IEEE Transactions on Magnetics, vol. Mag-11, no. 5, September 1975
WAVELENGTH RESPONSE IN AC BIASED RECORDING
H. Neal Bertram*
.5r\'1
A
ABSTRACT
TAPE
mTlNG
The w a v e l e n g t h d ep en d en ce of t h e a c b i a s e d recording s e n s i t i v i t y is c a l c u l a t e d u t i l i zi n g a model in
which both longitudinal a n d v e r t i c a l record f i e l d s are
c o n s i d e r e d . Even though t h e e f f e c t s of d emag n e tiz a tion
f ie ld s and spreadsin p ar t i cl e s wi t ch i n g f i el d s a r e negl e c t e d , t h e c a l c u l a t io n y i e l d s e x c e l l e n t q u an t i t ative
agreement with measurements on extremely well-oriente d CrO2 a n d 7 F e z 0 3 mag n et i c t a p e s .
INTRODUCTION
M o s t t h e o r i e s of t h e mag n et i c recording p r o c e s s
u t i l i z e models w h i c h c o n s i d e r only t h e component of
t h e recording field that l i e s in t h e d i r ect i o n of r ela tive
head-to-tape motion. T h i s s i mp l i f i cat i o n h a s been
ju s t if i e d for media of nomlnal t h i c k n e s s by noting tha t
magnetic t a p e s p o s s e s s a high d eg r ee of p ar t i cl e orienta tio n a n d t h u s s h o u l d r es p o n d primarily t o t h e s i n g l e
fleld component.
T h e r e i s e v i d e n c e that t h i s approximation i s t o o res t r i c t i v e . z In c a l c u l a t i o n s of a c b i a s e d recording
s e n s i t i v i t y , so - c a l l ed "l o n g i t u d i n al field" models d o not
y ie ld v e r y good a g r e e m n t with measurements .& In partic u la r , t h e s e models y i el d a r eco r d ed magnetization
which d o e s not v a r y wi t h d ep t h i n t o t h e t ap e. The contribution t o t h e wavelength d ep en d en ce of t h e reproduce
flux d u e t o t h i s t y p e of mag n et i zat i o n pattern i s t h e w e l l
known t h i c k n e s s r e s p o n s e , % T h e s u b s t a n t i a l d i s a gre e ment b e t w e e n t h i s t heo r et i cal r e s p o n s e a n d measurement i s u s u a l l y a t t r i b u t ed t o s u c h f act o r s a s t h e s p r e a d
in p a r t i c l e s w i t c h i n g f i e l d s , t h e d emag n et i zat i o n f i e l d ,
head-to-tape s p a c i n g , or r ed u ced s h o r t wavelength recording d e p t h . It i s ar g u ed h er e t h a t a s i m p l e model
w h ic h n e g l e c t s the a b o v e f act o r s but u t i l i z e s both components of t h e record field p r ed i ct s extremely w e l l t h e
small s i g n a l ac b i a s e d wav el en g t h r e s p o n s e of t a p e s
composed of hiqhly oriented u n l ax i al p ar t i cl es .
DETERMINATION OF THE RECORDED MAGNETIZATION
The model u t i l i zed h a s b een d i s c u s s e d in a r e c e n t
publication d e a l i n g on l y wi t h long wav el en g t h ac biase d recording.:
Briefly, t h e recording medium i s c o n s ide d t o p o s s e s s a n i so t r o p i c remanent co er ci v i t y a n d t o
exhibit a negligible s p r e a d in p ar t i cl e s wi t ch i n g f i e l d s .
The b i a s current a p p l i e d t o t h e record h ead i s t a k e n t o
be of infinite frequency, an d d emag n et i zat i o n f i e l d s a n d
phasing e f f e c t s a r e n eg l ect ed . At low s i g n a l levels e a c h
e le m e n t of t a p e is recorded when it e x p e r i e n c e s a dec r e a s i n g b i a s f i e l d amplitude e q u a l t o t h e t ap e coercivity. In Fig. 1 contours of c o n s t a n t t o t a l t o t a l field utiliz in g t h e Karlqvist approximatio&are s h o wn wh ic h
a p p l y t o both t h e b i as an d s i g n a l f i e l d s . For e a c h b i a s
current there c o r r e sp o n d s one co n t o u r p o s s e s s i n g a bia s
field amplitude e q u a l t o t h e co er ci v i t y of t h e medium
(termed here t h e c o er ci v i t y contour). For ex amp le , in
Fig. 1 , if t h e contour l a b e l e d . 7 r e p r e s e n t s a b i as field
e q u a l t o t h e t a p e c o e r ci v i t y , t h en t h e el emen t of t a p e
shown i s recorded when it passes t h e point l ab el ed
(X,Y).
M a n u s c r i p t r e c e i v e d February 27, 1975
*Ampex Corporation, Redwood C i t y , California 94063
' \ '
(N2)
TAPE.
'
MOTION
HEAD
Fig. 1, Contours of c o n s t a n t total fie ld l a b e l e d in perc e n t a g e of d e e p g a p field.
It i s a s s u m e d tha t recording oc c urs v i a the anhyst e r e t t c proc e ss a nd.tha t t h e dire c tion of the b i a s field is
irrelevant. Thus, t h e amplitude of t h e recorded longitud i n a l magnetization component i s g i v e n a t low recording
levels by*
In (1),X i s t h e longitudinal a n h y s t e r e t i c s u s c e p t i b i l i t y
w hic h i s measured in uniform fie lds with a d c s i g n a l
f i e l d , a nd Hx(y) is t h e amplitude of t h e s i g n a l field a t
a de pth y into t h e t a p e on the bia s c oe rc ivity contour.
The t a p e is thereby recorded only to a d e p t h corresponding t o t h e maximum de pth of penetration of t h e coercivity contour (ym in Fig. 1 for the .7 contour). Analogous
t o (1) a v e r t i c a l recorded m a gne tiz a tion component could
be w ritte n a s s u m i n g a sim ple te nsor form for t h e anhyst e r e t i c s u s c e p t i b i l i t y . However, s i n c e t a p e s a r e w e l l
orie nte d t h e a n h y s t e r e t i c s u s c e p t i b i l i t y m e a sured in the
v e r t i c a l dire c tion is extremely sm a ll. Thus, e v e n though
t h e v e r t i c a l component of t h e s i g n a l field c a n be signif i c a n t , the r e s u l t a n t v e r t i c a l m a gne tiz a tion may b e
ne gle c te d.
Maximum long w a ve le ngth s e n s i t i v i t y o c c u r s when
t h e bia s current i s s e t so tha t t h e c oe rc ivity contour
j u s t pe ne tra te s t o t h e back of t h e c oa ting. In Fig. 1
t h e .6 contour would correspond w ith ym = d . Using
the Karlqvist he a d field func tion, t h e longitudinal component of t h e s i g n a l field a t t h e back of t h e t a p e coating ( y = d ) a long t h e g a p c e nte rline (x=O) is given by,
N I i s the mmf a pplie d t o a 100% e f f i c i e n t record head
w i h g a p length g .
In the limit of infinite c o a t i n g t h i q n e s s t o g a p
le ngth ra tios (d/g+ m ) w e ha ve shown tha t a l o n g a
c ontour of c onsta nt t o t a l field t h e longitudinal field increases line a rly with d e p t h into t h e c oa ting. Numerical
a n a l y s i s show s tha t t h e line a r va ria tion s t i l l holds to
good approximation for finite c o a t i n g t h i c k n e s s t o g a p
le ngth ra tios a nd tha t e v e n for t h e ra the r s m a l l r a t i o of
d/g = 0.25 t h e average error i s only a bout 4%. It is in
this line a r va ria tion t h a t t h e e s s e n c e of this total field
model is e xhibite d. Thus, w e w ill a s s u m e here that t h e
*SI units a r e utiliz e d in which B = p
0
(H+ M).
11 76
Copyright 0 1 9 7 5 by The Institute of Electrical and Electronics Engineers, Inc
Printed in U.S.A. Annals No. 509MA906-3
c
v a r i a t i o n of t h e longitudinal f i el d component i s s t r i c t l y
l i n e a r a n d in t h e case of maximum long wav ele ngth se ns i t i v i t y i s g i v e n by
HX(y) = (y/d) Hx(d)
.
(3)
Combining (1) a n d (3) a recorded mag n et i zat ion i s obta i n e d which v a n i s h e s a t t h e t ap e surface an d inc re a se s
li n e a r l y t o a maximum a t t h e back of t h e co ating. This
behavior h a s been s e e n in l ar g e scale modeling a n d
qua1ita t ive l y e x pl ai n ed by Tjaded.
REPRODUCE FLUX AND EXPERIMENTAL COMPARISON
The flux a m p 1 i t u d e in a 100% ef f i c ie nt reprod u c e h e a d of t r a ck wi d t h w i s g i v e n by
(4)
where M (y) i s t h e amplitude of a s i n u s o i d a l l y varying
l o n g i t u d c a l m a gn et i zat i o n pattern of wavenumber
k = 2n/k. Equation (4) i s p ar t i cu l ar l y simple s i n c e t h e
p h a s e v a r i a t i o n wi t h d e p t h of t h e magnetization h a s been
n e g l e c t e d . In a d d i t i o n t h e u s u a l head-to-tape s p a c i n g
a n d reproduce g a p loss terms are not included. U sing
th e magnetization o b t ai n ed from (1) a n d (3), it i s e a s i l y
shown that
cp = powXxd Hx(d)
[1-e'kd
(1 +kd)] /(kdl2
(5)
0
-10
-20
-30
-401
a5
I
c
I
.5
.I
I
# , I
I
I
5
1
10
REMlcED W A K NUMBER (kd)
Fig. 2 , W a ve le ngth r e s p o n s e d u e t o a) a linearly inc r e a s i n g m a gne tiz a tion, a nd b) a c o n s t a n t magnetization w ith de pth into the ta pe . v i s t h e ta pe s p e e d , d is
the c oa ting t h i c k n e s s , g is t h e record g a p length a n d
k = 2 a/k.
oriented unia xia l p a r t i c l e s , t h e s e fa c tors c a n amount
only t o a re la tive ly small correction. The 6 dB per
o c t a v e a dditiona l loss in short w a ve le ngth response
a s w e l l a s the 6 dB loss in long w a ve le ngth a b s o l u t e
s e n s i t i v i t y r e s u l t dlre c tly from the l i n e a r variation of
recorded magnetization with de pth into t h e coating.
EFFECT OF BIAS AMPLITUDE ON THE REPRODUCE FLUX
where Hx(d) i s g i v e n by (2). Th i s e x p r e s s i o n g i v e s t h e
wavelength d e p en d en ce of t h e reproduce flux when the
b i a s current i s a d j u s t e d t o maximize t h e long wavelength se n s i t i v i t y . Th e implication of a l i n ea rly increas ing magnetization o n t h e wav el en g t h r e s p o n s e w a s
f i r s t s h o w n by T j a d e n g In t h e model p r es en te d he re ,
th e a b s o l u t e reproduce flux i s c a l c u l a t e d a n d i t s dep e n d e n c e on recording h ead an d t a p e parameters is
e x p l i c i t l y shown.
In Fig. 2a t h e wav el en g t h d ep en d en t term of (5) i s
plotted vs. t h e normalized wavenumber, kd. The experimental d a t a shown r e p r e s e n t s measurements of s m a l l
s i g n a l s e n s i t i v i t y on a C r 0 2 t a p e a n d a n extremely well
oriented Y Fe 2 0 3 tape. Both t a p e s ex h i b i t ed extremely
s q u a r e M-H l o o ps wi t h s q u a r e n e s s e s Z 0 . 8 6 . For e a c h
ta p e measurement t h e b i a s w a s s e t t o maximize the 104
w a v e l e n g t h s e n s i t i v i t y an d t h e s i g n a l w a s s e t so tha t
t h e third harmonic d i s t o r t i o n a t long wav el en gth did not
e x c e e d 1%. The d a t a h a s been co r r ect ed for recorder
e q u a l i z a t i o n and reproduce h ead r e s p o n s e so that it
c o r r e s p o n d s ( e x cep t for a p o s s i b l e s p a c i n g loss) t o t h e
flux defined by (4). S i n c e t h e r eco r d er w a s not calibrate d a b s o l u t e l y both s e t s of d at a h a v e b een a d j u s t e d vert i c a l l y t o match t h e long wav el en g t h as y mp t ote of t h e
theory. The a g r eemen t i s ex cel l en t : both t a p e s follow
th e t h e o r e t i c a l cu r v e c l o s e l y over t h e whole ra nge .
The a a r e e m en t of measured wav el en q t h r e s p o n s e
with t h e prediction of a t o t al f i el d model i s remarkable
when compared w i t h t h at d u e t o a purely longitudinal
field model. U t i l l zat i o n of longitudinal f i el d s for both
s i g n a l a n d b i a s yield a recorded mag n et i zat i on which is
c o n s t a n t with d ep t h into t h e magnetic coating. W ith (4)
s u c h a d e p e n d e n ce y i e l d s t h e w e l l known t h i c k n e s s re sponse which i s plotted in Fig. 2b. The extreme disagreement ( a s s e e n by comparing t h e s h i f t ed cu r v e w ith t h e
d a t a ) h a s been formerly at t r i b u t ed t o t h e f act ors mentioned in the introduction. On t h e b a s i s of t he model
p r e s e n t e d h e r e , as ap p l i ed t o t a p e s composed of well1177
U sing t h i s sim ple model, t h e e ffe c t on t h e reprod u c e flux spectrum of changing t h e d e e p g a p bias field
c a n be e a s i l y determined. In t h e previous s e c t i o n only
t h a t bia s field corresponding t o maximum long wavele ngth s e n s i t i v i t y w a s c onside re d. If t h e d e e p g a p bias
field i s re duc e d from t h i s v a l u e , t h e c oe rc ivity contour
w i l l not pe ne tra te through to t h e back of t h e coating.
The previous a n a l y s i s a p p l i e s , however if in (2) (5)
t h e c oa ting t h i c k n e s s d i s re pla c e d by the depth of recording Ym give n, utiliz ing the Karlqvist approximation,
by
-
m
'
=
(5)c o t a n
[>)
HO
w he re Hc, Hob are t h e t a p e c oe rc ivity a nd d e e p g a p bias
fie ld respectively.5 The de gre e of approximation in t h i s
e x p r e s s i o n should be vie w e d in the same s e n s e as ( 2 ) .
W ith som e manipulation of t h e previous e x p r e s s i o n s ,
t h e flux for t h i s ra nge c a n be shown t o be
($7
Q = ~oWXxN1s
n
1
(2;c7
)
tan
(2T)
b
*
(7)
If t h e d e e p g a p bia s field i s i n c r e a s e d beyond that
corresponding t o peak long w a ve le ngth s e n s i t i v i t y the
c oe rc ivity contour w ill pe ne tra te beyond t h e back of
t h e ta pe c oa ting. In t h i s ra nge t h e e x p r e s s i o n for t h e
s i g n a l field a long t h e recording c ontour (3) holds with
d re pla c e d by t h e larger v a l u e ym ( 6 ) . However, t h e
magnetization only e x t e n d s through t h e c oating s o
tha t (4) s t i l l holds with t h e c oa ting t h i c k n e s s d a s the
upper limit of integration. The reproduce flux for t h i s
ra nge is
(P2= (Pl[l
-e -kd (1 +kd)]/
[ 1-e-kym ( 1 +ky,)]
(8)
Equatiors(7), (8) g i v e t h e v ar i at i o n of s e n s i t i v i t y
with b i a s a t a l l w av el en g t h s . The long wavelength behavior h a s been examined in detail.5 At s h o r t wavele n g t h s (kd > 1) a n d b i a s levels s u c h t h at k y m7 1, (7)
a n d (8) become i d en t i cal . This condition a p p l i e s in
p r a c t i c e only for c o a t i n g t h i c k n e s s e s wh i ch a r e not t o o
s m a l l with r e s p e c t t o t h e g a p l en g t h . In t h i s case t h e
d e e p g a p b i a s f i e l d is a t l e a s t t w i c e a s l ar g e a s t h e coe r c i v i t y s o t h a t (7) a n d (8) imply t h at t h e s h o rt wavelength s e n s i t i v i t y wi l l v ar y approximately i n ve rse ly as
t h e s q u a r e of t h e b i a s current.
-I
Fig. 3 , o u t p u t vs.
b i a s for' Y F e 2 0 3 ta
-15
-201 I
0
I 1 1 ,
\
I
I I I 1 1 1 ,
5
10
BIAS ( dB 1
I
15
In Fig. 3 s e n s i t i v i t y vs. b i a s meas u r ed on a welloriented y F e z 0 3 t a p e i s plotted. The d a t a parameters
correspond t o t h o s e g i v e n in Fig. 2 . N o s i g n i f i c a n c e
s h o u l d be a t t a c h e d t o t h e v e r t i c a l d i s p l a c e m e n t of t h e
long a n d short w av el en g t h c u r v e s . At b i a s v a l u e s n e a r
th a t which maximizes t h e long wav el en g t h s e n s i t i v i t y
it i s clear t h a t t h e s h o r t wav el en g t h output v a r i e s a p w m a t e l y inversely a s t h e s q u a r e of t h e b i as current a s pred i c t e d by t h i s model. This behavior i s g en er a lly found
with t a p e s composed of well-oriented u n i ax i a l pa rtic le s.
It s h o u l d be noted t h a t a purely longitudinal field model
p r e d i c t s only a si mp l e i n v er s e r e l a t i o n s h i p which i s not
a t a l l i n d i c a t e d by t h e d a t a .
DISCUSSION
There are t w o important a s p e c t s of t h i s model w hic h
s h o u l d be s t r e s s e d . The f i r s t i s t h a t ( S ) , ( 7 ) , a n d (8)
predict t h e a b s o l u t e reproduce flux a t a l l wave le ngths
for a w i d e r a n g e of b i as cu r r en t s . The experimental
v e r i f i c a t i o n of t he r el at i v e wavelength r e s p o n s e i s pres e n t e d h e r e w h e r eas t h e corroboration of t h e a b s o l u t e
flux prediction with measurement i s d i s c u s s e d in Ref. 5.
Both t h e 6 dB loss in long wav el en g t h s e n s i t i v i t y d u e
t o a l i n e a r l y i n c reas i n g mag n et i zat i o n a n d t h e proport io n a l i t y between t h e a n h y s t er et i c s u s c e p t i b i l i t y a n d
recording s e n s i t i v i t y a r e ex p er i men t al l y verified t o good
approximation. The a n h y s t e r e t i c s u s c e p t i b i l i t i e s of t h e
CrO2 a n d y FegO3 t a p e s meas u r ed f o r Fig. 2 w e re approximately 35 a n d 20 r e s p e c t i v e l y , which predicts w e l l
t h e i r r e l a t i v e long wav el en g t h s e n s i t i v i t i e s .
The s e c o n d a s p e c t i s t h at t h e s e e x p r e s s i o n s repres e n t a n upper limit for t h e l o w level ac b i a s e d wavele n g t h r e s p o n s e wh i ch , as d i s p l a y e d in Fig. 2 , i s
a lm o st a c h i e v e d by t h e b e s t of p r es en t recording t a p e s .
On the b a s i s of t h i s model t h e s p r e a d in s wi t c hing
f i e l d s of t h a s e t a p e s is, i n f a c t , s m a l l en o u gh so tha t
further d e c r e a s e s c a n n o t s u b s t a n t i a l l y i n c r e a s e t h e
s m a l l s i g n a l reproduce flux o v er t h e r an g e of wavele n g t h s measured. In a d d i t i o n , t h e head-to-tape spacing must be e x c eed i n g l y s m a l l for t h e s e t a p e s .
M a n y s i m p l i f i cat i o n s are made in t h i s ca lc ula tion.
11 78
Ce rta inly t h e e ffe c t of t h e s p r e a d in pa rtic le switching
fie lds must be included in a c om ple te model of recording. Each e le m e nt of t a p e is not recorded when it expe rie nc e s a d e c r e a s i n g bia s field amplitude e q u a l t o its
c oe rc ivity but ra the r over a ra nge of fie lds. The head
field function t r a n s l a t e s t h i s c oe rc ivity s p r e a d into a
recording z one le ngth w hic h c a n re duc e the recorded
magnetization a t short w a ve le ngths. An e xpansion of
t h e model using t h e Pre isa c h formalism ve rifies t h e conc l u s i o n he re tha t this e f f e c t is s m a l l for modern t a p e s
w ith extremely good orientation. This i s e s p e c i a l l y
signific a nt s i n c e a tota l field model w ill e xhibit much
longer record z one le ngths tha n a purely longitudinal
model. Me a sure m e nts on t a p e s w ith re la tively broad
s p r e a d s in sw itc hing fie lds e xhibit s u b s t a n t i a l l y poorer
frequency r e s p o n s e tha n tha t predicted by t h i s simple
model a n d t h i s t o o is indic a te d by a n improved calculation.
Demagnetization fie lds ha ve also be e n neglected
in t h i s sim ple model. Their inc lusion re nders a n y model of recording extremely complicated. S i n c e t h e C r 0 2
a nd Y F e z 0 3 t a p e s e xhibite d s i m i l a r e xc e llent surface
sm oothne ss a nd virtua lly ide ntic a l M-H l o o p s h a p e s
w ith t h e only signific a nt diffe re nc e being t h e increased
c oe rc ivity of CrOZ, t h e s m a l l diffe re nc e in their wavele ngth r e s p o n s e could b e d u e t o t h e e f f e c t of demagnetiz a tion. However, t h e de gre e of isotropy of the remanent c oe rc ivity w a s not measured for t h e two t a p e s and
t h i s e f f e c t may b e important.
U tiliz a tion of a be tte r h e a d field approximation
(suc h as g i v e n in Ref. 2)show s tha t t h e de viation of
the longitudinal field from a s t r i c t l y line a r variation
a long a contour of t o t a l b i a s fie ld only r e d u c e s t h e overa l l output level a n d not t h e s h a p e of the r e s p o n s e function a t l e a s t for contours which pe ne tra te a d i s t a n c e
g/20 OT more into t h e c oa ting. Further, t h e prediction
of tha t bia s w hic h maximizes the short w a velength sens i t i v i t y i s improved somewhat over t h a t e n t a i l e d by (7)
w hic h y i e l d s a poor e s t i m a t e a t ve ry short wavelengths.
N o n e t h e l e s s , in c o n c l u s i o n , ove r a w ide ra nge of bias
v a l u e s , w a ve le ngths, g a p le ngths a n d c oa ting thickn e s s e s , t h e s i m p l e model pre se nte d he re predicts the
behavior of low level a c bia se d m a gne tic recordinq
extremely well.
ACKNOWLEDGEMENT
The a uthor would l i k e t o thank J a y McKnight for
many stim ula ting d i s c u s s i o n s during the c o u r s e of t h i s
work.
REFERENCES
1) D. Tj a d e n a n d J. Leyten, Philips Tech. Rev., 2,
319, 1963.
2) G. Bate a n d L. P. Dunn, IEEE Trans. M a g . , MAG1 0 , 667-669, 1974.
3)
D. M e e , " P h y s i c s of Ma gne tic Recording",
North-Holland Publishing C o . , 1964.
4) R. L. W a l l a c e , B e l l S y s t . Tech. 30, 1 1 4 5 , 1951.
5) H. N . Bertram, IEEE Trans. M a g . , MAG-10, 10391048, 1974
6) 0 . Karlqvist, K. Tekn, Hog. Handl.
1 , 1954.
7) W. K. W e s t m i j z e , Philips Re se a rc h Rept., 8 , 161,
1953.
8) D. L. T j a d e n , Third Int. Cong. A c o u s t i c s , Stuttgart,
758-760, 1959.
a,