the simultaneous equations: ˜ ˜ H = j(ω/k)E (59) ˜ ˜ −µj(ω/k)H = E . (60) 9 These relations are identical with those of section 1, and imply that ω = ωc = kc ˜ ˜ and E = −jη H. Note, it was the assumed form of the time dependence (i.e. ﬁelds linearly proportional to t) that forces ω = ωc – it was not that we assumed, at the outset, that the drive was ‘on resonance’. We compare the constant terms to give the simultaneous equations: ˜ ˜ ˜ e H = Ib + [E + j˜]/η (61) ˜ ˜ = e, −η[H + j h] ˜ (62) where we have used ωc /k = 1/η and µ ωc /k = η . These relations may be rearranged to obtain a more symmetrical form: ˜ ˜ ˜ η h − j˜ = η Ib + E e (63) ˜ ˜ ˜ e + jη h = −η H = −j E . ˜ (64) We may solve equations (63) and (64) to obtain e ˜ ˜ (˜ + jη h) = j Ib η /2 (65) η ˜ Ib ˜ E = − . (66) 2 2πaJ1 (P01 ) Evidently, the electric ﬁeld induced by the beam opposes its motion and acts to decelerate it. References o [1] G. Dˆme: RF systems: waveguides and cavities; CERN SPS/86-24 (ARF). [2] Ramo, Whinnery, Van Duzer: Fields and waves in communication electronics; John Wiley & Sons, New York. [3] S. Koscielniak: Introduction to RF systems in Accelerators; USPAS, San Diego, January 1996. http://decu10.triumf.ca:8080/uspas/ 10