Investigation of the bandpass properties of the local impedance of slow wave structures

1250
ISSN 1064-2269, Journal of Communications Technology and Electronics, 2008, Vol. 53, No. 10, pp. 1250–1258. © Pleiades Publishing, Inc., 2008.
Original Russian Text © S.V. Mukhin, D.Yu. Nikonov, V.A. Solntsev, 2008, published in Radiotekhnika i Elektronika, 2008, Vol. 53, No. 10, pp. 1324–1332.
INTRODUCTION
The interaction of an electron flow and the electro-
magnetic field in the resonator slow-wave structures
(SWSs) of power traveling-wave tubes (TWTs) is often
investigated with the use of RLC circuits or multiport
chains (excited by the current induced by the electron
flow) equivalent to SWSs. This representation makes it
possible to analyze the interaction characteristics
within and, sometimes, beyond the passbands of SWSs
and to perform calculation of a device in a reasonable
time. However, the choice of equivalent circuits of
SWSs and determination of the parameters of these cir-
cuits are a special problem necessitating more rigorous
theoretical methods.
Direct simultaneous numerical solution of the inho-
mogeneous Maxwell equations and the equations of
electromagnetics according to the available Karat,
Mafia, and Magic computer codes allows the analysis
and simulation of certain variants of devices. However,
the application of these equations for calculation of
power TWTs is impeded because real resonator SWSs
are complex and calculations necessitate substantial
computational resources.
The theory of excitation of periodic waveguides
uses a solution to the inhomogeneous Maxwell equa-
tions for the fields excited by the electron beam’s cur-
rent. The solution is decomposed in the eigenmodes of
a periodic waveguide, and the quasi-static electromag-
netic field is separated in the decomposition. Therefore,
this theory facilitates the solution of the above prob-
lems [1]. In this approach, it suffices to take into
account only (i) the fields of the waves that are synchro-
nous with the electron flow; (ii) the quasi-static field of
the spatial charge; and, sometimes, (iii) a small number
of asynchronous waves that provide for dynamic cor-
rections to this field. The equations of wave excitation
contain parameters that can be calculated from the
exact 3D simulation of an SWS or can be determined
approximately with the help of other methods for cal-
culating cold SWSs without electron beams [2]. Usu-
ally, the phase velocity and the coupling impedance of
the operating spatial field harmonic are involved in the
excitation equations. However, it is impossible to use
these parameters when a TWT operates at frequencies
near the passband of an SWS and when a transition to
the stopband occurs. The reason is that the coupling
impedance becomes infinite at the passband boundary.
It is known that this difficulty can be eliminated if we
consider the total excited field of forward and counter-
propagating SWS waves that, near the passband bound-
ary, have spatial harmonics synchronous with the beam.
The most general approach is based on the second-
order finite-difference equation of SWS excitation for
the total field [3, 4]. This equation contains, instead of
the standard coupling impedance for spatial harmonics,
the local coupling impedance that characterizes the
field intensity in interaction gaps. The analysis [4, 5] of
the general properties of this impedance has shown that
it is continuous at the boundaries of the passbands of an
SWS. A simple example of calculation of the local
impedance is presented in [5].
Here, we demonstrate that the local coupling imped-
ance can be determined through both the characteristics
of the electromagnetic field of an SWS and the param-
eters of a two-port chain simulating this structure. A
waveguide–resonator model (WRM) of an SWS is used
for investigation of the frequency properties of the local
coupling impedance for looped SWSs applied in power
TWTs. Such SWSs include interdigital lines, looped
waveguides, chains of coupled resonators with cou-
pling slots turned through 180°, and similar structures
Investigation of the Bandpass Properties of the Local Impedance
of Slow-Wave Structures
S. V. Mukhin, D. Yu. Nikonov, and V. A. Solntsev
Received April 8, 2008
Abstract—The properties of the local coupling impedance that determines the efficiency of the electron–wave
interaction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char-
acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-port
chain simulating the structure. The continuous behavior of the local coupling impedance in the passbands of
slow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the help
of a waveguide–resonator model.
PACS numbers: 41.20.Jb, 84.40.Dc
DOI: 10.1134/S1064226908100136
MICROWAVE
ELECTRONICS

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008
INVESTIGATION OF THE BANDPASS PROPERTIES 1251
in which the flux of the electromagnetic-wave energy
repeatedly crosses the axis of a structure. We show that
the local coupling impedance is continuous and finite
within, at the boundaries of, and beyond the passbands
of the considered SWSs.
1. LOCAL COUPLING IMPEDANCE
IN THE DIFFERENCE EQUATION
OF EXCITATION OF PERIODIC WAVEGUIDES
Consider sum (x, y, z, t) of the electric fields of the
forward (+s) and counterpropagating (–s) modes of a
periodic waveguide with period L:
(1)
According to the Floquet theorem, the eigenmode
field in a periodic waveguide has the form
(2)
where hs are wave numbers, is the amplitude of the
chosen field component at point (x0, y0, z0) at which the
distribution function of this component is unity, (x,
y, z) are distribution functions periodic in z, and the
time factor exp(–iωt) is omitted. According to the the-
ory of excitation of periodic waveguides [1, 3], the
excitation coefficients satisfy the equations
(3)
where Ns = – is the norm
of an eigenmode, S(z) is the cross section of a structure,
and (x, y, z) is the density of the exciting current at fre-
quency ω.
A second-order equation coupling the total fields in
three period-spaced sections z, z ± L can be obtained
with the use of first- and second-order finite differences
introduced according to the relationships
(4)
The calculation of the differences from (1)–(3) yields [3]
(5)
E
E x y z, ,( ) Cs z( )Es x y z, ,( ) C s– z( )E s– x y z, ,( ).+=
E s± x y z, ,( ) E s±
0
e s± x y z, ,( ) ihsz±( ),exp=
E s±
0
e s±
dC s±
dz
-----------
1
Ns
------ j x y z, ,( )E s+− x y z, ,( ) S,d
S z( )
∫±=
EsH s–[ ]{S z( )∫ E s– Hs[ ]}z0dS
j
∆±E E x y z L±, ,( ) E x y z, ,( ),–=
∆
2
E ∆+E ∆–E– E x y z L+, ,( )= =
– 2E x y z, ,( ) E x y z L–, ,( ),+
∆±C+s± C+s z L±( ) C+s z( ),–=
∆±C s–± C s– z L±( ) C s– z( ).–=
∆
2
E 2E 1 ϕscos–( )+ G,=
where ϕs = hsL is generally the period complex phase
shift and excitation function has the form
(6)
We investigate SWS excitation within passbands
and stopbands with the use of a 1D model of the inter-
action between the electron flow and the field. In this
case, the exciting current is aligned with the longitu-
dinal axis of the SWS. Function ψ(x, y) of the current
distribution over cross section Se of the beam is spec-
ified as
(7)
and normalized with the relationship
(8)
Then, J(z) is the HF current of the beam and S =
1/ is the effective area of its cross
section.
In the 1D model, the field and other quantities are
averaged according to the relationship
(9)
In [3, 5], the difference equation of excitation for the
field averaged according to this relationship is derived
in a different way. Here, we present this equation for the
case of the discrete interaction between the electron
beam and the field in SWS gaps. There are structures in
which full period L contains one interaction gap (e.g.,
comb structures, diaphragmatic waveguides). In a
looped SWS, there are two interaction gaps spaced by
the step D = L/2. It can be assumed that the field inter-
acting with the electron beam is in phase along the
SWS axis. By averaging field (2), we obtain for the qth
step
(11)
where (z) is the real distribution function averaged
over the section. This function is the same (e.g.,
because of the structure’s symmetry) for the forward
and counterpropagating waves.
By averaging (5), we can show [3, 4] that the aver-
aged total longitudinal field has the following form for
the qth step:
(12)
G
G ∆+C+s iϕs( )exp ∆–C+s iϕs–( )exp–( )E+s=
+ ∆+C s– i– ϕs( )exp ∆–C s– iϕs( )exp–( )E s– .
j x y z, ,( ) J z( )ψ x y,( )z0=
ψ x y,( ) Sd
Se
∫ 1.=
ψ
2
x y,( ) SdSe∫( )
1–
Ez z( ) ψ x y,( )Ez x y z, ,( ) S.d
Se
∫=
E s z,± z( ) E s±
0
e s± z( ) ihsz±( )exp=
= E s±
0
e z( ) iqϕs±( ),exp
e
Eq z( ) e z( )Eq,=

1252
MUKHIN et al.
where Eq satisfies the difference equation
(13)
In this equation,
(14)
is the specific coupling resistance at point (x0, y0, z0),
while
(15)
is the effective width of the equivalent plane gap. The
width is chosen such that the voltage across the gap is
equal to the step voltage:
(16)
where zq is the coordinate of the qth gap’s center.
The quantity
(17)
has the meaning of the current induced in the qth-step
interval.
The difference equation for the voltage is obtained
from (13) through simple multiplication by –d:
(18)
The local coupling impedance of the gap is intro-
duced according to the relationship
(19)
This impedance enters the right-hand side of differ-
ence equation (18) and characterizes the intensity of the
field excited by the current. Note that the local imped-
ance takes into account the distributions of the field and
∆
2
Eq 2Eq 1 ϕscos–( )+ iRs
0
ϕsJqd.sin–=
Rs
0 2E+s
0
E s–
0
Ns
-------------------–=
d e z( ) zd
zq D/2–
zq D/2+
∫=
Uq Eqd– Eq e z( ) zd
zq D/2–
zq D/2+
∫– E z( ) z,d
zq D/2–
zq D/2+
∫–= = =
Jq
1
d
--- J z( )e z( ) zd
zq D/2–
zq D/2+
∫=
∆
2
Uq 2Uq 1 ϕscos–( )+ iRs
0
Jqd
2
ϕs.sin=
Zs Rs
0
d
2
ϕs.sin=
of the HF current flowing across effective width d of the
gap and allows the uniform analysis of the SWS excita-
tion within, at the boundary of, and beyond the pass-
band because this impedance has singularities at the
boundary. Generally, local coupling impedance Zs can
be calculated via the above relationships upon process-
ing of the results of 3D simulation of the SWS fields or
via simpler models.
2. LOCAL COUPLING IMPEDANCE
OF A TWO-PORT CHAIN
Let us represent an SWS as a chain of two-ports
connected in series (Fig. 1) and consider the behavior
of Zs(ω) and dispersion characteristics ϕs(ω). This rep-
resentation is widely used in numerous studies on the
theory and calculation of TWTs. In order to express Zs
through the two-port parameters, we derive a difference
equation of excitation of form (18) by directly using the
equivalent circuit from Fig. 1. The problem of excita-
tion by a given current is solved under the assumption
that the motion of electrons in the gap is known and
does not depend on the voltage; i.e., it is assumed that
electron-beam current J(z) is specified. In this case, the
excitation currents of the two-port chain coincide with
the specified induced currents (see Fig. 1). Then, the
currents and voltages within one period are coupled
according to the following formulas:
(20)
Using the second-order finite differences ∆2U =
Uq + 1 – 2Uq + Uq – 1 and taking into account the reciproc-
Uk A11Uk 1+ A12Ik 1+ ,+=
Uk 1– A11Uk A12 Ik Jk
–
–( )+ ,=
Ik Jk
+
+ A21Uk 1+ A22Ik 1+ ,+=
Ik 1– A11Uk A22 Ik Jk
+
–( ),+=
Jk Jk
–
Jk
+
.+=
^^
q = Q
AAAA
q = k + 1
Uk + 1
q = k
Ik + 1
ZQ
Uk
Ik
Jk
–
Jk
+
Ik–1
q = 1 q = k–1
Uk–1
Jk
Z1 A
Fig. 1. Two-port chain modeling an SWS section of Q steps: Uq is the voltage across the interaction gap between the (q – 1)th and
qth two-ports, and Z1 and ZQ are the impedances of the loads at the beginning and end of the chain, respectively.

ity conditions A11A22 – A12A21 = 1 for two-ports and the
dispersion equation
(21)
we obtain from (20) an equation similar to (18):
(22)
The comparison of (18) and (22) yields the elemen-
tary expression
Zs = iA12. (23)
It should be emphasized that the two-ports forming
the chain (see Fig. 1) describe SWS cells of a complex
shape and the corresponding transmission matrices
may contain coefficients Aij that are expressed accord-
ing to complicated formulas, but satisfy the reciprocity
condition.
It is a common practice in the theory of TWTs to use
coupling resistance Ks, m of the mth spatial harmonic of
the sth wave. This resistance is related to Zs as follows [5]:
(24)
where ϕs, m = 2ϕs + 2πm is the phase shift of the mth-
harmonic field by full period L and |es, m| are the ampli-
tudes of spatial harmonics.
When, in the nonlinear theory of TWTs, an SWS is
modeled as a two-port chain, the voltage across the qth
gap is often represented in the form
(25)
where mutual impedances Zqk determine the contribu-
tion of the kth-gap current to the voltage across the qth
gap. These impedances are calculated through the two-
port parameters and impedances Z1 and ZQ of the input
and output loads with the help of the recurrence recal-
culation formulas. Using these formulas and forming
the second-order finite difference ∆2U = Uq + 1 – 2Uq +
Uq – 1 with allowance for (25), we can show that repre-
sentation of Uq in form (25) exactly satisfies second-
order difference equation (22) with specified boundary
conditions that are determined by loads Z1 and ZQ.
Therefore, expression (25) can be regarded as the rep-
resentation of a solution to Eq. (22) through the discrete
source function (the discrete Green’s function) deter-
mined by the set of impedances Zqk. System of linear
equations (22) can be solved with the help of other
methods, in particular, the sweep method. In the nonlin-
ear theory of TWTs, the method of solution is chosen
according to the possibilities of minimizing the compu-
tation time and necessary computational resources.
ϕscos
A11 A12+
2
----------------------,=
∆
2
Uk 2Uk 1 ϕscos–( )+ A12Jk.–=
Ks m, Zs
es m,
2
ϕs m,
2
ϕssin
-----------------------,=
Uq U1 i q 1–( )ϕs( )exp=
+ UQ i q Q–( )ϕs–( )exp Zqk Jk,
k 1=
Q
∑+
3. ANALYSIS OF THE BANDPASS PROPERTIES
OF THE LOCAL COUPLING
IMPEDANCE WITH THE USE
OF THE WAVEGUIDE–RESONATOR MODEL
According to the WRM proposed in [6], an SWS is
replaced with an equivalent structure made from
waveguide sections. In contrast to [7], in the WRM,
waveguide sections may be oriented along and across
the SWS axis in accordance with the direction of the
energy flux on a specific SWS section. The application
of the WRM in theVEGA code for TWT simulation has
shown that it is possible to describe the SWS properties
qualitatively well. This description can likewise be
quantitatively adequate if the waveguide dimensions
are appropriately chosen with the use of reference
points. In this study, the properties of the local coupling
impedance and the properties of dispersion in the SWS
passbands and stopbands are considered with the help
of an elementary WRM consisting of two or three
waveguide sections (Fig. 2). In each section, only one
wave (with wave number h and impedance Z) is taken
into account.
For a rectangular waveguide, this wave is the H10
mode with the components of the electric field
(26)
where k = ω/c and Z0 = are the wave number
and impedance in free space, respectively; kc = π/a is
the critical wave number of the H10 waveguide mode;
h = ; and the x, y, and z axes are oriented along
the wide wall with dimension a, the narrow wall with
dimension b, and the longitudinal axis of the consid-
ered waveguide section, respectively. Generally, the
WRM can be constructed with the use of waveguide
sections of arbitrary cross sections: H-shaped,
Π-shaped, or of other shapes.
Such a WRM approximately describes the main
SWS properties within at least the first two passbands
and stopbands for a looped SWS (a looped waveguide,
an interdigital line, a chain of coupled resonators with
coupling slots turned through the angle 180°) and for a
helical twisted waveguide.
We determine impedance Z of a rectangular
waveguide as the ratio of x-averaged voltage U to sur-
face current J flowing along the waveguide’s axis [8]:
Hz C kcx ihz( ),expcos=
Hx i
h
kc
----C kcx ihz( ),expsin–=
Ey iC
k
kc
----Z0 kcsin x ihz( ),exp=
Ex Ez 0, Hy 0,= = =
µ0/ε0
k
2
kc
2
–

1254
MUKHIN et al.
(27)
If loss is disregarded, impedance Z is real in the
waveguide’s passband when k > kc and imaginary
beyond the passband when k < kc. In the latter case, the
impedance is inductive for a decaying wave (h =
).
Then, the transmission matrix of the jth waveguide
section of geometric length lj and electric length θj = hjlj
has the form
(28)
The transmission matrix of the SWS section between
neighboring interaction gaps (see Fig. 2) is a product of
three matrices: Ä = Ä1Ä2Ä3. The first and third
waveguide sections are identical. Then, Ä1 = Ä3 and we
obtain the following expressions for the components:
Z
U
J
----
1
a
--- Eyb xd
0
a
∫–
Hx xd
0
h
∫
------------------------- Z0
b
a
---
k
h
---.= = =
i kc
2
k
2
–
Aj
θj( )cos iZj θj( )sin–
i
Zj
----- θj( )sin– θj( )cos
.=
A11 A22 θ1 θ2coscos
1
2
---
Z1
Z2
-----
Z2
Z1
-----+
⎝ ⎠
⎛ ⎞ θ1 θ2,sinsin–= =
A12 iZ1 θ1 2 θ1 θ2coscos
Z1
Z2
----- θ1 θ2sinsin–
⎝ ⎠
⎛ ⎞sin–=
(29)
where θ1 denotes the total electric length of the first and
third waveguide sections.
Along with (21), (23), and (27), expressions (29),
which are obtained with the use of the WRM, allow the
analysis of the dispersion and the local coupling imped-
ance in a frequency band. Let us consider looped SWSs
with allowance for the geometric rotation of the field
phase in the neighboring gaps. In a linear WRM that is
a two-port chain (see Fig. 1), this change of the phase
can be taken into account through introduction of
induced currents of opposite directions in the neighbor-
ing gaps. The dispersion is calculated for the first spa-
tial harmonic, which is used in such TWTs, according
to the formulas
(30)
where ϕs is the phase shift on step interval D, ϕs, 0 =
ϕs – π ≤ 0 is the phase shift of the fundamental spatial
harmonic with allowance for the geometric rotation,
ϕs, 1 = ϕs, 0 + 2π is the phase shift of the first spatial har-
monic, and hs, 1 = ϕs, 1/D is its wave number.
– iZ2 θ1 θ2,sincos
2
A21
i
Z1
----- θ1 2 θ1 θ2coscos
Z2
Z1
----- θ1 θ2sinsin–
⎝ ⎠
⎛ ⎞sin–=
–
i
Z2
----- θ1 θ2,sincos
2
ϕscos A11, ϕs 1, ϕs π,+= =
c
Vs 1,
---------
hs 1,
k
--------
ϕs 1,
kD
---------,= =
Electron
beam
q + 1
q
l
1
2
3
D
Fig. 2. Schematic of a looped SWS with an electron flow: (2) waveguide section (of the length l2 = D) corresponding to a coupling
slot and (1, 3) waveguide sections of the total length l-D with interaction gaps.

Local coupling impedance Zs is calculated with the
use of expression (23), where A12 is calculated from for-
mula (29), or through direct multiplication of three
matrices, Ä = Ä1Ä2Ä1. The correctness of single-mode
description (26) of waveguide sections and their imped-
ance (27) is checked through calculation of the trans-
mission matrix of three series-connected sections by
means of the ISFEL-3D code. As an example, coupling
resistance K and slowing factor n calculated through the
parameters of the transmission matrix (solid line) and
with the help of the ISFEL-3D code (dots) are depicted
in Fig. 3.
Consider certain variants that characterize the
behavior of the properties of looped SWSs.
A Homogeneous Looped Waveguide
of a Constant Section
In this case, from (29) and (30), we obtain
(31)
Here, λπ is the low-frequency boundary of the main
passband. This boundary corresponds to ϕs, 1 = π and is
determined by the critical frequency of a looped
waveguide:
A11 θ1 θ2+( ), ϕscos θ1 θ2+ hl,= = =
ϕs 1, θ π,+=
c
Vs 1,
---------
ϕs 1,
kD
---------
l
D
---- 1
λ
λπ
-----
⎝ ⎠
⎛ ⎞
2
–
λ
2D
-------+= =
=
c
V1 π,
----------
2l
λπ
----- 1
λ
λπ
-----
⎝ ⎠
⎛ ⎞
2
–
λ
λπ
-----+
⎝ ⎠
⎛ ⎞ .
(32)
High-frequency boundary λ2π of the main passband
corresponds to ϕs, 1 = 2π and is absent in an ideal reﬂec-
tion-free WRM. However, in a real structure, this
boundary exists because of reﬂections from the
waveguide’s bends and from the junctions of
waveguide sections.
We apply relationship (31) to estimate the maximum
possible width of the main passband of the structure.
Setting in (31) ϕs, 1 = 2π and taking into account (32),
we obtain
(33)
i.e., we always have λ2π < λπ.
Consider the possibility of equal slowing factors at
the edges of the main passband. This situation is real-
ized when
(34)
In this case, the width of the main passband is one
octave,
λπ λc, h 0,
c
V1 π,
----------
λπ
2D
-------.= = =
1
λ2π
2
-------
1
λπ
2
-----–
1
2l( )
2
-----------,
c
V1 2π,
------------ 2
λ2π
λπ
-------
c
V1 π,
----------;= =
c
V1 2π,
------------
c
V1π
--------, λπ 2λ2π, l
λπ
2 3
----------,= = =
D
3l
c/V1π
-------------.=
f 2π 2 f π, ∆f / f 0 2
f 2π f π–
f 2π f π+
------------------- 66.6%,= = =
3.02.82.52.32.01.8
6
5
4
3
2
1
0
120
100
80
60
40
20
n K, Ω
λ, cm
2
2
1
1
Fig. 3. (Curves 1) Coupling impedance K and (curves 2) slowing factor n calculated (solid line) on the basis the parameters of the
transmission matrix and (dots) with the use of the ISFEL-3D code.

1256
MUKHIN et al.
and the slowing factor is determined from the relation-
ship
The corresponding dispersion characteristic is
depicted in Fig. 4a (curve 1). Dispersion curves for the
looped waveguide are calculated for various l/λπ =
with disregard for reflections and presented in Fig. 4a in
the normalized slowing-factor and frequency variables
n/nπ = V1π/V1 and ω/ωπ = λπ/λ, respectively.
From (23), (27), (29), and (31), we can find an
elementary expression for the local coupling imped-
ance of the homogeneous looped waveguide for the
case Z2 = Z1:
(35)
c
V1
------
c
V1π
--------
1
3
------- 1
λ
λπ
-----
⎝ ⎠
⎛ ⎞
2
–
λ
λπ
-----+ .=
kπl
2π
-------
Zs Z1 ϕssin Z01
k
h
--- lh,sin= =
which coincides with the expression obtained in [5].
The frequency dependences of the local coupling
impedance that are depicted in Fig. 4b show that it
changes continuously when the passband-to-stopband
transition occurs. Depending on the waveguide’s
dimensions, the maximum of Zs may be located in dif-
ferent bands. The absolute value of Zs is independent of
the shape of the looped waveguide: In particular, Z01 =
= for a rectangular waveguide and, for an
H-shaped waveguide or a slot line, Z01 may substan-
tially increase, because a Ӷ λc/2.
An Inhomogeneous Looped Waveguide Formed
from Sections with Equal Critical frequencies and Dif-
ferent Impedances: kc2 = kc1 = kc, Z2 ≠ Z1.
This case corresponds to a looped waveguide such
that only width b of the narrow wall may be variable if
rectangular-waveguide sections are considered. This
case also describes the properties of an interdigital
SWS.
Then, we have
and the dispersion equation takes the form
(36)
In this case, there are gaps in a passband at ϕs, 1 = 2π,
the width of the first passband decreases simulta-
neously with the slowing factor in this passband, and
both normal and anomalous dispersions are possible
(Fig. 5a). The local coupling impedance remains a con-
tinuous function of frequency when the transition to the
stopband occurs. The presence of the stopband is due to
the superposition of reflections from the junctions of
waveguide sections, i.e., due to the longitudinal reso-
nance (the Bragg resonance) in the SWS (Fig. 5b). The
dispersion in the SWS remains unchanged when the
first and second waveguide sections are interchanged,
because β is the same for Z1 Z2. However, the fre-
quency dependences and the value of the local coupling
impedance change because the exciting electron-beam
current is introduced into different waveguide sections.
This effect was investigated earlier during the analysis
of the frequency dependences of the coupling imped-
ance [9].
An Inhomogeneous Looped Waveguide Formed
from Sections with Different Critical Frequencies and
Different Impedances: kc2 ≠ kc1 and Z2 = Z1.
This case is the most general and describes the tran-
sition from an interdigital SWS with a transmitting cou-
b
a
---Z0
2b
λc
------
1
2
---
Z1
Z2
-----
Z2
Z1
-----+
⎝ ⎠
⎛ ⎞ 1
2
---
Z01
Z02
-------
Z02
Z01
-------+
⎝ ⎠
⎛ ⎞ β const 1,≥= = =
ϕscos θ1( ) θ2( )coscos β θ1( )sin θ2( ).sin–=
2.01.51.00.50
1.8
1.6
1.4
1.2
1.0
n/nπ
ϕ = π
ϕ = 2π
ϕ = 3π
1
2
3
(a)
2.01.51.00.5
ω/ωπ
0
40
30
20
10
Zs/Z02
(b)
1
2
3
ω/ωπ
Fig. 4. Frequency dependences of (a) the normalized slow-
ing factor of the first spatial harmonic and (b) the local cou-
pling impedance for a homogeneous looped waveguide cal-
culated at l/λπ = (1) 0.29, (2) 0.5, and (3) 0.75.

pling slot to an SWS that is a coupled-resonator chain
with an evanescent coupling slot.
The values ϕs = 0, ϕs, 1 = π (transverse resonance)
and ϕs = π, ϕs, 1 = 2π (longitudinal resonance) corre-
spond to the boundary frequencies of the first passband.
We have performed calculations for a WRM formed
from rectangular-waveguide sections. Width a2 of the
coupling slot was decreased in comparison with the
width of the coupling slot used for Fig. 4 (curve 2), the
remaining parameters being retained.Accordingly, crit-
ical frequency ωc2 of the second waveguide section
describing the slot was increased; i.e., ωc2 > ωc1. The
calculation results presented in Fig. 6 show that the
passband decreases as ωc2 grows, but, in this case, the
local coupling impedance changes continuously when
the transition to the stopband occurs.
CONCLUSIONS
The local coupling impedance determines the effi-
ciency of the electron–wave interaction in a periodic
SWS with allowance for the forward and counterprop-
agating waves of the structure that are synchronous
with the electron flow near the cutoff frequencies. This
impedance is determined (i) in terms of the characteris-
tics of the electromagnetic field with the use of 3D and
simpler models of SWSs and (ii) in terms of the param-
eters of the two-port chain modeling an SWS. It has
been confirmed that the local coupling impedance is
2.01.51.00.50
2.0
1.5
1.0
0.5
n/nπ
ϕ = 2π
ϕ = 3π
ϕ = π
1, 3
1, 3
(a)
(b)
15
12
9
6
3
0 2.01.51.00.5
ω/ωπ
1
2
3
ω/ωπZs/Z02
pling impedance for an inhomogeneous looped waveguide
formed from waveguide sections with equal critical fre-
quencies and different impedances. The results are obtained
for l/λπ = 0.5; l/D = 1.875; and Z1/Z2 = (1) 2/3, (2) 1, and
(3) 3/2.
1.00.5 1.50
1.3
1.0
0.8
0.5
n/n2π
(a) ϕ = 2π
ϕ = 3π
ϕ = π
1
2
3
4
1
2
3
4
1.51.00.50
ω/ωπ
15
12
9
6
3
(b)
1
3
4
2
Zs/Z02 ω/ωπ
pling impedance for an inhomogeneous rectangular looped
waveguide. The results are obtained for l/λc1, 3 = 0.5, l/D =
1.875, and varying widths of the coupling slot: a2/a1, 3 = (1)
0.5, (2) 0.8, (3) 0.9, and (4) 1. Curves 1 correspond to a nar-
row coupling slot that is evanescent for all frequencies of
the main passband of a homogeneous looped waveguide.
Curve 4 is obtained for this waveguide.

1258
MUKHIN et al.
continuous within, at the boundaries of, and beyond
SWS passbands. Therefore, the linear and nonlinear
interactions between the electron ﬂow and the ﬁeld
within and beyond SWS passbands can be uniformly
described in a wide frequency range. The obtained
results can serve as the basis for 1D–3D simulation of
power TWTs with periodic SWSs.
ACKNOWLEDGMENTS
This study was supported by the Russian Founda-
tion for Basic Research, project no. 07-02-00947.
REFERENCES
1. L. A. Vainshtein and V. A. Solntsev, Lectures on Micro-
wave Electronics (Sovetskoe Radio, Moscow, 1973) [in
Russian].
2. R. A. Silin, Periodic Waveguides (Fazis, Moscow, 2002)
[in Russian].
3. V. A. Solntsev and S. V. Mukhin, Radiotekh. Elektron.
(Moscow) 36, 2161 (1991).
4. V. A. Solntsev, Lectures on Microwave Electronics and
Radiophysics (X Winter School–Seminar for Engineers)
(GosUNTs “Kolledzh”, Saratov, 1996) [in Russian].
5. V. A. Solntsev and R. P. Koltunov, Radiotekh. Elektron.
(Moscow) 53, 738 (2008) [J. Commun. Technol. Elec-
tron. 53, 700 (2008)].
6. S. V. Mukhin, O. E. Lomakin, and V. A. Solntsev,
Radiotekh. Elektron. (Moscow) 33, 1637 (1988).
7. R. A. Silin and I. P. Chepurnykh, Radiotekh. Elektron.
(Moscow) 35, 939 (1990).
8. I. V. Lebedev, Microwave Engineering and Devices
(Vysshaya Shkola, Moscow, 1970), Vol. 1 [in Russian].
9. S. V. Mukhin, D. Yu. Nikonov, and V. A. Solntsev, in
Thes. LIV Sci. Session of the Popov Society Dedicated to
the Radio Day, Moscow, Russia, 1999 (Ross. Nauch.-
Tekh. Obshch. Radiotekh. Elektron. Svyazi, Moscow,
1999), p. 42.

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