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The Effect of Tolerance on Interdigital Lines
2.5 The Effect of Tolerance on Interdigital
Lines
by M. C. PEASE
I. II. III. IV.
Continuous Small Periodic Errors Stepwise Systematic Error (Misregister) Stepwise Random Errors Conclusions List of Symbols References
88 89 92 96 96 97
I n t h e design a n d construction of extended interaction tubes, whether helix, interdigital, disc-loaded waveguide t y p e s or other, t h e problem of tolerances on t h e delay line h a s a critical importance. I t is more t h a n a problem of ease of manufacture; it m a y even determine whether t h e t u b e can be built a t all. F o r these structures are such t h a t even quite minor tolerances m a y accumulate t o a disastrous degree. Some consideration h a s been given t o this problem, although for a differ ent purpose, b y Moore (1) and b y Mullen and P r i t c h a r d (2). T h e i r papers are concerned with t h e r a n d o m combination of reflections t h a t m a y not b e individually small. F o r our purposes here, however, we m u s t be concerned with t h e resultant of a large n u m b e r of very small reflections whose loca tions are known. T h e t w o problems, although related, are different and require different t r e a t m e n t . I n w h a t follows we shall analyze t h e effect of tolerance and of construc tional errors in extended repetitive structures, particularly of t h e interdigital t y p e commonly used in backward-wave magnetrons. W e shall consider three t y p e s of variations: (a) a long t e r m cyclic error such as m a y occur from an indexing fault in hob grinding, (b) a stepwise periodic error due to misregister of t h e t w o halves of t h e line, a n d (c) stepwise r a n d o m error, either uncorrelated or locally correlated, resulting from t h e dislocation of individual teeth. These three cases will show t h e general consequences t o be expected from t h e inevitable lack of precision in t h e manufacture of delay lines. 87
88
Μ. C. PEASE
I. Continuous Small Periodic Errors
A continuous a n d periodic error m a y arise from indexing tolerances in t h e grinding of t h e hob used t o m a k e t h e line. T h e problem can b e handled as a perturbation one. T h e basic equations are t h e familiar ones for a line, e
- f
-
( i
>
b u t with characteristic impedance Ζ now being a function of x. W e shall assume t h a t Z, after normalization, is nearly constant a t unity, b u t with a small periodic variation: Ζ = 1 + δ sin kx
(3)
where k is a measure of t h e length of t h e periodicity of Z. If, now, we let 2
Ε = Eo + δΕι + δ Ε2 + · · · 2
/ = h + Βΐι + δ / 2 + · · ·
and substitute in (1), collecting b y powers of δ, t h e n we obtain a series of equations which can b e solved for t h e E's or I's separately. T h e first t w o equations for t h e E's, for example, a r e 2
Ei' + ß E0
= 0
(5) 2 Ei' + ß Ei = k cos kxEO These can be solved, either generally or under assumed b o u n d a r y condi tions. T h e conditions of principal interest here are t h a t t h e reflection co efficient shall vanish a t t h e end of t h e line, a t χ = 0, =
E ^ jßZE Ε + I jßZE
+ E' _ - Ε' "
= U
at
X
U
W
This condition is applied t o each order of perturbation separately. If we do this through t h e first order perturbation, a n d t h e n compute Γ t o t h e first power of δ, w e find Γ = ^2 _
fc2
s
n
i kx — k sin 2ßx — jk (cos kx — cos 2ßx)]
(7)
This level of approximation is sufficient for most purposes. 2 2 Of particular interest in this equation is i t s behavior as β —> k /4. E q u a t i o n (7) t h e n becomes indeterminate. I t m a y b e evaluated as a limit to give lim
ikx
Γ = τ (sin kx — kxe )
(8)
2 . 5 TOLERANCE EFFECT ON INTERDIGITAL LINES
89
of which t h e first t u r n can b e ignored if kx is large, i.e., t h e periodicity of t h e error is such t h a t there are several periods within t h e line. W e are left, then, with a reflection t h a t is proportional t o length (times a phase term) with a proportionality factor of kb/4. (Note: Formally, this equation allowed |Γ| t o become > 1 for large enough x. This, however, is d u e t o t h e approximations used, particularly t h a t Γ shall b e small.) T h e reflection coefficient, therefore, has a peak t h a t m a y b e quite large when t h e frequency is such t h a t t h e period of t h e error is a half wavelength long. Although n o t a t r u e cutoff, it h a s t h e same effect a n d m a y cause a " h o l e " in a n operating t u b e . W e shall see a similar effect in Section I I .
II. Stepwise Systematic Error (Misregister)
An interdigital line is formed b y interleaving two " c o m b s . " If these are not properly registered, t h e spaces between t h e t e e t h will b e alternately large and small, with comparable variation of t h e impedance. W e assume t h a t t h e effect of misregister is t o cause alternate sections of high a n d low impedance. W e c a n t h e n consider t h e line t o b e formed b y t h e η-tuple iteration of a unit, such a s is shown in Fig. 1. I n t h e drawing, 0/2
0/2
FIG. 1. Line formed by n-tuple iteration of a unit.
alternate sections a r e of impedance ζ a n d z' (normalized t o t h e nominal impedance so t h a t these are approximately b u t not exactly equal t o u n i t y ) , and t h e electrical length of each section is Θ. T h e matrix relating t h e voltage and current a t t h e i n p u t t o t h a t a t t h e o u t p u t of a unit is written Μ =
\A jC
jB D
(9)
Since we shall assume t h a t t h e line is lossless, A, B, C, a n d D are real. I t is convenient t o t r e a t such a matrix in t e r m s of i t s coefficients (3) which are defined a s : a 0 = | ( A + D)
a2 =
+ C)
ai
a 3 = i(A - D)
(10)
90
Μ . C. P E A S E
and which are related b y 2
2
2
2
a 0 + ai - a 2 - a 3 = 1
(11)
T h e n t h e reflection coefficient is a
Γ =
- ^ - & a 0 + jai
(12)
Since t h e coefficients are real for a lossless network, a n d since a 3 = 0 for a symmetrical one, (12), with (11) can be written
li? = rf^i
(13)
T h e η-tuple iteration of a network such as (9) gives a network whose coefficients are (4): .(ao)
=
n
(cti)n =
a0Un(ao)
—
Un-i(a0)
aiUn(a0)
( α 2 ) η = a 2C/n(a 0) ( α 3) Λ =
cizUn(ao)
where t h e second subscript on t h e coefficient denotes t h e degree of itera tion, a n d where Un is t h e Tschebysheff polynomial of t h e second kind a n d order η in t h e variable ao. These functions can b e written -1 «. sin η c o s a 0 ' TJ ( Un(a0)
=
—
r;
(15)
1
sin cos ao B y t h e usual methods we m a y calculate t h e matrix of t h e unit shown in Fig. 1 t h a t is t o b e iterated. T h e t w o coefficients of interest a r e : 2
2
(Mo) 2 = cos θ - ^ (J + Yj sin θ
" I " ^) + 7 + -l(^3)(f-i)-
( M 2) 2 =
2
)
s i n Θc o s θ
( ) 16
Z
(The subscript 2 because t h e unit of Fig. 1 already includes t w o teeth.) Let us take, as t h e nominal impedance of t h e line a n d its termination, t h e geometric mean of ζ a n d z', so t h a t these impedances, when norjnalized, are reciprocals. Since t h e errors are assumed small, let u s set z
=
1
+
δ
ζ
' = Γ + 1 =
1
-
δ
(17)
Then, t o t h e first order of δ, (Mo) 2 = cos 2Θ ( M 2 ) 2 = δ (sin 2Θ - 2 sin Θ)
(18)
2.5
TOLERANCE
EFFECT ON INTERDIGITAL
LINES
91
W e note t h a t (Af 2 ) n as given b y (14) is t h e product of ( M 2 ) 2 which is a slowly varying function of frequency, a n d of Un(a,o) which introduces t h e rapidly varying part. Hence t h e application of t h e approximation (17), t o t h e t e r m s of (16) is separate. W e h a v e t h e n from (13) a n d (14) t h a t 2
, 1
2| 2 n|
2
2
δ sin 2ηθ (sin 2 0 - 2 sin 0) 2 2 2 " sin 2Θ + δ sin 2ηθ (sin 2 0 - 2 sin 0)
=
2
(
'
A t 0 = π / 2 , this equation becomes indeterminate. T h e limit, however, can be evaluated as l i m
Ι *»Ι = 1 ΓΓ*2 + 4δ η2 Γ
Θ-+*/2
2
2
Λ
(20)
2
A t lower values of 0, t h e curve of |Γ| oscillates rapidly (period Δ0 = τ/η) between zero a n d a n envelope given b y replacing Un b y its m a x i m u m value 1/sin 20: 2
2
τ? ^ IT, 1 δ (1 - c o sL0 ) 22 Envelope 1 Γ 2 η 1 = — 22ü \ > 22 /1 —~r2 cos 0 + δ (1 — cos 0) 2
(21)
,
|Γ|βην, t h e square root of (21) h a s been plotted in Fig. 2 for a 1 % error, i.e., δ = 0.01. I t is seen t o be negligible except in t h e neighborhood of 90°.
o.i II 1/
Ii //
I
//
II Ί !
STE >WISE -
CONriNuouj
// //
******* 30°
60°
90°
θ
FIG. 2. Envelope of fr] for δ — 0.01.
At 90° t h e limit of t h e envelope is 1, b u t t h e limit of t h e function itself is given b y (20), which is always less t h a n 1. For comparison, t h e comparable curve for t h e continuous case is also plotted a s t h e dotted line. T h e t w o cases are m a d e comparable b y setting
/ 0 1
Μ. C. PEASE
92
θ = ßl kl = IT χ = nl where I is t h e length of t h e tooth. T h e n (7) becomes 2
2
• l2 _ / 2πθδ Χ / s i n ηθ 1 1 2 2 ~ \TT - 4(97 \ c o s n ö
if η is e v e n \ if η is odd /
. * { '
a n d t h e first factor is t h e envelope. I t is this formula t h a t is plotted in Fig. 2 as t h e dotted line. T h e difference between it a n d (21) is n o t large. T h e extreme sensitivity of (19) t o very slight misregister a t θ = 7r/2, i.e., when t h e frequency is such t h a t each finger is a quarter-wavelength long, introduces a n effective cutoff a t this value. T h a t this was so, and t h e physical reason for it, w a s pointed o u t b y J. Hull some time ago. T h e present analysis gives a q u a n t i t a t i v e value for it. If, for example, we w a n t |Γ| ^ i (VSWR ^ 2) for a n 80-fingered line, t h e n (24) requires t h a t δ ^ 0.002. F o r most line dimensions used, this is wholly impractical a n d t h e effect acts t o limit t h e range of application of t h e line t o frequencies less t h a n this "pseudo cutoff." Since, in this case, θ is half t h e periodicity of t h e error, this effect is similar t o t h a t noted in Section I for a continu ously varying error. T h e "cutoff" occurs a t t h a t frequency for which t h e period of t h e error is a half-wavelength.
III. Stepwise Random Errors
Among other possible errors in a n interdigital line, t h e most likely a r e r a n d o m errors, either uncorrelated or correlated only with immediate neighbors. T h e latter m a y b e considered as a n approximation t o t h e b e n t t o o t h case, or t o b e a n exact description of t h e displaced t o o t h problem. I n either case a n error leads t o too high a n impedance on one side and t o o low a n impedance on t h e other. T h e correlation is therefore negative a n d exists only between neighboring sections, b u t is complete over this range. Since r a n d o m errors destroy t h e constancy of t h e unit elements, t h e iteration formula employed in t h e last section cannot b e used. Instead, we m u s t compute t h e total matrix directly. F o r this purpose, it is most con venient t o use t h e wave matrix r a t h e r t h a n t h e transmission matrix. W e obtain this b y a similarity transformation: W =
1 1
l
P~ MP
1 - 1
Μ
1 1
1 - 1
(23)
where Μ is t h e transmission m a t r i x of a single section as used before. If now, we set ZN
= 1 + δη
(24)
2.5
TOLERANCE EFFECT ON INTERDIGITAL LINES
93
for the nth unit, numbered from the termination and if the 8n's are small, then, to the second order of the 5's:
(25)
where we have retained only the squared terms in δ η. W e choose this limit of approximation because ( a ) the mean value of the d's will be nearly zero while the root mean square may be significant, and, in fact, may be ex pected to be the dominant effect, and ( b ) while we will consider two-term correlation coefficients (i.e., 8j8k) we will not consider higher order terms (δ7δ*δ, etc.). W e compute, then, a sequence of column matrices defined by
(26)
Sn is the column matrix that describes the forward and backward waves that would be observed were the η sections of the line terminated in 1Ω and "looked at" with a 1Ω detector. B y computing the first few to find the pattern, and proving the pattern by induction, we can find
(27)
Then Γ is the ratio of the t w o terms of (27) (28) Since the numerator contains no term in less than δ and the denominator 3 contains no δ terms, the denominator adds only δ and higher terms. P r o viding, then, that Σδι? and the autocorrelation sums are small compared to 1, we can approximate (29) This may be interpreted 2
as a sequence of random walks of length 2k l)e
(dk — %Sk ) sin Θ along the vectors e^ ~
(and the whole rotated by
2]ne
e- ).
Μ. C. PEASE This is obtained by considering each δ* as having a probability distribution. T h e direct interpretation, therefore, gives t h e distribution of Γ, a t a given Θ, over a universe of lines. T h e curve of Γ vs Θ t h a t we obtain by applying probability theory t o this t h e n describes t h e probable behavior within this universe of lines. W e can, however, interpret this curve further b y consideration of t h e "ergodic t h e o r e m " used, for example, in gas kinetic theory. T h e variation of θ causes t h e 5's t o shift their mode of combination very rapidly. Let u s consider a (2n + 1)-dimensional space in which each of t h e 2n axes cor responds t o one of t h e 8's times t h e real and imaginary p a r t s of its phase term. T h e final axis corresponds t o 0. For t h e universe of lines t h e average Γ is t h e n t h e average over t h e cross section of this space a t t h e given 0. Since we assume t h a t all t h e 8's h a v e t h e same probability distribution, this space has a high degree of s y m m e t r y a b o u t t h e 0 axis (but not com plete symmetry, since there m a y be local correlation). A n y one line a t a given 0 is represented b y a point in this space. As 0 varies, however, this point moves in some elaborate spiral a b o u t t h e 0 axis. And if η is large, it moves around t h e 0 axis very rapidly compared t o its motion along t h e 0 axis. I t appears evident t h a t this generalized spiral will pass near all acces sible regions of this space in accordance with their probabilities; a n d t h a t , therefore, t h e average over t h e universe of lines m a y be t a k e n as t h e average over t h e fine details of t h e p a t h t h a t a particular line traces out. T h e r e will, of course, be fine s t r u c t u r e superimposed on this c o m p u t e d average, b u t t h e average level of t h e standing wave is computable b y this approach. Let us, then, consider (29). B y t h e s y m m e t r y of t h e configuration space, 2 t h e average value of Γ is zero (in spite of t h e presence of δ* t e r m s since these are "washed o u t " b y t h e phase terms). W e must, then, consider t h e 2 absolute value of Γ. W e find t h a t , neglecting t e r m s higher t h a n δ , 2
2
2
|Γ| = sin 0 { Σ δ* + 2 V U =l
k= l
cos 2/c0 V
δΑ+Λ
(30)
J
h= l 2
I n doing this, we h a v e dropped out t h e effect of t h e δ t e r m of (29). M o s t of t h e complexity of (28) has t h u s been lost. I n fact, (30) could h a v e been obtained b y keeping only t h e t e r m s in δ in (25). T h i s is equivalent t o t a k i n g as t h e total reflection simply t h e sum of t h e individual reflections. W e h a v e gone t h r o u g h this r a t h e r elaborate development, however, for two reasons. I n t h e first place, it was not evident t o us t h a t this l a t t e r approximation could be m a d e w i t h o u t losing some of t h e second order factors of (30). W e h a v e m a d e t h e approximations only a t those times when it was rigorously clear t h a t t h e y were justified. W e h a v e also wished t o establish basic formulas t h a t could be used in other w a y s when t h e approximations t h a t we h a v e used m a y not be permissible. T h e problem
2.5
95
TOLERANCE EFFECT ON INTERDIGITAL LINES
of stepwise systematic errors, treated in t h e preceding section, is a good example of this. I t s development from (30) leads t o a n incorrect answer. If, now, there is no correlation, t h e second s u m of (30) becomes essen tially zero a n d WJ
2
2
= A s sin 0
(31)
where δΓΓη8 is t h e root mean square value of t h e d's a n d t h e b a r over t h e 2 IΓ I indicates t h e average in t h e sense discussed above (i.e., over t h e fine structure). 5 r m8 = { - 2 ^ 4
(32)
This solution is one extreme situation. T h e other extreme is when t h e only errors are d u e t o t h e displacement of individual t e e t h (or, as a n approxi mation, t o bent t e e t h ) . T h e n t h e autocorrelation s u m of (30) h a s t h e v a l u e : - 2 ühüh+i = η h
— Ö^
MS
(33)
2
minus because a displaced t o o t h means t h e impedance is large on one side, and small on t h e other; a n d t h e factor £ because each nonzero t e r m of t h e autocorrelation s u m contributes t w o t e r m s t o t h e r m s sum. T e r m s of t h e autocorrelation s u m with k other t h a n u n i t y are assumed t o add u p essentially t o zero, since longer range correlation is assumed n o t t o be present. T h e n (30) becomes 4
f ! \ p = 2nbvL sin θ
(34) 4
2
T h i s differs from (31) in t h e factor 2 a n d in t h e sin for t h e sin term. T h e latter means a different dependence on frequency. Figure 3 shows plots of (31) a n d (34) for η = 75 (a reasonable figure) and 5 r m8 = 0.01 (which is a greater accuracy t h a n can often be obtained). 0.2
LOCAL CORRELAT
CORRELATE D
1
30°
60°
90°
FIG. 3. |Γ| vs θ for η = 75,firms= 0.01. Locally correlated and uncorrelated.
Μ . C. P E A S E
96
I n b o t h cases t h e effect rises t o a broad m a x i m u m around θ = π/2. I n t h e case of local correlation, this m a x i m u m is somewhat narrower a n d rises t o a higher maximum. B u t in b o t h cases t h e effect is spread across t h e band.
IV. Conclusions
W e h a v e developed methods for t h e analysis of various t y p e s of con structional errors t h a t m a y occur in periodic structures such as those used in backward wave magnetrons. Because of t h e long length of t h e lines used, and t h e large n u m b e r of iterated units t h a t are involved, such lines become highly sensitive t o constructional errors. A n error of misregister leading t o only a 1 % error in impedance, in a 75-unit interdigital line gives a V S W R a t t h e "pseudo cutoff," where t h e t o o t h length becomes a quarter-wave length, of a b o u t 10 t o 1. E v e n r a n d o m errors whose root mean square value is this same small m a g n i t u d e will lead t o a probable reflection coefficient of a b o u t 0.1 (VSWR a b o u t 1.2) if uncorrelated, or a b o u t 0.2 ( V S W R a b o u t 1.5) if locally correlated. B u t whereas t h e effect of misregister is sharply confined t o t h e neighborhood of t h e pseudo cutoff, t h e effect of r a n d o m errors is spread over most of t h e useful band. This, then, indicates t h e extraordinary precision t h a t is required in these devices. I n m a n y applications, t h e precision t h a t is demanded is greater t h a n can b e reasonably expected, even on a laboratory basis. While these cases can b e handled u p t o a point b y building t o t h e greatest possible precision a n d t h e n compensating, t h e considerations discussed here place a practical limit on w h a t can b e accomplished. List of Symbols
Ε I Ζ ζ ß k χ θ δ δη δ' rτιm s Γ
voltage amplitude current amplitude characteristic impedance characteristic impedance, normalized t o nominal value ζ for n t h section propagation constant space periodicity constant length of section distance variable electrical length perturbation p a r a m e t e r δ for n t h section root mean square δ voltage reflection coefficient
2.5 TOLERANCE EFFECT O N INTERDIGITAL
LINES
Γη
Γ i n t o η - t u p l e u n i t , t e r m i n a t e d as specified
Μ
transmission (A BCD)
W
wave matrix
97
matrix
Ρ
similarity t r a n s f o r m a t i o n m a t r i x , Μ t o W
ABCD
elements of Μ
do, αϊ, a,2, ci3
coefficients of o p e r a t o r as defined
Sn
w a v e v e c t o r i n t o η sections, t e r m i n a t e d as specified
Un
Tschebysheff polynomial of second k i n d References
1. R . K . MOORE, The effects of reflections from randomly spaced discontinuities in transmission lines. IRE Trans, on Microwave Theory Tech. 5, 1 2 1 ( 1 9 5 7 ) . 2. J. A. MULLEN AND W. L. PRITCHARD, The statistical prediction of voltage standingwave ratio, IRE Trans, on Microwave Theory Tech. 5, 1 2 7 ( 1 9 5 7 ) . 3. M. C . PEASE, The analysis of broadband microwave ladder networks. Proc. IJI.E. (Inst. Radio
Engrs.)
38, 1 8 1 - 1 8 3
(1950).
4. M. C. PEASE, The iterated network and its application to differentiators. I.R.E. (Inst. Radio Engrs.) AO, 7 0 9 - 7 1 1 ( 1 9 5 2 ) .
Proc.
Lines
by M. C. PEASE
I. II. III. IV.
Continuous Small Periodic Errors Stepwise Systematic Error (Misregister) Stepwise Random Errors Conclusions List of Symbols References
88 89 92 96 96 97
I n t h e design a n d construction of extended interaction tubes, whether helix, interdigital, disc-loaded waveguide t y p e s or other, t h e problem of tolerances on t h e delay line h a s a critical importance. I t is more t h a n a problem of ease of manufacture; it m a y even determine whether t h e t u b e can be built a t all. F o r these structures are such t h a t even quite minor tolerances m a y accumulate t o a disastrous degree. Some consideration h a s been given t o this problem, although for a differ ent purpose, b y Moore (1) and b y Mullen and P r i t c h a r d (2). T h e i r papers are concerned with t h e r a n d o m combination of reflections t h a t m a y not b e individually small. F o r our purposes here, however, we m u s t be concerned with t h e resultant of a large n u m b e r of very small reflections whose loca tions are known. T h e t w o problems, although related, are different and require different t r e a t m e n t . I n w h a t follows we shall analyze t h e effect of tolerance and of construc tional errors in extended repetitive structures, particularly of t h e interdigital t y p e commonly used in backward-wave magnetrons. W e shall consider three t y p e s of variations: (a) a long t e r m cyclic error such as m a y occur from an indexing fault in hob grinding, (b) a stepwise periodic error due to misregister of t h e t w o halves of t h e line, a n d (c) stepwise r a n d o m error, either uncorrelated or locally correlated, resulting from t h e dislocation of individual teeth. These three cases will show t h e general consequences t o be expected from t h e inevitable lack of precision in t h e manufacture of delay lines. 87
88
Μ. C. PEASE
I. Continuous Small Periodic Errors
A continuous a n d periodic error m a y arise from indexing tolerances in t h e grinding of t h e hob used t o m a k e t h e line. T h e problem can b e handled as a perturbation one. T h e basic equations are t h e familiar ones for a line, e
- f
-
( i
>
b u t with characteristic impedance Ζ now being a function of x. W e shall assume t h a t Z, after normalization, is nearly constant a t unity, b u t with a small periodic variation: Ζ = 1 + δ sin kx
(3)
where k is a measure of t h e length of t h e periodicity of Z. If, now, we let 2
Ε = Eo + δΕι + δ Ε2 + · · · 2
/ = h + Βΐι + δ / 2 + · · ·
and substitute in (1), collecting b y powers of δ, t h e n we obtain a series of equations which can b e solved for t h e E's or I's separately. T h e first t w o equations for t h e E's, for example, a r e 2
Ei' + ß E0
= 0
(5) 2 Ei' + ß Ei = k cos kxEO These can be solved, either generally or under assumed b o u n d a r y condi tions. T h e conditions of principal interest here are t h a t t h e reflection co efficient shall vanish a t t h e end of t h e line, a t χ = 0, =
E ^ jßZE Ε + I jßZE
+ E' _ - Ε' "
= U
at
X
U
W
This condition is applied t o each order of perturbation separately. If we do this through t h e first order perturbation, a n d t h e n compute Γ t o t h e first power of δ, w e find Γ = ^2 _
fc2
s
n
i kx — k sin 2ßx — jk (cos kx — cos 2ßx)]
(7)
This level of approximation is sufficient for most purposes. 2 2 Of particular interest in this equation is i t s behavior as β —> k /4. E q u a t i o n (7) t h e n becomes indeterminate. I t m a y b e evaluated as a limit to give lim
ikx
Γ = τ (sin kx — kxe )
(8)
2 . 5 TOLERANCE EFFECT ON INTERDIGITAL LINES
89
of which t h e first t u r n can b e ignored if kx is large, i.e., t h e periodicity of t h e error is such t h a t there are several periods within t h e line. W e are left, then, with a reflection t h a t is proportional t o length (times a phase term) with a proportionality factor of kb/4. (Note: Formally, this equation allowed |Γ| t o become > 1 for large enough x. This, however, is d u e t o t h e approximations used, particularly t h a t Γ shall b e small.) T h e reflection coefficient, therefore, has a peak t h a t m a y b e quite large when t h e frequency is such t h a t t h e period of t h e error is a half wavelength long. Although n o t a t r u e cutoff, it h a s t h e same effect a n d m a y cause a " h o l e " in a n operating t u b e . W e shall see a similar effect in Section I I .
II. Stepwise Systematic Error (Misregister)
An interdigital line is formed b y interleaving two " c o m b s . " If these are not properly registered, t h e spaces between t h e t e e t h will b e alternately large and small, with comparable variation of t h e impedance. W e assume t h a t t h e effect of misregister is t o cause alternate sections of high a n d low impedance. W e c a n t h e n consider t h e line t o b e formed b y t h e η-tuple iteration of a unit, such a s is shown in Fig. 1. I n t h e drawing, 0/2
0/2
FIG. 1. Line formed by n-tuple iteration of a unit.
alternate sections a r e of impedance ζ a n d z' (normalized t o t h e nominal impedance so t h a t these are approximately b u t not exactly equal t o u n i t y ) , and t h e electrical length of each section is Θ. T h e matrix relating t h e voltage and current a t t h e i n p u t t o t h a t a t t h e o u t p u t of a unit is written Μ =
\A jC
jB D
(9)
Since we shall assume t h a t t h e line is lossless, A, B, C, a n d D are real. I t is convenient t o t r e a t such a matrix in t e r m s of i t s coefficients (3) which are defined a s : a 0 = | ( A + D)
a2 =
+ C)
ai
a 3 = i(A - D)
(10)
90
Μ . C. P E A S E
and which are related b y 2
2
2
2
a 0 + ai - a 2 - a 3 = 1
(11)
T h e n t h e reflection coefficient is a
Γ =
- ^ - & a 0 + jai
(12)
Since t h e coefficients are real for a lossless network, a n d since a 3 = 0 for a symmetrical one, (12), with (11) can be written
li? = rf^i
(13)
T h e η-tuple iteration of a network such as (9) gives a network whose coefficients are (4): .(ao)
=
n
(cti)n =
a0Un(ao)
—
Un-i(a0)
aiUn(a0)
( α 2 ) η = a 2C/n(a 0) ( α 3) Λ =
cizUn(ao)
where t h e second subscript on t h e coefficient denotes t h e degree of itera tion, a n d where Un is t h e Tschebysheff polynomial of t h e second kind a n d order η in t h e variable ao. These functions can b e written -1 «. sin η c o s a 0 ' TJ ( Un(a0)
=
—
r;
(15)
1
sin cos ao B y t h e usual methods we m a y calculate t h e matrix of t h e unit shown in Fig. 1 t h a t is t o b e iterated. T h e t w o coefficients of interest a r e : 2
2
(Mo) 2 = cos θ - ^ (J + Yj sin θ
" I " ^) + 7 + -l(^3)(f-i)-
( M 2) 2 =
2
)
s i n Θc o s θ
( ) 16
Z
(The subscript 2 because t h e unit of Fig. 1 already includes t w o teeth.) Let us take, as t h e nominal impedance of t h e line a n d its termination, t h e geometric mean of ζ a n d z', so t h a t these impedances, when norjnalized, are reciprocals. Since t h e errors are assumed small, let u s set z
=
1
+
δ
ζ
' = Γ + 1 =
1
-
δ
(17)
Then, t o t h e first order of δ, (Mo) 2 = cos 2Θ ( M 2 ) 2 = δ (sin 2Θ - 2 sin Θ)
(18)
2.5
TOLERANCE
EFFECT ON INTERDIGITAL
LINES
91
W e note t h a t (Af 2 ) n as given b y (14) is t h e product of ( M 2 ) 2 which is a slowly varying function of frequency, a n d of Un(a,o) which introduces t h e rapidly varying part. Hence t h e application of t h e approximation (17), t o t h e t e r m s of (16) is separate. W e h a v e t h e n from (13) a n d (14) t h a t 2
, 1
2| 2 n|
2
2
δ sin 2ηθ (sin 2 0 - 2 sin 0) 2 2 2 " sin 2Θ + δ sin 2ηθ (sin 2 0 - 2 sin 0)
=
2
(
'
A t 0 = π / 2 , this equation becomes indeterminate. T h e limit, however, can be evaluated as l i m
Ι *»Ι = 1 ΓΓ*2 + 4δ η2 Γ
Θ-+*/2
2
2
Λ
(20)
2
A t lower values of 0, t h e curve of |Γ| oscillates rapidly (period Δ0 = τ/η) between zero a n d a n envelope given b y replacing Un b y its m a x i m u m value 1/sin 20: 2
2
τ? ^ IT, 1 δ (1 - c o sL0 ) 22 Envelope 1 Γ 2 η 1 = — 22ü \ > 22 /1 —~r2 cos 0 + δ (1 — cos 0) 2
(21)
,
|Γ|βην, t h e square root of (21) h a s been plotted in Fig. 2 for a 1 % error, i.e., δ = 0.01. I t is seen t o be negligible except in t h e neighborhood of 90°.
o.i II 1/
Ii //
I
//
II Ί !
STE >WISE -
CONriNuouj
// //
******* 30°
60°
90°
θ
FIG. 2. Envelope of fr] for δ — 0.01.
At 90° t h e limit of t h e envelope is 1, b u t t h e limit of t h e function itself is given b y (20), which is always less t h a n 1. For comparison, t h e comparable curve for t h e continuous case is also plotted a s t h e dotted line. T h e t w o cases are m a d e comparable b y setting
/ 0 1
Μ. C. PEASE
92
θ = ßl kl = IT χ = nl where I is t h e length of t h e tooth. T h e n (7) becomes 2
2
• l2 _ / 2πθδ Χ / s i n ηθ 1 1 2 2 ~ \TT - 4(97 \ c o s n ö
if η is e v e n \ if η is odd /
. * { '
a n d t h e first factor is t h e envelope. I t is this formula t h a t is plotted in Fig. 2 as t h e dotted line. T h e difference between it a n d (21) is n o t large. T h e extreme sensitivity of (19) t o very slight misregister a t θ = 7r/2, i.e., when t h e frequency is such t h a t each finger is a quarter-wavelength long, introduces a n effective cutoff a t this value. T h a t this was so, and t h e physical reason for it, w a s pointed o u t b y J. Hull some time ago. T h e present analysis gives a q u a n t i t a t i v e value for it. If, for example, we w a n t |Γ| ^ i (VSWR ^ 2) for a n 80-fingered line, t h e n (24) requires t h a t δ ^ 0.002. F o r most line dimensions used, this is wholly impractical a n d t h e effect acts t o limit t h e range of application of t h e line t o frequencies less t h a n this "pseudo cutoff." Since, in this case, θ is half t h e periodicity of t h e error, this effect is similar t o t h a t noted in Section I for a continu ously varying error. T h e "cutoff" occurs a t t h a t frequency for which t h e period of t h e error is a half-wavelength.
III. Stepwise Random Errors
Among other possible errors in a n interdigital line, t h e most likely a r e r a n d o m errors, either uncorrelated or correlated only with immediate neighbors. T h e latter m a y b e considered as a n approximation t o t h e b e n t t o o t h case, or t o b e a n exact description of t h e displaced t o o t h problem. I n either case a n error leads t o too high a n impedance on one side and t o o low a n impedance on t h e other. T h e correlation is therefore negative a n d exists only between neighboring sections, b u t is complete over this range. Since r a n d o m errors destroy t h e constancy of t h e unit elements, t h e iteration formula employed in t h e last section cannot b e used. Instead, we m u s t compute t h e total matrix directly. F o r this purpose, it is most con venient t o use t h e wave matrix r a t h e r t h a n t h e transmission matrix. W e obtain this b y a similarity transformation: W =
1 1
l
P~ MP
1 - 1
Μ
1 1
1 - 1
(23)
where Μ is t h e transmission m a t r i x of a single section as used before. If now, we set ZN
= 1 + δη
(24)
2.5
TOLERANCE EFFECT ON INTERDIGITAL LINES
93
for the nth unit, numbered from the termination and if the 8n's are small, then, to the second order of the 5's:
(25)
where we have retained only the squared terms in δ η. W e choose this limit of approximation because ( a ) the mean value of the d's will be nearly zero while the root mean square may be significant, and, in fact, may be ex pected to be the dominant effect, and ( b ) while we will consider two-term correlation coefficients (i.e., 8j8k) we will not consider higher order terms (δ7δ*δ, etc.). W e compute, then, a sequence of column matrices defined by
(26)
Sn is the column matrix that describes the forward and backward waves that would be observed were the η sections of the line terminated in 1Ω and "looked at" with a 1Ω detector. B y computing the first few to find the pattern, and proving the pattern by induction, we can find
(27)
Then Γ is the ratio of the t w o terms of (27) (28) Since the numerator contains no term in less than δ and the denominator 3 contains no δ terms, the denominator adds only δ and higher terms. P r o viding, then, that Σδι? and the autocorrelation sums are small compared to 1, we can approximate (29) This may be interpreted 2
as a sequence of random walks of length 2k l)e
(dk — %Sk ) sin Θ along the vectors e^ ~
(and the whole rotated by
2]ne
e- ).
Μ. C. PEASE This is obtained by considering each δ* as having a probability distribution. T h e direct interpretation, therefore, gives t h e distribution of Γ, a t a given Θ, over a universe of lines. T h e curve of Γ vs Θ t h a t we obtain by applying probability theory t o this t h e n describes t h e probable behavior within this universe of lines. W e can, however, interpret this curve further b y consideration of t h e "ergodic t h e o r e m " used, for example, in gas kinetic theory. T h e variation of θ causes t h e 5's t o shift their mode of combination very rapidly. Let u s consider a (2n + 1)-dimensional space in which each of t h e 2n axes cor responds t o one of t h e 8's times t h e real and imaginary p a r t s of its phase term. T h e final axis corresponds t o 0. For t h e universe of lines t h e average Γ is t h e n t h e average over t h e cross section of this space a t t h e given 0. Since we assume t h a t all t h e 8's h a v e t h e same probability distribution, this space has a high degree of s y m m e t r y a b o u t t h e 0 axis (but not com plete symmetry, since there m a y be local correlation). A n y one line a t a given 0 is represented b y a point in this space. As 0 varies, however, this point moves in some elaborate spiral a b o u t t h e 0 axis. And if η is large, it moves around t h e 0 axis very rapidly compared t o its motion along t h e 0 axis. I t appears evident t h a t this generalized spiral will pass near all acces sible regions of this space in accordance with their probabilities; a n d t h a t , therefore, t h e average over t h e universe of lines m a y be t a k e n as t h e average over t h e fine details of t h e p a t h t h a t a particular line traces out. T h e r e will, of course, be fine s t r u c t u r e superimposed on this c o m p u t e d average, b u t t h e average level of t h e standing wave is computable b y this approach. Let us, then, consider (29). B y t h e s y m m e t r y of t h e configuration space, 2 t h e average value of Γ is zero (in spite of t h e presence of δ* t e r m s since these are "washed o u t " b y t h e phase terms). W e must, then, consider t h e 2 absolute value of Γ. W e find t h a t , neglecting t e r m s higher t h a n δ , 2
2
2
|Γ| = sin 0 { Σ δ* + 2 V U =l
k= l
cos 2/c0 V
δΑ+Λ
(30)
J
h= l 2
I n doing this, we h a v e dropped out t h e effect of t h e δ t e r m of (29). M o s t of t h e complexity of (28) has t h u s been lost. I n fact, (30) could h a v e been obtained b y keeping only t h e t e r m s in δ in (25). T h i s is equivalent t o t a k i n g as t h e total reflection simply t h e sum of t h e individual reflections. W e h a v e gone t h r o u g h this r a t h e r elaborate development, however, for two reasons. I n t h e first place, it was not evident t o us t h a t this l a t t e r approximation could be m a d e w i t h o u t losing some of t h e second order factors of (30). W e h a v e m a d e t h e approximations only a t those times when it was rigorously clear t h a t t h e y were justified. W e h a v e also wished t o establish basic formulas t h a t could be used in other w a y s when t h e approximations t h a t we h a v e used m a y not be permissible. T h e problem
2.5
95
TOLERANCE EFFECT ON INTERDIGITAL LINES
of stepwise systematic errors, treated in t h e preceding section, is a good example of this. I t s development from (30) leads t o a n incorrect answer. If, now, there is no correlation, t h e second s u m of (30) becomes essen tially zero a n d WJ
2
2
= A s sin 0
(31)
where δΓΓη8 is t h e root mean square value of t h e d's a n d t h e b a r over t h e 2 IΓ I indicates t h e average in t h e sense discussed above (i.e., over t h e fine structure). 5 r m8 = { - 2 ^ 4
(32)
This solution is one extreme situation. T h e other extreme is when t h e only errors are d u e t o t h e displacement of individual t e e t h (or, as a n approxi mation, t o bent t e e t h ) . T h e n t h e autocorrelation s u m of (30) h a s t h e v a l u e : - 2 ühüh+i = η h
— Ö^
MS
(33)
2
minus because a displaced t o o t h means t h e impedance is large on one side, and small on t h e other; a n d t h e factor £ because each nonzero t e r m of t h e autocorrelation s u m contributes t w o t e r m s t o t h e r m s sum. T e r m s of t h e autocorrelation s u m with k other t h a n u n i t y are assumed t o add u p essentially t o zero, since longer range correlation is assumed n o t t o be present. T h e n (30) becomes 4
f ! \ p = 2nbvL sin θ
(34) 4
2
T h i s differs from (31) in t h e factor 2 a n d in t h e sin for t h e sin term. T h e latter means a different dependence on frequency. Figure 3 shows plots of (31) a n d (34) for η = 75 (a reasonable figure) and 5 r m8 = 0.01 (which is a greater accuracy t h a n can often be obtained). 0.2
LOCAL CORRELAT
CORRELATE D
1
30°
60°
90°
FIG. 3. |Γ| vs θ for η = 75,firms= 0.01. Locally correlated and uncorrelated.
Μ . C. P E A S E
96
I n b o t h cases t h e effect rises t o a broad m a x i m u m around θ = π/2. I n t h e case of local correlation, this m a x i m u m is somewhat narrower a n d rises t o a higher maximum. B u t in b o t h cases t h e effect is spread across t h e band.
IV. Conclusions
W e h a v e developed methods for t h e analysis of various t y p e s of con structional errors t h a t m a y occur in periodic structures such as those used in backward wave magnetrons. Because of t h e long length of t h e lines used, and t h e large n u m b e r of iterated units t h a t are involved, such lines become highly sensitive t o constructional errors. A n error of misregister leading t o only a 1 % error in impedance, in a 75-unit interdigital line gives a V S W R a t t h e "pseudo cutoff," where t h e t o o t h length becomes a quarter-wave length, of a b o u t 10 t o 1. E v e n r a n d o m errors whose root mean square value is this same small m a g n i t u d e will lead t o a probable reflection coefficient of a b o u t 0.1 (VSWR a b o u t 1.2) if uncorrelated, or a b o u t 0.2 ( V S W R a b o u t 1.5) if locally correlated. B u t whereas t h e effect of misregister is sharply confined t o t h e neighborhood of t h e pseudo cutoff, t h e effect of r a n d o m errors is spread over most of t h e useful band. This, then, indicates t h e extraordinary precision t h a t is required in these devices. I n m a n y applications, t h e precision t h a t is demanded is greater t h a n can b e reasonably expected, even on a laboratory basis. While these cases can b e handled u p t o a point b y building t o t h e greatest possible precision a n d t h e n compensating, t h e considerations discussed here place a practical limit on w h a t can b e accomplished. List of Symbols
Ε I Ζ ζ ß k χ θ δ δη δ' rτιm s Γ
voltage amplitude current amplitude characteristic impedance characteristic impedance, normalized t o nominal value ζ for n t h section propagation constant space periodicity constant length of section distance variable electrical length perturbation p a r a m e t e r δ for n t h section root mean square δ voltage reflection coefficient
2.5 TOLERANCE EFFECT O N INTERDIGITAL
LINES
Γη
Γ i n t o η - t u p l e u n i t , t e r m i n a t e d as specified
Μ
transmission (A BCD)
W
wave matrix
97
matrix
Ρ
similarity t r a n s f o r m a t i o n m a t r i x , Μ t o W
ABCD
elements of Μ
do, αϊ, a,2, ci3
coefficients of o p e r a t o r as defined
Sn
w a v e v e c t o r i n t o η sections, t e r m i n a t e d as specified
Un
Tschebysheff polynomial of second k i n d References
1. R . K . MOORE, The effects of reflections from randomly spaced discontinuities in transmission lines. IRE Trans, on Microwave Theory Tech. 5, 1 2 1 ( 1 9 5 7 ) . 2. J. A. MULLEN AND W. L. PRITCHARD, The statistical prediction of voltage standingwave ratio, IRE Trans, on Microwave Theory Tech. 5, 1 2 7 ( 1 9 5 7 ) . 3. M. C . PEASE, The analysis of broadband microwave ladder networks. Proc. IJI.E. (Inst. Radio
Engrs.)
38, 1 8 1 - 1 8 3
(1950).
4. M. C. PEASE, The iterated network and its application to differentiators. I.R.E. (Inst. Radio Engrs.) AO, 7 0 9 - 7 1 1 ( 1 9 5 2 ) .
Proc.