nductance, pro-
vides some degree of immunity to power
supply and ground noise, and can pro-
vide higher output power in the case of
Matching Loop Antennas
To Short-Range Radios
The tapped or transformer-matched loop
antenna must be properly matched to
differential drives in many short-range
radio designs to achieve optimum
performance.
hort-range radios are invaluable wireless systems for data,
telemetry, and voice communications. But to achieve opti-
mum operation, the radio electronics must be properly
matched to the antenna. Part 6 of this design series on
short-range radios will address the design of tapped or
transformer-matched loop antennas and show how they
can best be matched to differential drives. The final
installment of this multipart series will
appear later this year and will cover
practical issues in board design such
as layout, shielding, cost control, and
regulatory compliance.
Previously, Parts 1, 2, 3, 4, and 5 of
this series (see Microwaves & RF,
September and October 2001 and Febru-
ary, March, and July 2002, respec-
tively) explored short-range radio design,
including link budgeting, regulatory
issues, device fabrication, and loop-
antenna design. Part 5 offered an intro-
duction to loop-antenna design and the
s
JAN VAN NIEKERK
RF Applications Engineering
Manager
(480) 792-4150, e-mail: jan.van.
niekerk@microchip.com
FARRON L. DACUS
RF Architecture Manager
(480) 792-7017, e-mail: farron.dacus
@microchip.com
STEVEN BIBLE
Principal RF Applications Engineer
(480)792-4298, e-mail: steven.bible@
microchip.com
Microchip Technology, Inc., 2355
West Chandler Blvd., Chandler, AZ
85224-6199; Internet: www.micro
chip.com.
MICROWAVES & RF
72
AUGUST 2002
DESIGN
C
w
Lb
Secondary
Primary
La
Ls = 0.25w
13. The typical
printed-circuit-
board (PCB) imple-
mentation of the
transformer
matched loop
antenna is shown
here.
MWaug72 8/2/02 11:13 AM Page 72 some device limits.
Although the term tapped
loop is common, this type
of antenna will be referred to
here as the transformer loop
antenna in reference to what
is actually its fundamental
mode of operation. Under-
standing this matching method
requires resorting to the under-
lying electromagnetics (EMs).
Once the new model is grasped
it leads directly to under-
standing the harmonic per-
formance of the tapped/trans-
former loop antenna.
As shown in
Fig. 13,
a
small loop is placed near
(usually actually sharing a
side with) the radiating loop
antenna. The radiating loop
still contains a tuning capac-
itor C. The two loops actu-
ally form a loosely cou-
pled transformer, though
there is a strong tendency
among circuit designers to
want to view this structure as
a tapped inductor (no mutu-
al coupling) or autotransformer (tapped
inductor with mutual coupling). The cor-
rect model will be shown here to be a
separated transformer, though under-
standing this will require a bit of an
effort on the part of the reader. The
transformer model seems counter-intu-
itive, even to experienced RF designers,
since they are trained to think in lumped-
component terms and not in terms of
the underlying EMs upon which lumped
models are based. Thus they normally
conceive of a segment of trace as hav-
ing complete inductance all by itself in
the absence of a return path, which
leads them to misinterpret Fig. 13 as a
tapped inductor or autotransformer.
No less an authority than Fujimoto
9
in his well-respected work on small
antennas mistakenly analyzes tapped-
loop antenna matching as an auto-
transformer, and this common error
incorrectly influences the design of loop
antennas to this day. The mistaken
mental model has at its root the failure
to understand that only closed current
loops have inductance or mutual induc-
tance. It is exacerbated by the fact that
the form of transformer exhibited by Fig.
13 is not one that the engineer has
encountered in his basic trainingno
class ever showed a separated trans-
former model for a situation where pri-
mary and secondary currents actually
share a path segment.
An open mind and a review of the
underlying EMs will allow the short-range
radio designer to add this important
form of transformer antenna to his tool
kit and gain an appreciation of the EM
effects in circuits that the designers
first EMs professor probably intended.
To set about developing the correct
first-order understanding of this struc-
ture, the authors shall state the basic EMs
upon which transformer-model argu-
ment is based with minimal explanation,
leaving the reader to review their basic
undergraduate e-mag text for verifica-
tion. However, the authors will inter-
pret these EMs with respect to this new
situation, the loop antenna of Fig. 13,
in some detail to make the
model fully clear and gen-
erate the correct mental model
in the reader's mind.
First consider, as back-
ground, the definition of a
voltage [as electromotive
force, (EMF)] as the closed
line integral of electric field,
which is the field form of
Kirchoffs Voltage Law
(KVL):
Next, consider the fact
that electric flux through a
surface is provided as the
surface integral of flux den-
sity over that surface:
Flux is integrated up over
an areanot over a line seg-
ment. Next, Faradays Law
offers voltage (EMF) as a
function of flux:
Comparing Eq. 71 and 73, we note
that the voltage around a loop is equal
to the negative of the time derivative of
flux through the loop.
Amperes Law gives current as the
closed-line integral of magnetic field:
Where magnetic field H is related to
flux density B by:
When terminal voltage and current
can be calculated and impedance deter-
mined as their ratio, a circuit model
results. The EM equations above pro-
vide the means to determine current
and voltage relationships in terms of phys-
ical geometry. Eq. 74, Amperes Law,
relates flux and current over a closed
line integral that provides current con-
B
H
=
µ
0
75
(
)
I
H
dL
=
(
)
74
emf
d
dt
=
(
)
73
=
B
dS
S
(
)
72
emf
E
dL
=
(
)
71
|
SHORT-RANGE RADIOS, PART 6
|
MICROWAVES & RF
74
AUGUST 2002
DESIGN
DESIGN
La
IS
Lb
Loffset
z
dS
14. Setting up the integration of flux density that enables calculat-
ing the mutual inductance between a closed primary and a wire
secondary is illustrated here.
MWaug74 8/2/02 11:13 AM Page 74 tained within the
closed-path caused
by flux. From
Amperes Law, cur-
rent can be found
from H or B, or H
and B can be found
from current. When
B is known, the
total flux can be
found from Eq. 72,
and then with flux
known, voltage can
be found from Eq. 73. Conceptually,
the full information needed for the cir-
cuit model is available, and from Eqs.
71 to 74 it can be seen that this always
relies upon closed paths around current
or field, and not upon a line segment.
Alternately, the definitions of inductance
and mutual inductance provided by
Eqs. 76 and 77 can be used to make this
conceptual process a bit shorter:
where:
N = the number of filamentary loops
of current (one in Fig. 13) and
I = the current linked by the flux,
meaning the current that surrounds the
area the flux density is integrated over
to get the total flux.
In Eq. 77, M
12
is the mutual induc-
tance where flux produced by closed (or
infinite) path I
1
links current in closed
or infinite path I
2
. It is also true that M
12
= M
21
. Parameters L and M result in cir-
cuit equations of the form:
where:
V
1
= the total voltage through a self-
inductance with current I
1
that is also
linked to a second current I
2
sharing mutu-
al inductance M with the current path
described by I
1
.
Note that in Eqs. 75 and 76 induc-
tance cannot be calculated for a segment
of line. It requires a closed path around
V
L
dI
dT
M
dI
dT
1
1
2
78
=
+
(
)
M
N
I
12
2 12
1
77
= (
)
L
N
I
=
(
)
76
77
DESIGN
a surface to obtain the total flux quan-
tities as the surface integral of flux den-
sity. This is why a tapped inductor or
autotransformer model of Fig. 13 is
simply wrongit does not satisfy the
definition of inductance. But an inte-
gration over a closed surface, such as
the primary and secondary shown in Fig.
13, gives total flux linking a closed cur-
rent path, which then by Eqs. 76 and
77 allows calculation of self- and mutu-
al inductance that enables writing cir-
cuit equations.
With the preconceived circuit-design
model altered to take these fundamen-
tals into account, it is now possible to
find a correct (transformer-based) cir-
cuit model for Fig. 13.
Figure 14
shows
a loop intended as the primary wind-
ing of inner dimension L
a
and L
b
linked
by the flux generated by an infinitely long,
thin, round wire. The loop is also con-
sidered to be made of thin round wire
and its inner dimension is separated
from the center of the infinite wire by
distance L
offset
. Of course, most anten-
nas will be fabricated with printed-cir-
cuit-board (PCB) materials having a
flat trace, but the round wire model is
simpler analytically and is a good
approximation of an antenna formed
of circuit traces, and so is used here. Most
basic EM texts go through the small exer-
cise needed to use Amperes Law (Eq.
74) to obtain radial H and B fields
around the infinite round wire induced
by current I
s
in the wire. This yields:
SEE EQ. 79 ON P. 79
Using Eq. 72 and the differential
area element dS shown to integrate
MICROWAVES & RF
77
AUGUST 2002
DESIGN
p72,74,77,79,