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A Spinor Model for Cascading Two Port Networks In Conformal Geometric Algebra

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Abstract and Figures

Building on the work in [1], this paper shows how Conformal Geometric Algebra (CGA) can be used to model an arbitrary two-port network as a rotation in four dimensional space, known as a spinor. This spinor model plays the role of the wave-cascading matrix in conventional network theory. Techniques to translate two-port scattering matrix in and out of spinor form are given. Once the translation is laid out, geometric interpretations are given to the physical properties of reciprocity, loss, and symmetry and some mathematical groups are identified. Methods to decompose a network into various sub-networks, are given. Since rotations in four dimensional minkowski space are Lorentz transformations, our model opens up up the field of network theory to physicists familiar with relativity, and vice versa. The major drawback to the approach is that Geometric Algebra is relatively unknown. However, it is precisely the Geometric Algebra which provides the insight and universality of the model.
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A Spinor Model for Cascading Two Port Networks
In Conformal Geometric Algebra
Alex Arsenovic1
1Eight Ten Labs LLC, Stanardsville Va, 22973
Abstract—Building on the work in [1], this paper
shows how Conformal Geometric Algebra (CGA) can
be used to model an arbitrary two-port network as a
rotation in four dimensional space, known as a spinor.
This spinor model plays the role of the wave-cascading
matrix in conventional network theory. Techniques to
translate two-port scattering matrix in and out of
spinor form are given. Once the translation is laid
out, geometric interpretations are given to the phys-
ical properties of reciprocity, loss, and symmetry and
some mathematical groups are identified. Methods to
decompose a network into various sub-networks, are
given. Since rotations in four dimensional minkowski
space are Lorentz transformations, our model opens
up up the field of network theory to physicists familiar
with relativity, and vice versa. The major drawback to
the approach is that Geometric Algebra is relatively
unknown. However, it is precisely the Geometric Alge-
bra which provides the insight and universality of the
I. Introduction
Two-port networks play an important role in microwave
engineering, control theory, quantum mechanics, and sev-
eral other disciplines. From a modeling standpoint, two-
port networks can be thought of as operators which act
on loads, or as quantities of interest. Microwave networks
are traditionally represented by various matrix formats,
such as the scattering (S), impedance (Z), and admittance
(Y) matrices. Choosing a given format is equivalent to
choosing a basis in which to frame a transformation. Be-
cause the different bases have different physical interpre-
tations, one may be more natural for a given problem than
another. For example, at high frequencies power is more
easily measured than impedance so scattering matrices are
generally used. When several two-port networks are cas-
caded together, matrices such as the wave-cascading (T)
and (ABCD) matrix are used, which implements network
cascading through matrix multiplication. The cascading
matrix algebra has been very successful in impedance
matching, filter theory [2], and calibration problems [3].
An alternative to matrices is to use Geometric Algebra
(GA) to model two-port networks as spinors. As shown in
[1], the fundamental relations of transmission line theory
become linearized by using a tool known as Conformal
Geometric Algebra (CGA). This construction allows op-
erations such as adding impedance and admittance, or
changing line impedance to be implemented with rotations
in a four dimensional minkowski space, otherwise known as
Lorentz transformations. While the work in [1], gave some
spinor representations for fundamental circuit elements,
this paper presents a method to translate an arbitrary two-
port s-matrix into a spinor. While essentially equivalent
to the wave-cascading matrix, the CGA spinor approach
provides unique geometric insight and basis invariance,
neither of which are possible with linear algebra. Using
spinors opens up the field of network theory to physicists
familiar with Lorentz transformations, and allows tech-
niques to be translated between the two disciplines. The
major drawback to this new approach is the mathematical
sophistication required.
The paper is divided into two main parts. Sections II
and III deal with translating scattering matrix represen-
tation of two-port networks in and out of CGA Spinor
representation. This ability is required in order to interface
existing infrastructure, and also provides a way to migrate
one’s understanding into the new CGA formalism. Section
IV-B describes the geometry of some special cases of
networks, while section V, demonstrates how to decom-
pose two-port networks into sub-networks based on the
physical properties of reciprocity, loss, and symmetry. In
contrast with the conventional characterization based on
matrix conditions, each decomposition is given a concrete
geometric interpretation.
A. What is Not Done Here.
This paper does not present a direct geometric interpre-
tation of the S, Z, or Y-matrix itself, nor show how such
a model would be related to the wave cascading spinor
model presented here. These things are currently being
worked on.
II. CGA Spinor For Reciprocal Networks
A. Geometry of Reflectometry
Our network spinors are geometric objects which repre-
sent the wave cascading matrix. The cascading matrix is
useful because it models reciprocal networks as rotations,
while the scattering matrix only represents lossless net-
works as rotations. Additionally, as it’s name implies, cas-
cading networks is accomplished by multiplication. There
are several ways to setup the matrix-to-spinor translation.
The most direct route is to translate the wave-cascading
matrix directly into a spinor, and replace the matrix
Figure 1. Circuit diagram of an arbitrary two-port highlighting the
reflection coefficients on either side. The left port is labeled port 1
and the right side port 2.
product with the geometric product. However, we choose
to take an engineering approach and assemble our model
from the s-matrix elements using physical arguments from
reflectometry. While this may seem archaic to mathemati-
cians, it was how our model was developed.
The basic problem in reflectometry is that some ar-
bitrary two-port network is located between a load of
interest and the measurement device, as shown in Figure
1. By measuring a set of known loads the two-port network
can be sufficiently characterized, thereby allowing un-
known loads to be measured. The S-matrix is a commonly
used representation for the intervening two-port as it’s
elements are reflection and transmission coefficients and
these quantities are more easily measured than impedance
at microwave frequencies. Starting with the conventional
formulation [2], the reflection coefficient ais transformed
by a two-port network into b, according to the formula,
b=s11 +s12s21 a
Where the sij are the various elements of the S-matrix
for the two-port, and all variables are complex numbers.
This formula can be re-arranged as,
b=s11 +s12s21 a1s22 1.(2)
Writing the equation in this form illustrates how the
relation can be broken up into a series of simpler functions,
each of which is geometrically interpretable. In sequence:
the function a1s221is known as a transversion,
the s12s21 term affects a rotation/dilation, and finally,
s11 performs a translation. Writing this as a sequence of
V=Ts11 D|s21s21 |Rs12 s21 Ks22 .(3)
Where Vis the total transformation, and the elemen-
tary transformations are represented by a transversion K,
rotation R, dilation D, and translation T. The subscripts
of each operator correspond to the matrix element, or
component thereof, which parameterizes it. For reciprocal
networks the transmission coefficients are equal s21 =s12
and so the dilation factor can be written as |s21|2with a
rotation angle is of 2s21. An advantage of casting this re-
lationship in operator form is that group theory can be put
to use. The decomposition of eq 3 is identical to that of a
general conformal transformation as given in [4]. Once this
is recognized, much of network theory can be abstracted
to group theory. In CGA, each operator is represented by
a multivector which enacts a rotation, known as a rotor.
An un-normalized rotor is called a spinor. By finding the
rotors for the elementary transformations in eq (3), any S-
matrix can be converted into a CGA rotor, allowing two-
port networks to be analyzed with the CGA framework.
Since the rotors for all of the operations in eq (3) are well
known, we could just write down the total rotor. However,
first we must we give an introduction to CGA to make the
notation clear. More information can be found in [1].
This section provides a brief introduction to the CGA
and notation employed in this paper. Start by representing
reflection coefficient as a vector within a plane spanned by
two orthonormal vectors e1and e2of positive signature.
These can be thought of as the real and imaginary axes of
the complex plane. Next, add a third dimension of positive
signature (e3) and a fourth of negative signature (e4). The
orthonormal vector basis for the conformal space is given
4= 1.(4)
The basis generates a geometric algebra containing the
following blades:
, ei
, eij
, eijk
, i
Here the e12-plane is identified as the original 2D space,
and e34-plane contains the added dimensions. Due to the
signature of the added space, the e34-plane is known as
the Minkowski plane, which is commonly labeled E0,
It is convenient to further define a null basis.
These two vectors represent the points of infinity and
zero, as their subscripts suggest. They have the properties,
= 0 (9)
eeo=1 + E0.(10)
In terms of the null basis, a vector xin the original space
of e12 is mapped upwards to a conformal vector X, by the
X=(x) = x+1
The inverse, downwards map, is the made by normal-
izing the conformal vector then rejecting it from the
Minkowski plane.
x=(X) = XE0
In the above formula and all others, we adhere to
the convention that the inner and outer products take
precedence over the geometric product. Now that the CGA
has been laid out, rotors representations for the operations
in eq (3) can be expressed.
C. S-Matrix to Rotor
Derived in [4], the CGA rotors for the operators in eq
(3) as expressed in the basis defined above are,
2eoxe2=e23Txe23 (14)
Dρeln ρ
2e12 .(16)
Where the subscripts are either vectors in e12 or scalars
which parameterize the rotor. The transversion rotor K
is slightly different than that given in [4] because the
complex inversions in eq (2) add additional reflections, so
the Kin eq (14) could be called a complex transversion.
By defining these rotors, we have an explicit formula
for translating a S-matrix into a CGA rotor, through
eq (3). For implementation it is important to avoid any
unnecessary multiplying or division of complex numbers.
For example, the rotation operator
Rs12s21 =e
2e12 ,(17)
is better implemented as,
Rs12s21 =e(s12 +s21 )e12 .(18)
The reverse procedure of translating a rotor into a S-
matrix is discussed next.
D. Rotor to S-matrix
Translating a CGA rotor into a S-matrix is equivalent
to finding the reflection and transmission coefficients for
the two-port network when the network is terminated in
matched impedances. A procedure to accomplish this for
reciprocal networks can be developed by employing some
concepts from microwave reflectometry. First, we express
the relation in Figure (1) in the language of CGA,
b=V a ˜
V . (19)
Where aand bare the up-projected null vectors of the
reflection coefficients, and Vis the rotor representing two-
port network. By definition, s11 is the reflection coefficient
at port 1 when port 2 is matched, so this quantity is found
by letting Vact on a match, ie a=eo.
s11 =V eo˜
This determines Tin (3). Geometrically this corre-
sponds to determining how the origin is translated (also
known as the directivity). From an operator perspective,
determining Tis possible because the origin is invariant to
operations of K, D and R. It is also obvious by inspecting
eq (2). Next, we can determine s22 in an identical way
by flipping the network, and then terminating it with a
match. The operation of physically flipping a network,
which will exchange port indices’s is denoted with an
underbar and can be defined,
Ve14V1e14 .(21)
A proof that this operator does permutes port 1 and 2
is given in Appendix (VII-A). By using this flip operator,
s22 is found in a similar way as s11,
s22 =Veo˜
Which determines Kin (3). Once Tand Kare found,
they can be removed from Vto leave only the rotation
and dilation operators.
T V ˜
K=DR (23)
The parameters of DR can be found by taking the
logarithm as described in [5]. Which completes the deter-
mination of the S-matrix.
E. Other Bases and Conversions
To convert the wave-cascading matrix to another for-
mat, such as ABCD-matrix, a basis transform must take
place at some point. With the conventional treatment,
two-port networks in different bases are related through
a bilinear matrix equation such as ,
S= (ZI) (Z+I)1(24)
Where Sand Zare complex NxNmatrices containing
the scattering and impedance parameters, and Iis the
identity matrix. While concise, this equation has no ge-
ometric interpretation. In contrast, CGA allows different
basis representations to be related through simple, geo-
metrically interpretable rotations as demonstrated in [1].
The basis rotors work in the same way for the cascading
matrices, such as ABCD or T-matrices. For example, a
rotor can be transformed from the reflection and wave-
cascading spinor Vinto the ABCD spinor (A) with a
simple π
2-rotation in the e13 plane.
4e13 V e π
4e13 .(25)
The rotors are repeated below, and these can be used
to transform a CGA versor to and from the desired basis.
Rzs eπ
4e13 (26)
Rzy eπ
2e23 (27)
Rsy =eπ
22(e23+e21 )(28)
However, since CGA provides a basis invariance such
transformations are not as important. Next we extend the
model to include non-reciprocal networks.
III. Non-reciprocal Networks
A. The Model
So far we have shown how reciprocal two-port networks
can be modeled as rotations in Conformal Geometric
Algebra. This sections extends that model to include
non-reciprocal networks using whats known as a duality
spinor [6]. Developing the theory in this way makes sense
because; 1) reciprocal networks are far more common, 2)
areciprocity has a geometrically distinct interpretation,
and 3) because the areciprocity can be easily separated
and removed. Any model for areciprocity must be imper-
ceivable from the perspective of reflectometry because the
transmission coefficients in eq 1 are not individually ob-
servable. As described in part 4 of [6], the transformation
of any vector in the Dirac algebra of G1,3can be written,
p0=V pS (29)
A Lorentz transformation preserves the length of vec-
p02=V pSV pS =p2(30)
This will hold for any vector ponly if SV either
commutes or anti-commutes with p. The only grade of
elements in G1,3which fulfill this property are scalars and
pseudo-scalars, so we write,
SV =α+βi =ρeθi .(31)
Where iis the pseudoscalar. Equation (31) implies
so that a non-reciprocal two-port network has the fol-
lowing spinor representation,
p0=±ρeθiV pV 1.(33)
This form cleanly separates the reciprocal from the
areciprocal; the areciprocal part is a duality rotation, while
the reciprocal part is a bivector rotation. Using geometri-
cally distinct objects for physically distinct network prop-
erties provides insight into the physics of 2-port networks
that is difficult to attain with matrix representations.
Reflecting on the proposed duality spinor model, we note
that it has many of the required features of areciprocity;
it inverts with the flip operation, it commutes when
several two-ports are cascaded together, and it requires
two-independent parameters. The duality spinor is the
geometric representation of the determinant of the wave
cascading matrix.
B. S-matrix Areciprocity to Spinor
To convert a non-reciprocal s-matrix into a spinor, first
compute Vby way of equation (3). This requires the
parameters s11,s22 , and the product s12s21. Next, the
duality spinor Pis determined by the complex ratio of
s12 to s21,
P=ρe θ
s21 |(35)
Once Pis found the total non-reciprocal versor is
created by multiplying Vwith P.
A=P V (37)
The half-angle θ
2and ρare used because we choose to
implement the duality spinor in a double-sided formula.
This way we just keep track of A, instead of Pand
Vseparately (but this would work too). Converting a
duality spinor to a complex number is done by reversing
this procedure. This requires a technique to separate the
duality spinor from the total spinor, which is given in the
next section.
C. Areciprocity Spinor to S-matrix
Any versor which represents a non-reciprocal network
can be broken up into reciprocal and areciprocal parts by
separating the duality spinor from the bivector rotor. This
is necessary since a bivector rotation in four dimensions
will have scalar and psuedo-scalar components. A tech-
nique to accomplish this separation can be developed by
exploiting each part’s behavior in regard to reversion [6].
The spinor for a an arbitrary two-port network can be
A=P V. (38)
Where Pis an areciprocal duality spinor and Vis a
reciprocal rotor generated by the bivector U.
P=ρe θ
To separate Ainto Pand V, first determine Pand then
remove it. Start by forming,
A=P V ˜
V P =P2=ρeθi .(40)
Once Pis found, the rotor Vcan be found using,
V=P1A. (41)
This section has given formulas needed to convert a s-
matrix to and from a spinor representation in a four di-
mensional minkowski space. In the next section reciprocal
networks are further decomposed based on the physical
attributes of loss, symmetry and matched-ness.
IV. Special Cases and Groups
A. Reciprocal Networks
Since cascading any number of reciprocal networks
yields a reciprocal network it is clear that they form a
mathematical group which is a sub-group of all two-port
networks. With CGA this group structure is represented
geometrically by the fact that reciprocal networks are rota-
tions, and non-reciprocal networks are spinors. Extending
this logic, several other groups can be anticipated. For ex-
ample, lossless networks form a group, as well as reflection-
less or matched networks. Since two-port networks are
modeled as rotations in CGA, we can analyze their group
structure by identifying the planes of their rotations.
B. Lossless Networks
It is well known that lossless networks are represented
by unitary scattering matrices, but there properties in
cascading matrices or spinors is different. A mathematical
representation for lossless networks can be made by con-
sidering the following physical argument. A lossless load
has a reflection coefficient magnitude of unity, which can
be visualized as a vector confined to the unit circle. In
CGA, lines and circles are represented by tri-vectors and
can be defined by taking the outer product of three null
vectors which lay on the line/circle [7]. In this way, lines
and circles become geometric objects in the algebra, as
opposed to equations regarding their coordinates. Using
this construction, the unit circle can be defined by the
(e1)∧ ↑ (e1)∧ ↑ (e2) = e124.(42)
Cascading a lossless two-port network in front of a
lossless load preserves the magnitude of the reflection
coefficient, for there is no way for the power to dissipate.
This is only true for lossless loads. Algebraically, this
means that a lossless versor leaves e124 invariant.
V e124 ˜
V=e124 (43)
Equivalently, we can say that a lossless versor commutes
with e124. The only generators which have this quality are
those contained within the subspace defined by e124, which
can be interpreted as the lossless subspace. As shown in
[1], the generators of the discrete element group which are
contained within this subspace are those for reactance X,
suscpetance B, and an impedance transformer N,
Xe12 e24 (44)
Be12 +e24 (45)
Ne14 (46)
These are by definition lossless elements. A figure illus-
trating the lossless subspace of CGA, with the bivector
rotation planes labeled is shown in Figure 2. By summing
infinitesimal rotations in both Xand Bequally, a rota-
tion in the e12-plane (labeled L) can be created, which
Figure 2. Lossless subspace of CGA with relevant vectors and
bivector planes labeled.
extends the list of lossless elements to include matched
transmission lines [1]. Mismatched transmission lines can
be modeled as cascading impedance transformers on either
side of a matched line, or by summing infinitesimal rota-
tions in Xand Bunequally. This exact model for a lossless
subspace was published in the 1950’s by E.F. Bolinder [8],
albeit through a different approach. Next we identify some
special cases of reciprocal networks.
C. Matched Networks
Matched networks are defined by having no reflection
coefficient at either port, meaning the diagonal elements
of their s-matrix are zero ,
s11 =s22 = 0.(47)
This special class of networks also forms a group, from
the same physical argument given above. The matched
condition reduces eq 3 to rotations and dilations, which
are generated by rotations in e12 and e34 respectively, ie
2e12ln ρ
2e34 .(48)
D. Symmetric Networks
Symmetric networks don’t form a group because cascad-
ing two symmetric networks can produce a non-symmetric
network. By definition, symmetric networks are invariant
to a flip, which we can write,
V=V=e14V1e14 .(49)
The only rotors which fulfill this property are those
which don’t contain e14 or e23.
E. The Structure of Two-port Networks
Now that some special cases of networks have been
identified by various means, it starts to become clear that
two-port networks can be systematically classified based
on the planes of rotation. This is most easily done by
inspecting the bivectors present in their generators, as
Property Bivectors in Generator
Reciprocal e12 , e23, e34 , e14, e23 , e14
Symmetric e12, e34 , e13, e24
Asymmetric e12, e23 , e34, e14
Lossless e12, e24 , e14
Non-propagating e13, e34 , e23
Matched e12, e34
Table I
List of network classifiers and the bivectors present in
their generators. Groups are emboldened.
Ma tch ed (2)
Lo s sl e ss ( 3) Non-propagating(3)
Symmetric(4) Asymmetric(4)
1 2
1 2
1 2
1 2
1 2
Reciprocal (6)
1 2
Figure 3. Graph illustrating bivector generators present in various
network classifications. Combinations of classifications can be accom-
plished with the set theory intersection operation, between the group
is done to classify Lie Groups in [9]. A list of different
classifiers and the bivectors present in their generators
is given in Table I. We find a graph more helpful to
visualize this structure. The graph shown in Figure 3
represents vectors as nodes, and bivectors as connecting
edges, with labels to indicate the group and degrees of
freedom. The generators for various physical classifiers is
visualized as subsets of edges. Different classifiers can be
combined through the intersection operator of set-theory.
For example, in reference to Figure 3, a symmetric, lossless
network requires two parameters and it’s generator con-
tains e12 and e24. Visually, this result can be determined
by overlaying the two Cayley graphs and take the union.
V. Decomposition Methods
A. The Projective Split
A fundamental component in the Space Time Alge-
bra (STA) is the concept of a projective split, where it
represents the relationship between the Dirac and Pauli
algebras [4], [6]. Since the geometry of CGA for two-
port networks is identical to STA, we might suspect the
split to be useful in network theory as well. It turns out
that through a series of splits with various directions,
reciprocal two-port networks can be decomposed based on
their physical properties. Some of these physical properties
form mathematical groups.
The concept is best illustrated with an example, so we
revisit the lossless subgroup to demonstrate. As shown in
section IV-B, lossless two-port networks for a 3-parameter
group which can be identified as bivectors belonging to
the e124-subspace of CGA. This subgroup can also be
generated by employing a projective split with e3, in which
case the bivectors of G1,3are mapped into vectors and
bivectors in G1,2. The bivectors not containing e3map
into bivectors and form the group, while the bivectors
containing e3map into vectors which do not form a group.
The map can be defined as follows,
eij eij , i, j 6= 3.(51)
In this way a lossy network can be decomposed into
lossless and phase-less or non-propagating parts. The term
non-propagating, although awkward, seems to be the most
accurate description of the antithesis of lossless. If the non-
propagating vector part is thought of as representing a
translation and the lossless bivector part is thought of as
a rotation, then two-port networks become a hyperbolic
motion in a three-space. This means two-port’s could be
studied with a minkowskian motor algebra, an interesting
The lossless group can be split once more, this time sep-
arating the group into symmetric and asymmetric parts.
By choosing e2as the splitting vector, the bivectors in
G1,2are mapped into the vectors and bivectors of G1,1
according to,
e14 =e12.(53)
Here, the vectors in the minkowski plane represent the
symmetric part, and the bivector represents the asym-
metric part. As we have shown, employing the projec-
tive split decomposes networks by mapping bivectors into
different grade objects in a sub-algebra. By choosing dif-
ferent splitting vectors, it is possible to achieve different
decompositions and sub-algebras. An alternative approach
to decomposition which does not leave G1,3is described
B. Elemental Rotations
Since CGA allows any two-port network to be modeled
as a rotation in four dimensions, decomposing a network
into simpler sub-networks can also be modeled as de-
composing a rotation into a series of elemental rotations.
Theorems about four dimensional rotations are well devel-
oped thanks to work done to characterize Lorentz trans-
formations. From this we know that any rotation can be
decomposed into a rotation which leaves a specified vector
invariant, followed by a rotation in a plane containing
that vector [10]. This fact is used in [6] to decompose
a Lorentz transformation into time-like and space-like
rotations. An exact translation of this decomposition in
our basis amounts to choosing e4as the invariant vector.
While possible, we have yet to find a use in network theory
for such a decomposition. However, as we have shown in
the last two sections, decomposing a rotation based on e3
allows a network to be separated into lossless and non-
propagating parts, so we revisit this dichotomy here as
Start with a lossy rotor Vand assume it can be broken
up into a non-propagating part Hand a lossless part Uin
V=HU (54)
Like V, both Hand Uare rotations, so
U= 1.(55)
The lossless part will leave e3invariant,
and the non-propagating part will contain e3,
Next, form the quantity,
V e3˜
V e3,(58)
Then express this in terms of Hand U, and insert
factors of e2
3strategically to find,
V e3˜
V e3=HUe3˜
So Hcan be found from Vif the square root of H2can
be computed. Since His simple so is H2, and the formula
for the square root of a simple rotor is [10],
R=(1 + R)
2 (1 + hRi).(63)
A direct formula for Hin terms of Vis thus,
H=1 + V e3˜
V e3
21 + V e3˜
V e3.(64)
Once His determined, Ucan be found from Vby
removing H.
HV (65)
Which completes the determination of Hand Ufrom
V. The same procedure can be used to further decompose
the lossless rotation into symmetric and asymmetric parts
by choosing e2as the invariant vector. Simply use Ufor
V, and replace all e3’s with e2.
VI. Conclusion
We have presented a spinor version of the wave-
cascading matrix and developed methods for translating
between the matrix and CGA Spinor representations.
Geometric approaches to two-port network decomposition
based on the physical characteristics of reciprocity, loss,
and symmetry were presented and associated rotation
groups were identified. An arbitrary network can be de-
composed into specific parts either with the projective
split shown in Section V-A, or the elemental rotational
decomposition given in Section V-B. The advantage of
using Geometric Algebra for modeling networks is the
geometric meaning given to various physical attributes,
and the ability to unify results with other fields. We
are currently working on applying the spinor model to
applications in filter theory, calibration, and even-odd
mode analysis.
VII. Appendix
A. Proof of Flip Operator
Flipping a two-port network is defined as interchanging
its ports. In regard to a s-matrix, this has the effect of
swapping the following elements,
s12 s21 (66)
s11 s22.(67)
We seek the geometrical equivalent of this operation.
Given that nonreciprocal network can be expressed as the
A=P V. (68)
Where Pis an areciprocal duality spinor and Vis a
reciprocal rotor. The flip operation is
Ae14A1e14 .(69)
a) Proof: The flip operation will effect the areciprocal
and reciprocal parts of the network differently, so each is
analyzed separately. Since a reciprocal network is a bivec-
tor rotor, inversion is equivalent to reversion, V1=˜
Additionally, the areciprocal part Pcommutes with the
e14’s, which annihilate each other, so the flip operations
reduces to inversion of P. Combining these facts, allows
us to write,
A=e14 (P V )1e14 =P1e14 ˜
V e14.(70)
Given the relation of Pto the s-matrix defined in the
III-B, inverting Pswaps s12 with s21. Whats left is to swap
the s11 and s22 elements. By inspecting eq (3), it can be
seen that the flip operation exchanges the parameter of
the transversion with that of the translation. The proof
that this can be accomplished by reversion combined with
a reflection in e14 is below,
(TtDdRrKk)e14 (71)
=e14 ˜
Tte14 (72)
=e14 ˜
| {z }
e14 ˜
| {z }
e14 ˜
| {z }
e14 ˜
| {z }
Here we have used the fact that e2
14 = 1 to insert pairs of
e14 where convenient, and computed the result of reflecting
each operator in e14.
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Full-text available
In this paper we present the application of a projective geometry tool known as Conformal Geometric Algebra (CGA) to transmission line theory. Explicit relationships between the Smith Chart, Riemann Sphere, and CGA are developed to illustrate the evolution of projective geometry in transmission line theory. By using CGA, fundamental network operations such as adding impedance, admittance, and changing lines impedance can be implemented with rotations, and are shown to form a group. Additionally, the transformations relating different circuit representations such as impedance, admittance, and reflection coefficient are also related by rotations. Thus, the majority of relationships in transmission line theory are linearized. Conventional transmission line formulas are replaced with an operator-based framework. Use of the framework is demonstrated by analyzing the distributed element model and solving some impedance matching problems.
Full-text available
1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.
Full-text available
It is shown that every Lie algebra can be represented as a bivector alge- bra; hence every Lie group can be represented as a spin group. Thus, the computa- tional power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thor- oughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
Full-text available
Conformal transformations are described by rotors in the conformal model of geometric algebra (CGA). In applications there is a need for interpolation of such transformations, especially for the subclass of 3D rigid body motions. This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA. It also classifies the types of conformal transformations and their orbits. To derive the results, we employ a novel polar decomposition for the even subalgebra of 3D CGA and an associated norm-like expression.
Full-text available
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.
As leading experts in geometric algebra, Chris Doran and Anthony Lasenby have led many new developments in the field over the last ten years. This book provides an introduction to the subject, covering applications such as black hole physics and quantum computing. Suitable as a textbook for graduate courses on the physical applications of geometric algebra, the volume is also a valuable reference for researchers working in the fields of relativity and quantum theory.
Conference Paper
A unified network theory is presented. It consists of three parts: a network formalization, a geometrical model, which is the Minkowski model of Lorentz space, and a mathematical tool, Clifford algebra. The latter is well suited in dealing with rotations in Lorentz space. The rotations can be represented by the exponentials of a single infinitesimal isometry or a single Clifford bivector. Special emphasis is put on the parabolic rotations. Through the work of M Riesz we now know how to deal with these. The network theory is applied to some simple synthesis examples starting with a given insertion loss power ratio. The entire procedure is performed in Lorentz space. Transformations to Flatland, the impedance plane, for example, are done by simple projective transformations. The examples chosen are: 1) Butterworth-3 network (parabolic), 2) Chebyshev-1 network (hyperbolic-parabolic), 3) simple stepline (hyperbolic-elliptic), and 4) exponentially tapered line (finite continuous transformation group).
A general theory for performing network analyzer calibration is presented. Novel calibration procedures are derived which allow for partly unknown standards. The most general procedure derived is called TAN and allows for five unknown parameters in the three calibration standards. The values of the unknown parameters are determined during the calibration procedure via eigenvalue conditions. The good performance of all the procedures has been shown experimentally. This wide spectrum of procedures using different calibration standards makes it possible to choose an optimal algorithm for any environment