# Field analysis of a dielectric-loaded rectangular waveguide accelerating structure. Page: 2 of 3

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A ky cos-(x+-)cosky,,y,0<y<a

E = W 2

B1 kIW cos- 2(x+-)sinnk'(b-y),a<y<b

w w 2

AEkk= sin w(x+-)sinky")y,0<y<a

Bk,,kk, sin-(x+-)cosk'>)(b- y),a < y <b

A,,(-j,,,)k( m sin (x+r)cosk( )y,0<y<a

w 2 (3)

B,,(-j,,,)k>, sin - (x+-)sink) (b-y),a<y<b

Tr- ,o, sin -(x+ -)sin k 0 y,0 <y< a

H = W 2

-B,,CoEoe,/3, sin (x+-)cosk) ,(b-y),a<y<b

W 2

mA mA 2

A, (jwE0)-cos-(x+-)sink y,0 <y <a

H w w 2

Bi (JWEEr) cos W(x+ )cosk 21(b- y),a < y <b

Lw w 2

where kemn2=(m-/w)2+-mn2. Note that exp(-j-mnz) has

been omitted in these expressions for simplicity.

At the interface between the region 0 and I, the

tangential electric and magnetic field components are

continuous:

Ak cos k a = B1 k' sin k"' (b - a), (4a)

A_ sin k0 a = B_ e, cos k0 (b - a), (4b)

Dividing (4a) by (4b) yields the dispersion relation,

kt', tan(ky21,(b - a))= Ek0 cot(k20) a) (5)

This is a transcendental equation of the general form

of a complex function of Jlmn and f For the

inhomogeneous guide considered here, the dispersion

relation must be solved numerically. An infinite

number of discrete solutions exist.

From (4), the coefficient ratio A/B can be found,

and, when substituted into (3), completes the solution.

3. PROPERTIES OF THE LSM , AS

ACCELERATION MODE

In general, the relative advantages of accelerating

structures can be understood with reference to figures

of merit such as the ratio of the peak surface electric

field to the axial acceleration field E/EO, the group

velocity vg, the attenuation constant *, and R/Q which

measures the efficiency of acceleration in terms of the

given stored energy, etc. The definitions of these

parameters will be quantified here. In this section,

only the results pertaining to the LSM11 mode are

given, because the LSM11 mode is the lowest luminalmode in our considered structures. And also because

its symmetrical field distribution in x and y-direction.

3.1 The ratio of surface field E, to accelerating field EoE, E,(0,b) + + w

E E 0,0) k sin k a = sin( a)

E E, ky/

w(6)

It is desirable to have this ratio as small as possible

since the surface field contributes nothing to

accelerating the beam but is responsible for breakdown

of the structure.

3.2 The group velocity vgP 1

v =- P=-JJE H dxdy

g U 2(7)

2L ( 0 r (y)(E2 +Ey +E2)+p0(H2 +Hz ))dxdydz

where P is the power flow over the half section under

consideration and U is the stored energy per unit length

in the half structure.

3.3 The quality factor Q (ignoring dielectric losses)

and the attenuation constant-

_ U

Q ossP,, = R, sf ds

(8)

R = and 5s, 2

S* a- p ao

where - is the conductivity of the metal wall.

CO P

a loss (9)

2Qvg 2P

3.4 The normalised shunt impedance R/Q

R E v

ep" (10 )

where Eo is the maximum accelerating field on the z-

axis.

4. NUMERICAL RESULTS

Figure 2 shows the dispersion curves of the lower

LSMm1 and LSEm, modes for an X-band structure.

Because the LSM11 mode is the lowest luminal mode

and its field distribution pattern, it will be chosen as the

accelerating mode. Figure 3 gives the normalized

longitudinal electric field distributions of the LSM

synchronous mode as a function of x and y using the

field analysis method.

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### Reference the current page of this Article.

Xiao, L.; Gai, W. & Sun, X. Field analysis of a dielectric-loaded rectangular waveguide accelerating structure., article, July 16, 2001; Illinois. (digital.library.unt.edu/ark:/67531/metadc723231/m1/2/: accessed December 13, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.