# Automatic Measurement Technique of Electromagnetic Rotation in a
Nonreciprocal Medium

Swadesh Poddar, Alexander M. Holmes, , and George W. Hanson, Manuscript
received xxx; revised xxx.S. Poddar, (email<EMAIL_ADDRESS>
###### Abstract

This article aims at developing a simple, automated test bench procedure to
measure polarization rotation of an electromagnetic wave upon propagation
through a nonreciprocal medium. A comprehensive measurement approach is
developed from the scattering matrix. The proposed measurement procedure is
demonstrated using an electronically tunable nonreciprocal metamaterial, and
the accuracy of the proposed method is compared with the well-accepted
conventional measurement technique.

###### Index Terms:

Faraday rotation, non-reciprocity, polarization rotation.

##  I Introduction

Polarization rotation, the change in orientation of the plane of polarization
about the microwave/optical axis of a linearly polarized wave as it propagates
through certain materials, is an important concept in both reciprocal and
nonreciprocal systems. The precise determination of the polarization state of
an electromagnetic wave is fundamental for a wide variety of applications in
microwaves, radio frequency front ends, astronomy, optics and terahertz
technology, and remote sensing, to name just a few. A linearly polarized
electromagnetic wave may be represented as the sum of two circularly polarized
waves (right-and left-handed circular polarized waves) with opposite
handedness and equal amplitude. Faraday rotation occurs when electromagnetic
waves propagate through a nonreciprocal medium, whereupon the right-handed and
left-handed circularly polarized waves (L/RHCP) propagate with different phase
velocities, and experience different impedance and refractive indices. This
LHCP and RHCP waves start to accumulate a phase difference as they propagate
through the medium, therefore, with respect to time and distance, the
direction of linear polarization changes. The greater the deviation in phase
velocity, the greater the rotation [1, 2, 3].

The robust practical measurement of polarization rotation in a nonreciprocal
medium is very challenging. Various approaches have been investigated to
accurately measure Faraday rotation in various frequency regimes [4, 5, 6, 7,
8, 9] and the most conventional approach is to measure polarization rotation
by rotating an antenna [10]. However, manual approaches to measure Faraday
rotation lack precision and are time consuming. Therefore, a simple,
realizable, cost effective, accurate test-bench model is of immense importance
to measure Faraday rotation. In this work, we have developed a simplified and
automatic S-parameter translation to measure polarization rotation in a
nonreciprocal environment using a network analyzer. We apply our proposed
measurement method to one of our previously developed nonreciprocal
metamaterials and we observe good agreement with manual measurement.
Furthermore, our proposed model will work for a non reciprocal slab of finite
thickness.

The paper is organized as follows. In Section II, we provide insight to the
mechanism of EM nonreciprocity and various polarization rotations, in Section
III, we present the S parameter analysis leading to simplified automatic
Faraday rotation measurement and, in Section IV, we benchmark our proposed
measurement model with a conventional approach on one of our previously
implemented tunable nonreciprocal meta-surfaces.

##  II Electromagnetic Non-reciprocity and Polarization Rotation

A system can be subdivided into two broad categories based on its interaction
with electromagnetic waves: reciprocal and non-reciprocal. A reciprocal system
is one in which the fields created by the source at the observation point is
the same when source and observation points exchange position, which can be
represented by the scattering parameter equality $\mathbf{S}=\mathbf{S^{T}}$.
Devices such as antennas, passive RLC electrical circuits, and most RF
components are reciprocal. On the other hand, nonreciprocity, achieved by
breaking time reversal symmetry, dictates that the field created by the source
and measured at the observation point is different when source and observation
points are interchanged, and can be represented by $\mathbf{S}$ $\neq$
$\mathbf{S^{T}}$ [11, 12, 3]. As an example, nonreciprocal components such as
gyrators, isolators, and circulators, are very important because of their
impact in military applications, telecommunications and so on [11, 13]. There
are various reported approaches to achieve nonreciprocity since the 1950s such
as using ferrite materials biased by a static magnetic field, and the latest
concept of achieving nonreciprocity using time modulation [12]. In all
reported cases of achieving nonreciprocity, the fundamental concept is
breaking time reversal symmetry [3, 2].

Figure 1: Matrix representation of various media and classification of non-
reciprocal and reciprocal media

Polarization rotation is directly related to material properties, and system
conditions. For example, polarization rotation in various material and
structural systems, such as chiral materials, wire grid polarizers, ferrites,
plasmas, birefringents, and gyro-tropic mediums are different. If a linearly
polarized wave travels forward through a nonreciprocal medium by a distance
$l$, gets reflected, and travels back to the starting point, the polarization
rotation angle can be represented as [14].

$\phi=\frac{1}{2}(k_{+}-k_{-})l,$ (1)

where wavenumber is $k_{\pm}=\omega\sqrt{\epsilon\mu_{\pm}}$, impedance is
$\eta_{\pm}=\sqrt{\frac{\mu_{\pm}}{\epsilon}}$ and permeability is
$\mu_{\pm}=\mu_{1}\pm\mu_{2}$. Therefore, the LHCP and RHCP waves exhibit
different phases, impedances, and refractive indices.

Figure 2: Criteria for time reversal symmetry and assymmetry. (a) Isotropic
medium (b) Chiral medium (Reciprocal). (c) Faraday system by unaltered
external magnetic bias, hence TRS breaking, non-reciprocal system.

Polarization rotation, the phase difference between cross polarized
components, and transmission parameters ($S_{21}$, $S_{12}$) are key
performance indicators to characterize nonreciprocity. For example,
nonreciprocal gyrotropy is the usual way to achieve nonreciprocity via having
a certain type of off-diagonal elements in the $\mu$ or $\epsilon$ tensor.
Figure 1 represents the matrix form of various reciprocal and nonreciprocal
cases. In bulk media nonreciprocity requires asymmetric electric permittivity
and magnetic permeability which can be classified as gyroelectric
(${\varepsilon}\neq\mathbf{{\varepsilon}^{T}}$) and gyromagnetic
(${\mu}\neq\mathbf{{\mu}^{T}}$), respectively where in Fig.1, k in the off-
diagonal terms represent the gyrotropic parameters. Nonreciprocal gyrotropic
and gyromagnetic materials exhibit off diagonal tensor elements, whereas,
reciprocal cases such as uniaxial or biaxial material don’t have any off
diagonal elements, hence gyrotropic response of the magnetized ferrites is
evident. In addition because of off diagonal tensor components, nonreciprocal
materials exhibit cross polarized phase and magnitude difference [15]. It is
important to note here that systems with symmetric tensors
${\varepsilon}=\mathbf{{\varepsilon}^{T}}$ or ${\mu}=\mathbf{{\mu}^{T}}$ with
nonzero off-diagonal elements exhibit polarization rotation (optical
activity), however, in a reciprocal way [16].

To characterize material interactions with electromagnetic waves, in Fig. 2 we
show three different systems that can be classified with respect to time
reversal symmetry and asymmetry [3]. Figure 2 (a) defines an isotropic medium
where there is not any polarization rotation and hence, is classified as
reciprocal. Figure 2 (b) represents a chiral system, where the x polarized
wave travels through the medium and experiences a polarization rotation of
$\phi^{0}$ at the receiver end, and in the reverse direction it will
experience a polarization rotation of $-\phi^{0}$. As a result, the field
polarization of a chiral system symmetrically returns to its original state,
hence, this is a reciprocal system. Figure 2 (c) depicts Faraday rotation in
the presence of a static $\textbf{B}_{0}$ field. The applied static magnetic
bias breaks time-reversal symmetry, making the process time-reversal
asymmetric [3, 17, 2]. In comparison to the chiral case, the wave experiences
$\phi^{0}$ rotation, then in the reverse direction it experiences another
$\phi^{0}$ rotation, revealing Faraday rotation [18]. To recapitulate, the
process of polarization rotation can be reciprocal or nonreciprocal. In the
case of Faraday rotation the sign (clockwise or anticlockwise) is determined
relative to the axis along which time-reversal (TR) symmetry is broken,
whereas in chiral media exhibits reciprocal magneto-electric coupling, the
sign of rotation is determined relative to the propagation direction [19, 15].
The next section will be focused on developing a realistic model for the
measurement of polarization rotation in a transparent medium.

Figure 3: A two port network model where medium 1 and 3 are reciprocal and
medium 2 is nonreciprocal.

##  III Principle of Operation and Practical Realization

Figure 4: (left panel) Automatic Faraday rotation measurement in 4 steps
leading to combined numerical calculation. (right panel a,b,c,d) Measured
S-parameter spectra for Co-Pol and Cross-Pol configurations used to calculate
Faraday rotation.

Figure 3 depicts a two-port network segmented into three parts used for our
mathematical modeling of Faraday rotation, where segments 1 and 3 represent
reciprocal media and segment 3 represents a nonreciprocal medium. Superscript
+/- represents forward/backward waves. For practical realization, focus will
be on measuring $E_{1}^{+}$, $E_{1}^{-}$, $E_{3}^{+}$ and $E_{3}^{-}$ in terms
of S matrices which can be represented by reflection and transmission
components as

$\overrightarrow{\mathbf{S}}=\begin{bmatrix}\overrightarrow{\mathbf{r}}^{(-)}&\overrightarrow{\mathbf{t}}^{(-)}\\\
\overrightarrow{\mathbf{t}}^{(+)}&\overrightarrow{\mathbf{r}}^{(+)}\end{bmatrix}=\begin{bmatrix}\overrightarrow{\mathbf{S}}_{11}&\overrightarrow{\mathbf{S}}_{12}\\\
\overrightarrow{\mathbf{S}}_{21}&\overrightarrow{\mathbf{S}}_{22}\end{bmatrix},$
(2)

where in terms of medium 1, 2 and 3, each component can be described as

$\begin{split}\overrightarrow{\mathbf{S}}_{11}=\overrightarrow{\mathbf{r}}(1\rightarrow{2}\rightarrow{1})=\overrightarrow{\mathbf{r}}^{(-)}\\\
\overrightarrow{\mathbf{S}}_{12}=\overrightarrow{\mathbf{t}}(1\rightarrow{2}\rightarrow{3})=\overrightarrow{\mathbf{t}}^{(+)}\\\
\overrightarrow{\mathbf{S}}_{21}=\overrightarrow{\mathbf{t}}(3\rightarrow{2}\rightarrow{1})=\overrightarrow{\mathbf{t}}^{(-)}\\\
\overrightarrow{\mathbf{S}}_{22}=\overrightarrow{\mathbf{r}}(3\rightarrow{2}\rightarrow{3})=\overrightarrow{\mathbf{r}}^{(+)}\end{split}$
(3)

Therefore, Eq. 2 can be represented as

$\overrightarrow{\mathbf{S}}=\begin{bmatrix}S^{yy}_{11}&S^{xy}_{11}&S^{yy}_{12}&S^{xy}_{12}\\\
S^{yx}_{11}&S^{xx}_{11}&S^{yx}_{12}&S^{yy}_{12}\\\
S^{yy}_{21}&S^{xy}_{21}&S^{yy}_{RR}&S^{xy}_{22}\\\
S^{yx}_{21}&S^{xx}_{21}&S^{yx}_{22}&S^{yy}_{22}\end{bmatrix},$ (4)

where the wave is being transmitted in the Z direction and the first and
second subscript/superscript represents port number/polarization of receiver
and transmitter antenna, respectively. We obtain Faraday rotation when the
cross polarized transmission component from port 1 and port 2 are not the
same. We can easily calculate Faraday rotation by using Eq. 5 as described in
the appendix [1, 20]

$\theta_{F}=\frac{1}{2}\tan^{-1}(\frac{t_{\circlearrowleft\circlearrowleft}}{t_{\circlearrowright\circlearrowright}}),$
(5)

where $t_{\circlearrowleft\circlearrowleft}$ defines the transmission
component due to both incident and received waves being LHCP. Similarly,
$t_{\circlearrowright\circlearrowright}$ defines the transmission component
due to an incident and received waves being RHCP. Therefore, from Eq. 4
Faraday rotation can be represented in terms of S parameters in linear
polarization as

$\theta_{F}=\frac{1}{2}\tan^{-1}\left(\frac{S_{21}^{xx}+S_{21}^{yy}-i(S_{21}^{xy}-S_{21}^{yx})}{S_{21}^{xx}+S_{21}^{yy}+i(S_{21}^{xy}-S_{21}^{yx})}\right).$
(6)

##  IV Simulation and Measurement

In our previous work [21], we calculated and measured the Faraday rotation of
a normally incident plane wave on an electronically tunable non-reciprocal
metasurface. Tunable non-reciprocity in the metasurface was introduced using a
field effect transistor (FET) with variable gate-source and drain-source bias
conditions. Here, we elaborate on the measurement procedure used in [21], and
show agreement between the calculated and manually measured results for a set
bias condition where the drain-source current is saturated.

Figure 5: Process flow of proposed automatic Faraday rotation as a function of
frequency

Figure 5 shows a block diagram of the measurement procedure to calculate
Faraday rotation from the measured S-parameters over the operational frequency
range. The diagram depicts the fabricated metasurface placed equidistant
between the antennas, in the far field. A 30V 5A HY3005F-3 linear DC power
supply is used to bias the metasurface. Once the the mechanical setup and
calibration (S-Short, O-Open, L-Load, T-Through) of the Keysight E8361 vector
network analyzer (VNA) is completed, a measurement is performed to obtain the
co-polarized and cross-polarized S-parameter coefficients, which are then fed
into Eq. (6) to calculate Faraday rotation with respect to operating
frequency. In our proposed transmission based measurement, four S-parameter
coefficients (i.e., $S_{21}$) are measured for different transmit and revieve
polarizations. For example, the cross-polarized S-parameter $S_{21}^{xy}$
($S_{21}^{yx}$) is described as the transmission coefficient from antenna 1 to
antenna 2 when the incident wave is y (x) polarized and the received wave is x
(y) polarized. Similarly, the co-polarized S-parameter $S_{21}^{yy}$
($S_{21}^{xx}$) is described as the transmission coefficient from antenna 1 to
antenna 2 when the incident wave is y (x) polarized and the received wave is y
(x) polarized. A non-reciprocal system will exhibit a phase difference in the
cross-polarization, however, the co-polarized states exhibit no phase
difference. This principle holds for both Faraday (transmission based) and
Kerr (reflection based) rotation [21, 15].

Figure 6: Measurement of Faraday rotation based on automatic and manual
process

Figure 4(a)-(d) depicts the various co-polarized and cross-polarized states of
the receiver with respect to the transmitter, while Fig. 4(e)-(h) shows the
measured S-parameters associated with each case in (a)-(d), respectively. From
the measurement, it is clear that the co-polarized measurements (a)-(b) show
very little sign of isolation while the cross-polarized measurements (c)-(d)
exhibit a large amount of isolation at the resonant frequency. A major
challenge in measuring Faraday rotation is that while transmission is maximum
at resonance, it is still very low. To account for Faraday rotation
accurately, the signal from antenna 1 to antenna 2 and vice versa must pass
through the metasurface only. As a result the metasurface at far field
distance from both antennas must be sufficiently large. If the board is not
large enough relative to the antenna beamwidth at the metasurface position, a
PEC material at the operating frequency can be used surrounding the
metamaterial to block any direct signal transmission from antenna 1 to antenna
2. Once the co-polarized and cross-polarized S-parameter coefficients are
measured, an in house script is used to perform the Faraday rotation
calculation with respect to frequency.

For comparison, a manual method to measure Faraday rotation at different
operating frequencies points can be used (see Ref. [21] for details regarding
the measurement setup). Figure 6 shows the measured Faraday rotation obtained
using the automatic (calculated) and manual measurement with respect to the
frequency range of operation. To manually measure Faraday rotation, the
receive antenna is rotated manually until maximum transmission (i.e., maximum
$\left|S_{21}\right|$ or $\left|S_{12}\right|$) is achieved.

TABLE I: COMPARISON OF THE RECENTLY PROPOSED MEASUREMENT OF POLARIZATION ROTATION Criteria | [22] | [10] | [8] | This Work
---|---|---|---|---
| Measurement
---
Type
Discrete | Discrete | Continuous | Continuous
Setup |  | Open
---
Space
| Open
---
Space
Waveguide |  | Open
---
Space
| Measured
---
Component
Tranmission | Reflection | Transmission | Transmission
| Measured
---
Co-Cross Pol
No | No |  | Not
---
Reported
Yes
Process | Manual | Manual |  | Automatic
---
(Backfitting)
Automatic
Tunability | Partial | Partial | Full Scale | Full Scale
Bias | Electronic | Electronic | Magnetic | Electronic

Table 1 shows the comparison between the proposed automated system and the
manual rotation measurement. In comparison with other works, our proposed
measurement procedure offers a very simple, cost effective, precise
measurement of polarization rotation. In addition, in our proposed model we
can calculate Faraday rotation over all frequencies whereas most of the
conventional approaches are based on manual measurement of Faraday rotation by
rotating the antenna at various frequency points and post interpolation
between those discrete data points.

##  V Conclusion

To summarize, we have presented a S parameter model leading to a very simple
and accurate test-bench setup to measure polarization rotation in an
anisotropic media using a nonreciprocal metasurface. The numerical modelling
has been analyzed and verified. We envision that this proposed model will add
a new building block in the measurement of polarization rotation and can be
implemented in any types of polarization rotation system utilizing two
linearly-polarized antennas and a network analyzer.

##  Appendix A Mapping between Transmission Matrix in linear and circular
basis

The transmission matrix in a linear basis can be mapped to a transmission
matrix in a circular basis by following the advanced Jones calculus [23] as

$\left[\begin{array}[]{cc}\mathbf{t}_{\circlearrowright\circlearrowright}&\mathbf{t}_{\circlearrowright\circlearrowleft}\\\
\mathbf{t}_{\circlearrowleft\circlearrowright}&\mathbf{t}_{\circlearrowleft\circlearrowleft}\end{array}\right]=\frac{1}{2}\left[\begin{array}[]{cc}P&Q\\\
R&S\end{array}\right],$ (7)

where
$P=t_{xx}+t_{yy}+i(t_{xy}-t_{yx})$
$Q=t_{xx}-t_{yy}-i(t_{xy}+t_{yx})$
$R=t_{xx}-t_{yy}+i(t_{xy}\text{ }+\text{ }t_{yx})$
$S=t_{xx}+t_{yy}-i(t_{xy}-t_{yx})$
and $\mathbf{t}_{\circlearrowright\circlearrowright}$=Incident RHCP to
recieved RHCP, $\mathbf{t}_{\circlearrowright\circlearrowleft}$=Incident LHCP
to recieved RHCP, $\mathbf{t}_{\circlearrowleft\circlearrowright}$=Incident
RHCP to recieved LHCP, and
$\mathbf{t}_{\circlearrowleft\circlearrowleft}$=Incident LHCP to recieved
LHCP.

## Acknowledgment

We thank Dr. Md. Tanvir Hasan and Ragib Shakil Rafi for helpful discussions
and guidance.

## References

  * [1] B. Lax, K. Button, K. Button, K. Button, and M. I. of Technology. Lincoln laboratory, _Microwave Ferrites and Ferrimagnetics_ , ser. Lincoln Laboratory publications. McGraw-Hill, 1962. [Online]. Available: https://books.google.com/books?id=LAhTAAAAMAAJ
  * [2] V. S. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” _arXiv: Applied Physics_ , 2020.
  * [3] C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” _Physical Review Applied_ , vol. 10, no. 4, Oct 2018. [Online]. Available: http://dx.doi.org/10.1103/PhysRevApplied.10.047001
  * [4] C. L. Hogan, “The ferromagnetic faraday effect at microwave frequencies and its applications: The microwave gyrator,” _The Bell System Technical Journal_ , vol. 31, no. 1, pp. 1–31, 1952.
  * [5] J. A. Stamper and B. H. Ripin, “Faraday-rotation measurements of megagauss magnetic fields in laser-produced plasmas,” _Phys. Rev. Lett._ , vol. 34, pp. 138–141, Jan 1975. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.34.138
  * [6] C. Caloz and T. Itoh, _Electromagnetic metamaterials: transmission line theory and microwave applications_. John Wiley & Sons, 2005.
  * [7] T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” _Applied Physics Letters_ , vol. 99, no. 3, p. 031114, 2011. [Online]. Available: https://doi.org/10.1063/1.3615688
  * [8] D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” _Applied Physics Letters_ , vol. 102, no. 19, p. 191901, 2013\. [Online]. Available: https://doi.org/10.1063/1.4804437
  * [9] G. Lavigne, T. Kodera, and C. Caloz, “‘perfect’ faraday-rotation metasurface,” in _2020 Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials)_ , 2020, pp. 156–158.
  * [10] T. Kodera, D. L. Sounas, and C. Caloz, “Switchable magnetless nonreciprocal metamaterial (mnm) and its application to a switchable faraday rotation metasurface,” _IEEE Antennas and Wireless Propagation Letters_ , vol. 11, pp. 1454–1457, 2012.
  * [11] A. Nagulu, N. Reiskarimian, and H. Krishnaswamy, “Non-reciprocal electronics based on temporal modulation,” _Nature Electronics_ , vol. 3, no. 5, pp. 241–250, May 2020. [Online]. Available: https://doi.org/10.1038/s41928-020-0400-5
  * [12] A. Kord, D. L. Sounas, and A. Alù, “Microwave nonreciprocity,” _Proceedings of the IEEE_ , vol. 108, no. 10, pp. 1728–1758, 2020.
  * [13] T. Kodera and C. Caloz, “Unidirectional loop metamaterials (ulm) as magnetless artificial ferrimagnetic materials: Principles and applications,” _IEEE Antennas and Wireless Propagation Letters_ , vol. 17, no. 11, pp. 1943–1947, 2018.
  * [14] S. J. Orfanidis, _Electromagnetic waves and antennas_. Rutgers University Press, 2016. [Online]. Available: https://www.ece.rutgers.edu/ orfanidi/ewa/
  * [15] S. Poddar, “Design and analysis of fully-electronic magnet-free non-reciprocal metamaterial,” _Theses and Dissertations. 2578_ , 2020, https://dc.uwm.edu/etd/2578.
  * [16] S. V. Kutsaev, A. E. Krasnok, S. N. Romanenko, A. Y. Smirnov, K. Taletski, and V. Yakovlev, “Up‐and‐coming advances in optical and microwave nonreciprocity: From classical to quantum realm,” 2020.
  * [17] C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “What is nonreciprocity? part ii,” 2018.
  * [18] C. Caloz and A. Sihvola, “Electromagnetic chirality,” 2019.
  * [19] D. Floess, J. Y. Chin, A. Kawatani, D. Dregely, H.-U. Habermeier, T. Weiss, and H. Giessen, “Tunable and switchable polarization rotation with non-reciprocal plasmonic thin films at designated wavelengths,” _Light: Science & Applications_, vol. 4, no. 5, pp. e284–e284, May 2015. [Online]. Available: https://doi.org/10.1038/lsa.2015.57
  * [20] I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, _Electromagnetic waves in chiral and bi-isotropic media_. Artech House, 1994\.
  * [21] S. Poddar, A. M. Holmes, and G. W. Hanson, “Design and analysis of an electronically tunable magnet-free non-reciprocal metamaterial,” _IEEE Transactions on Antennas and Propagation_ , pp. 1–1, in press, 2022.
  * [22] T. Kodera, D. L. Sounas, and C. Caloz, “Faraday rotation by artificial electric gyrotropy in a transparent slot-ring metamaterial structure,” in _Proceedings of the 2012 IEEE International Symposium on Antennas and Propagation_ , 2012, pp. 1–2.
  * [23] C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced jones calculus for the classification of periodic metamaterials,” _Phys. Rev. A_ , vol. 82, p. 053811, Nov 2010. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.82.053811


# Wang Algebra: From Theory to Practice

Bob Ross, Life Member, IEEE, and Cong Ling, Member, IEEE This work was
presented in part in Asian IBIS Summit 2007 and Virtual European IBIS Summit
2022. B. Ross is with Teraspeed Labs, 10238 SW Lancaster Road, 97219, USA,
(e-mail: bob@teraspeedlabs.com). C. Ling is with the Department of Electrical
and Electronic Engineering, Imperial College London, London SW7 2AZ, UK
(e-mail: cling@ieee.org).

###### Abstract

Wang algebra was initiated by Ki-Tung Wang as a short-cut method for the
analysis of electrical networks. It was later popularized by Duffin and has
since found numerous applications in electrical engineering and graph theory.
This is a semi-tutorial paper on Wang algebra, its history, and modern
applications. We expand Duffin’s historic notes on Wang algebra to give a full
account of Ki-Tung Wang’s life. A short proof of Wang algebra using group
theory is presented. We exemplify the usefulness of Wang algebra in the design
of T-coils. Bridged T-coils give a significant advantage in bandwidth, and
were widely adopted in Tektronix oscilloscopes, but design details were
guarded as a trade secret. The derivation presented in this paper, based on
Wang algebra, is more general and simpler than those reported in literature.
This novel derivation has not been shared with the public before.

###### Index Terms:

Bandwidth extension, bridged-T networks, matrix determinant, peaking, T-coil,
Wang algebra, wideband amplifier.

##  I Introduction

Wang algebra is a commutative algebra $\mathbb{W}$ with the properties

$x+x=0,\ \ x^{2}=0$ (1)

for all $x\in\mathbb{W}$. It was proposed by Ki-Tung Wang in 1934 as a
convenient rule to simplify the analysis of electrical networks [1]. In
essence, it is a clever method to compute the determinant of a symmetric
matrix in the context of solving a system of simultaneous linear equations. A
usual approach in the pre-computer era would be to use Cramer’s rule, but
calculating the determinants by hand is not only tedious, but also highly
likely to run into mistakes, due to the many terms involved. In Wang algebra,
a large number of terms vanish due to (1), thus greatly reducing the
calculation complexity [2, 3, 4, 5, 6]. Wang algebra has been a useful method
to design electronic circuits and modern integrated circuits (ICs), in
particular T-coils and interconnects [7, 8]. Yet, the impact of Wang algebra
goes much beyond electrical engineering. It has also found applications in
networking and graph theory [9]. Historically, Wang algebra was studied in
great detail by famous mathematicians Wei-Liang Chow [4] and Richard Duffin
[10, 11]111The works by Chow [4] and Duffin [10] themselves deserve special
remarks. Wei-Liang Chow, who had studied algebraic geometry under the great
algebraist van der Waerden, was running a business in Shanghai, China, to
support his family. Shanghai was occupied by Japanese troops during World War
II, and as a result he had lost his academic job in the National Central
University. Chow could hardly do any research during this war period
(1937-1945), and the study of Wang algebra [4] was an exception (and in fact
his only work on an engineering problem). Amazingly, after almost a decade in
business, he made a strong return to research and became a prominent algebraic
geometer in the US. Chow ring, a fundamental concept in algebraic geometry, is
named after him. Richard Duffin is well known for his contributions to
electrical networks and geometric programming. Together with Albert Schaeffer,
he introduced the concept of frames, which play an important role in signal
processing. The Duffin-Schaeffer conjecture, concerning rational approximation
of real numbers, is a famous conjecture in number theory that had stood open
for nearly 80 years. It was finally proved by Dimitris Koukoulopoulos and
James Maynard in 2019. Maynard won a Fields Medal partly because of this
proof..

Somewhat mysteriously, Duffin gave the following comment in [12]:

“K. T. Wang managed an electrical power plant in China, and in his spare time
sought simple rules for solving the network equations… Wang could not write in
English so his paper was actually written by his son, then a college student.”

However, the comment by Duffin seemed incomplete and did not convey Ki-Tung
Wang’s full story. The first aim of this article is to provide an expanded
biography of Ki-Tung Wang.

###  I-A Biography of Ki-Tung Wang

Figure 1: Ki-Tung Wang (1875-1948), Chinese mathematician, electrical engineer
and philosopher. Photo courtesy of Dr Jin-Hai Guo, Institute for the History
of Natural Sciences, Chinese Academy of Sciences.

Ki-Tung Wang (1875-1948), shown in Fig. 1, was a Chinese mathematician,
electrical engineer and philosopher. Believed to be the first Chinese scholar
to publish a mathematical paper in an international journal, he is well known
for his work on Wang algebra, as well as investigation on the relationship
between sciences and Buddhism [13].

Ki-Tung Wang was born into a prominent family in Suzhou, Jiangsu Province in
1875. His ancestor Ao Wang (1450-1524) was ranked no. 3 in the Imperial
Examination and later became a Grand Secretary of the Cabinet (equivalent to
Prime Minister) of the Ming Dynasty. His father Song-Wei Wang (1849-1895) was
also a Jinshi (Imperial Scholar), the highest degree of Imperial Examination
in China.

In 1895, Ki-Tung Wang graduated from the Imperial Tungwen College222It
literally means “Multilingual College”. (modern-day Peking University), then
was hired as a mathematical lecturer there. He had already published several
Chinese articles on mathematics, and investigated the relation between
classical Chinese mathematics and modern mathematics [14].

In 1909, Ki-Tung Wang served as an administrator of Chinese students in
Europe, then he did internships at the English Electrical Company and Siemens
[13]. During this period, he published a paper on the differentiation of
quaternionic functions333Quaternions are extension of complex numbers, with
lots of engineering applications such as astronautics, robotics, computer
visualization, animation, special effects in movies, navigation, etc. It is
worth mentioning that differentiation of quaternionic functions is highly
nontrivial—even more involved than the Cauchy-Riemann condition for complex
functions. in the Proceedings of the Royal Irish Academy [15], which is
believed to be the first paper published by Chinese mathematicians in
international journals [16, 13].

In 1912, Ki-Tung Wang was offered a job at the Ministry of Education, then in
as part of the newly founded Republic of China [13].

In 1914, Ki-Tung Wang went to industry and became an electrical engineer at
the Zhenjiang Power Plant, Jiangsu Province [13].

In 1928, Ki-Tung Wang was appointed Research Fellow444Equivalent to Principal
Scientist nowadays. at the National Research Institute of Engineering,
Academia Sinica [13]. He proposed a new method to derive the impedance of
electrical networks, which is sometimes advantageous to the traditional method
[1, 17]. In 1950, Duffin and Bott recognized that his rules form an algebra
and presented this method to the American Mathematical Society, under the
title “The Wang algebra of networks” [10]555Duffin himself cited in [10] H. W.
Becker’s 1948 unpublished notes on Wang algebra.. It is unclear how Duffin
learned the story that Wang could not write in English666One possibility is
that Duffin might have met Ki-Tung Wang’s daughter or son. Several children of
Ki-Tung Wang earned Ph.D. degrees in the US; in particular, his son Shou-Jin
Wang (1904-1984) obtained his Ph.D. in physics from Columbia University,
worked at Peking University and retired from MIT Lincoln Lab
(https://www.guokr.com/article/441034/). It is likely that Duffin had met
Shou-Jin Wang.. This was unlikely to be true, since Ki-Tung Wang had learned
English at the Imperial Tungwen College [18], worked in Britain, and published
an English paper [15] before. Nevertheless, it is possible that Ki-Tung Wang
needed some help on English writing, since it had been more than 20 years
since the publication of [15].

Ki-Tung Wang was very interested in philosophy beyond the limits of modern
sciences. He had several publications on the relationship between sciences and
Buddhism, including a book Comparative Study of Buddhism and Sciences [19].

###  I-B From Old Theory to Modern Applications

Nearly 90 years have passed since Wang algebra was proposed. It has found
numerous applications in areas ranging from graph theory to electrical and
electronic engineering. This article serves as a survey of the history, theory
and modern applications of Wang algebra.

As an example of applications, we will focus on T-coils, which have been used
for many years for wideband amplifier designs with distributed amplification,
peaking777In this paper, peaking refers to the technique to improve bandwidth
by using passive elements (such as a small inductor or a T-coil). See [20,
Section 12.3] for more details. networks, termination networks, and
transmission line equalization [21, 22, 20, 23, 24, 25, 26, 27, 28, 29, 30,
31, 7, 32, 33, 34]. In recent years, high-speed IC designs have added T-coils
because of three useful properties: 1) constant input resistance (constant-R),
2) usually over double the bandwidth extension, and 3) a second order transfer
function.

Tektronix pioneered the use of T-coils for bandwidth enhancement in 1960s.
Bridged T-coils give a significant advantage in bandwidth over conventional RC
design (more precisely $2\sqrt{2}$ times improvement). They were widely
adopted in Tektronix oscilloscopes, but design details were guarded as a trade
secret [35, 20, 36]. It was not until 1990 that former Tektronix engineer
Feucht provided the T-coil design equations [26], but no derivation was given.
Lee published a tedious derivation for the standard symmetric T-coil [20,
Chap. 12].

In this paper, a general method is presented to design constant impedance
bridged-T networks in terms of symbolic elements. General formulas for the
element relationships are derived using Wang Algebra. The constant impedance
constraint is used by adding an element to balance a bridge circuit. This
simplifies calculating the transfer function, and it is presented in a reduced
form in terms of a Thévenin equivalent circuit. When applied to a constant
input resistance (constant-R) network, such as a bridged T-coil that connects
to a capacitor load, extended designs with optional resistors in series with
inductors and resistors in parallel with capacitors exist for both symmetrical
and asymmetrical configurations. The process is to substitute known element
values in the symbolic formulas and to solve for the remaining elements. All
transfer functions are second order, and it is convenient to parameterize the
formula in terms of a bridging capacitor. Compared with an RC circuit, the
bandwidth is usually more than doubled with selected peaking. Previously
published and new configurations can be derived from these formulas.

###  I-C Organization

The rest of this paper is organized as follows. In Section II, we give an
introduction to Wang’s algebra, as well as a one-page proof based on group
theory. Section III is a survey of the applications of Wang’s algebra to
electrical engineering and graph theory. Section IV-A is devoted to the design
of T-coils using Wang algebra. Conclusions are given in Section V.

##  II Wang Algebra

In essence, Wang algebra gives a method to calculate the determinant of a
symmetric matrix, which can be more convenient sometimes. A proof was already
outlined in the original article [1]. Duffin’s proof [10] was based on
Grassmann algebra, which is 10-pages long. Chow’s proof [4] based on matrix
theory is also quite tricky. In the following, we will present a short proof
using group theory, for completeness. Familiarity with group theory is
assumed, in particular symmetric group $S_{n}$ of order $n$ [37]. Readers
uninterested in the proof may simply skip it.

######  Theorem 1 (Wang Algebra).

Let $\mathbf{A}=[a_{ij}]_{n\times n}$ be a symmetric matrix, i.e.,
$a_{ij}=a_{ji}$, where $1\leq i,j\leq n$. Write the diagonal elements of
$\mathbf{A}$ as

$a_{ii}=a^{\prime}_{ii}-\sum_{j\neq i}a_{ij}.$

Then the determinant $\det(\mathbf{A})$ can be computed as

$\det(\mathbf{A})=\prod_{i=1}^{n}\left(a^{\prime}_{ii}-\sum_{j\neq
i}a_{ij}\right)$ (2)

in Wang algebra $\mathbb{W}$.

It should be pointed out that the product (2) must be computed symbolically,
for it is certainly not true that $\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}$
numerically. In other words, we take $a^{\prime}_{ii},a_{ij}\ (j\neq i)$ as
symbols $\in\mathbb{W}$ in the computation of (2). The key point here is that
many terms disappear thanks to the rule of Wang algebra (1). Once the product
(2) has been computed, numerical values $a_{ij}\in\mathbb{C}$ can be
substituted in to find the determinant. Readers are referred to Section III
for examples.

###### Proof.

We will prove Theorem 1 in two steps. We use a standard formula of the matrix
determinant:

$\det(\mathbf{A})=\sum_{\sigma\in
S_{n}}\mathrm{sgn}(\sigma)\prod_{i=1}^{n}a_{i\sigma(i)}$ (3)

where $\sigma$ is a permutation in the symmetric group $S_{n}$, and
$\mathrm{sgn}(\sigma)=\pm 1$ denotes the signature of $\sigma$.

######  Lemma 1.

Let $\mathbf{A}=[a_{ij}]_{n\times n}$ be a symmetric matrix. Then, except the
term corresponding to the main diagonal, all terms of Eq. (3) contain either a
square or a factor $2$.

###### Proof.

Firstly, the factor $2$ is due to the symmetry of the matrix: associated with
any term $\prod_{i=1}^{n}a_{i\sigma(i)}$ in Eq. (3), there is another one

$\prod_{i=1}^{n}a_{\sigma(i)i}=\prod_{j=1}^{n}a_{j\sigma^{-1}(j)}.$ (4)

Since $a_{i\sigma(i)}=a_{\sigma(i)i}$, the two terms are equal. Moreover,
$\mathrm{sgn}(\sigma)=\mathrm{sgn}(\sigma^{-1})$. Thus we obtain the factor
$2$ if the two terms are distinct.

The argument above fails if and only if the two terms are in fact the same
(e.g., the term $a_{11}a_{22}\cdots a_{nn}$ corresponding to the main
diagonal). This happens if and only if

$\sigma=\sigma^{-1},$ (5)

i.e., $\sigma^{2}=1$. This implies $\sigma$ has order $1$ or $2$.

The case of order $1$ corresponds to the main diagonal.

The case of order $2$ consists of one or more disjoint cycles of length $2$,
i.e., transpositions, which looks like $\sigma=(i,j)(k,l)$ etc. In other
words, such a product contains square(s) $a_{ij}a_{ji}=a_{ij}^{2}$,
$a_{kl}a_{lk}=a_{kl}^{2}$ etc. ∎

######  Lemma 2.

Let the diagonal elements of $\mathbf{A}$ be given by

$a_{ii}=a^{\prime}_{ii}-\sum_{j\neq i}a_{ij}.$ (6)

Then any terms of (3) involving an off-diagonal element are cancelled out.

######  Remark 1.

Lemma 2 is slightly more general than Wang algebra, since it does not assume
that $\mathbf{A}$ is symmetric.

###### Proof.

Without loss of generality, consider an off-diagonal element $a_{ij}$ where
$i\neq j$. The sum of all terms involving $a_{ij}$ is given by

$\sum_{\sigma:\sigma(i)=j}\mathrm{sgn}(\sigma)a_{ij}\prod_{k=1,k\neq
i}^{n}a_{k\sigma(k)}.$ (7)

We claim that all these terms will be cancelled out by those in the following
sum

$\sum_{\sigma^{\prime}:\sigma^{\prime}(i)=i}\mathrm{sgn}(\sigma^{\prime})a_{ii}\prod_{k=1,k\neq
i}^{n}a_{k\sigma^{\prime}(k)}.$ (8)

Since $\sigma(i)=j$, indexes $i$, $j$ must belong to a certain cycle
$(i_{1},i_{2},\ldots,i_{l})$ of length $l>1$. Let us extract from (7) those
containing this cycle:

$\displaystyle\sum_{\sigma:(i_{1},i_{2},\ldots,i_{l})\in\sigma}\mathrm{sgn}(\sigma)\prod_{k\in\\{i_{1},i_{2},\ldots,i_{l}\\}}a_{k\sigma(k)}\prod_{k\notin\\{i_{1},i_{2},\ldots,i_{l}\\}}a_{k\sigma(k)}.$
(9)

There exist corresponding terms in (8) with $\sigma^{\prime}$ fixing indexes
$i_{1},i_{2},\ldots,i_{l}$:

$\displaystyle\sum_{\sigma^{\prime}:\sigma^{\prime}(i_{1})=i_{1},\cdots,\sigma^{\prime}(i_{l})=i_{l}}\mathrm{sgn}(\sigma^{\prime})\prod_{k\in\\{i_{1},i_{2},\ldots,i_{l}\\}}a_{kk}\prod_{k\notin\\{i_{1},i_{2},\ldots,i_{l}\\}}a_{k\sigma^{\prime}(k)}.$
(10)

Clearly, $\sigma$ is a composition of $\sigma^{\prime}$ and
$(i_{1},i_{2},\ldots,i_{l})$.

Note that a negative counterpart of $a_{k\sigma(k)}$ exists in
$a_{kk}=a^{\prime}_{kk}-\sum_{m\neq k}a_{km}$, so the sign of those terms in
(10) is

$\mathrm{sgn}(\sigma^{\prime})(-1)^{l}.$

But the sign of those terms in is (9)

$\mathrm{sgn}(\sigma)=\mathrm{sgn}(\sigma^{\prime})(-1)^{l-1},$

they must have different signs, therefore being cancelled out completely. ∎

The proof of Theorem 1 is completed by combining the two lemmas. ∎

######  Remark 2.

In [1], Ki-Tung Wang treated terms containing a square and terms containing
factor 2 separately, which correspond to 2-cycles $(i_{1},i_{2})$ and longer
cycles $(i_{1},i_{2},\ldots,i_{l})$ in the above proof, respectively.

An astute reader may wonder if there is any gain to use Wang algebra to
compute the matrix determinant. After all, applying the formula (3) crudely
requires $n\cdot n!$ multiplications, while computing (2) may require $n^{n}$
multiplications in the worst case. By Stirling’s formula $n!\approx(n/e)^{n}$,
the latter is asymptotically much greater. Nevertheless, there are some
advantages of Wang algebra:

  * •

The number of terms is greatly reduced by applying Wang’s rule (1), thus it
can be more convenient for small dimensions $n$;

  * •

Even if there are faster numerical algorithms to compute matrix
determinant888For example, the elimination method (aka Gauss elimination) can
be used to compute matrix determinant with $O(n^{3})$ complexity. The
elimination method was invented by ancient Chinese mathematicians to solve
systems of linear equations in The Nine Chapters on the Mathematical Art (see
[38] for an English translation), predating the work of Gauss by about 2000
years. The Nine Chapters on the Mathematical Art is a textbook taught in the
National University during the Han Dynasty of China (202 BCE – 220 CE). Ki-
Tung Wang was aware of the elimination method, since classical Chinese
mathematics was taught at the Imperial Tungwen College [18, 14]., the symbolic
formulas given by Wang algebra can lead to considerable insights.

We will demonstrate these advantages in the following sections.

##  III Applications

###  III-A Applications to Electrical Networks/Circuits

Figure 2: Bridged-T network used to derive the constant-Z constraint by loop
equations and Wang algebra.

For simplicity, let us assume planar networks, i.e., networks which can be
drawn on a plane such that no branches cross each other. An example of planar
electrical network is shown in Fig. 2, with impedances $a,b,c,d,e$. In fact,
this is a model of the bridged T-coil which will be analyzed in detail in the
following section. The problem is to determine the joint impedance $Z$.
Applying Kirchhoff’s voltage law to three loops shown, we obtain the following
system of linear equations:

$\begin{pmatrix}a+d&-a&-d\\\ -a&a+b+c&-b\\\
-d&-b&b+d+e\end{pmatrix}\begin{pmatrix}I\\\ I_{2}\\\
I_{3}\end{pmatrix}=\begin{pmatrix}V\\\ 0\\\ 0\end{pmatrix}.$ (11)

Note that the above matrix is always symmetric and exhibits the structure
given in (6), by selecting the orientations of the loops properly. Denote by
$\mathbf{M}_{1}$ the $3\times 3$ matrix on the left-hand side of (11), and by
$\mathbf{M}$ the $2\times 2$ submatrix on its bottom right. Cramer’s rule is a
standard approach to computing the joint impedance $Z$:

$\frac{1}{Z}=\frac{I}{V}=\frac{\det(\mathbf{M})}{\det(\mathbf{M}_{1})}.$ (12)

To proceed, we firstly compute the numerator:

$\displaystyle\det{(\mathbf{M})}$
$\displaystyle=\det{\begin{pmatrix}a+b+c&-b\\\ -b&b+d+e\end{pmatrix}}$
$\displaystyle=ab+ad+ae+b^{2}+bc+bd+be+cd+ce$
$\displaystyle=ab+ad+ae+bc+bd+be+cd+ce$ (13)

where $b^{2}=0$ in Wang algebra $\mathbb{W}$. Then we multiply it with $a+d$
to compute the denominator:

$\displaystyle\det{(\mathbf{M}_{1})}$
$\displaystyle=(a+d)(ab+ad+ae+bc+bd+be+cd+ce)$
$\displaystyle=2abd+abc+abe+acd+ace+ade+bcd+bde+cde$
$\displaystyle=abc+abe+acd+ace+ade+bcd+bde+cde.$

where $2abd=0$ in Wang algebra $\mathbb{W}$. Many product terms are 0 and do
not have to be listed. Note the saving it brings in this case: 18 initial
terms in the denominator are reduced to 8 in the end.

For the reason to be clear in the next subsection, the determinant
$\det{(\mathbf{M})}$ is called the mesh determinant of the electrical network.

We can see that another statement of Wang algebra for computing network
determinants is:

######  Corollary 1 (Wang’s Rule).

The determinant of a planar network does not contain any terms containing a
square or a factor $2$. Moreover, all its terms have coefficient $+1$.

Ki-Tung Wang initially considered planar networks [1]. In follow-up works by
Chinese researchers [2, 3, 4], the restriction of planar networks was removed.
Wang algebra $\mathbb{W}$ still holds for non-planar networks, but the rule
stated above needs to be modified slightly. More precisely, a term coefficient
may be an odd number; therefore, the coefficients are calculated modulo $2$ as
in Wang algebra $\mathbb{W}$.

Duffin proved that the so-called just discriminants, whose coefficients are
all equal to $+1$ and which do not necessarily come from Kirchhoffian
networks, can also be evaluated in Wang algebra [10]. For example, he showed
that Wang algebra holds for a network lying on the surface of a torus. But
Grassmann algebra needs to be used for non-just discriminants, in a similar
way to Wang algebra for just discriminants.

###  III-B Applications to Graph Theory

Of course, electrical networks are examples of graphs. Since matrix theory and
graph theory are closely related, it is not surprising that Wang algebra is
also useful in graph theory. Specifically, it gives an algebraic method to
enumerate the trees and cotrees of a graph. Readers are referred to [9] for an
introduction to graph theory and its engineering applications.

A graph consists of a set of nodes together with a set of edges. The
electrical network shown in Fig. 2 is an example of graph. It contains 4
nodes, as well as 5 edges labelled by $a$, $b$, $c$, $d$ and $e$.

A (spanning) tree of a graph is a set of edges which connect all nodes and
which do not contain any loops. For example, edges $\\{a,b,e\\}$ form a tree
in Fig. 2. The complement of a tree in a graph in called a cotree; in other
words, it is a set of edges when removed leave no loops. For example,
$\\{c,d\\}$ form a cotree, since it is the complement of $\\{a,b,e\\}$. Wang
algebra gives a handy method to enumerate trees/cotrees.

####  III-B1 Enumerating Trees/Cotrees

For convenience, let the admittances be $A=1/a$, $B=1/b$, $C=1/c$, $D=1/d$,
$E=1/e$ in Fig. 2. By Kirchhoff’s current law, we have another set of
equations for Fig. 2:

$\begin{pmatrix}A+C&-A&-C\\\ -A&A+B+D&-B\\\
-C&-B&B+C+E\end{pmatrix}\begin{pmatrix}V\\\ V_{2}\\\
V_{3}\end{pmatrix}=\begin{pmatrix}I\\\ 0\\\ 0\end{pmatrix}.$ (14)

The above matrix also exhibits the structure given in (6). Denote by
$\mathbf{S}$ the $3\times 3$ matrix on the left-hand side of (14), and by
$\mathbf{S}_{1}$ the $2\times 2$ submatrix on its bottom right. The
determinant $\det{(\mathbf{S})}$ is called the node determinant of the
electrical network.

Applying Cramer’s rule yields the joint impedance

${Z}=\frac{V}{I}=\frac{\det(\mathbf{S}_{1})}{\det(\mathbf{S})}.$ (15)

Again, calculation is greatly simplified by using Wang algebra. The numerator
can be calculated as

$\displaystyle\det{(\mathbf{S}_{1})}$ $\displaystyle=(A+B+D)(B+C+E)$
$\displaystyle=AB+AC+AE+BC+BE+BD+CD+DE.$

Multiplying it with $A+C$ yields the denominator

$\displaystyle\det{(\mathbf{S})}$ $\displaystyle=(A+C)\det{(\mathbf{S}_{1})}$
$\displaystyle=ABE+ABD+ACD+ADE$ $\displaystyle+ACE+BCE+BCD+CDE.$

The saving is the same as before: 18 initial terms in the denominator are
reduced to 8 in the end.

A tree, respectively cotree, product is the product of the edges of the tree,
respectively cotree. It is known that the terms of node determinant are tree
products [9, Theorem 2.28], while the terms of mesh determinant are cotree
products [9, Theorem 2.29]. In fact, the 8 terms of node determinant
$\det{(\mathbf{S})}$ are precisely the trees, while those of the mesh
determinant (13) are the cotrees of the graph shown in Fig. 2.

The following relation holds between the mesh determinant and node
determinant:

$\det(\mathbf{M}(a,b,c,d,e))=abcde\cdot\det(\mathbf{S}(A,B,C,D,E)).$ (16)

This agrees with the one-to-one correspondence between trees and cotrees.

####  III-B2 Counting Trees/Cotrees

Obviously, setting $a=b=c=d=e=1$, we obtain the number of trees/cotrees of a
graph:

$\det(\mathbf{M}(1,1,1,1,1))=\det(\mathbf{S}(1,1,1,1,1)).$ (17)

For example,

$\mathbf{S}(1,1,1,1,1)=\begin{pmatrix}2&-1&-1\\\ -1&3&-1\\\
-1&-1&3\end{pmatrix}.$ (18)

It is easy to check that $\det(\mathbf{S}(1,1,1,1,1))=8$, implying there are 8
trees in the graph, as expected. In fact, $\det(\mathbf{S}(1,1,1,1,1))$ is a
cofactor of the Laplacian matrix of the graph, and this is a well-known method
in graph theory to calculate the number of trees. This fact can be used to
check if we have obtained the correct number of terms in network determinants.

##  IV Bridged T-coils

In this section, we demonstrate how Wang algebra can be used to greatly
simplify the design of bridged T-coils. T-coils are an old technology, but
featured with modern applications. Constant-R T-coils provide ideal load or
termination. They offer $2.73$ improvement for acceptable $0.4\%$ overshoot to
ideal step input. Now T-coils are used in high-speed buffer design and
electrostatic discharge (ESD) compensation with bandwidth improvement. A
review of historical applications at Tektronix is given in Section IV-A.

For peaking applications, T-coils are designed to drive capacitive loads that
model input transistor structures (or in a reciprocal manner, current source
transistor drivers with output capacitance). T-coil extensions for loads with
series or parallel resistor elements have been published [25, 26, 27, 29, 30,
31, 7]. Also, more detailed models of the physical structure reveal resistive
losses in the branches [32, 33, 34]. With the formulas in this section,
extensions for losses or resistive elements can be included in the design
process. Extended T-coil configurations are shown in Section IV-B.

Several methods have been reported to derive the transfer function to a
capacitor load [21, 20, 23, 24, 25], and this remains an algebraically tedious
process, especially for the extended configurations. Most of these methods
first derive fourth order equations. These are reduced to second order
equations after identifying common factors. The calculations (not shown) are
particularly tedious for the general configurations reported later [7].

For generality, a symbolic method using Wang algebra for simplicity is used in
Section IV-C to derive the general constant input impedance (constant-Z)
constraint relationship. A symbolic approach (based on an augmented Thévenin
equivalent circuit) is used along with the constant-Z constraint to produce a
transfer function that is already of reduced order (or easy to reduce).

Section IV-D gives two sets of T-coil formulas for the most general
symmetrical and asymmetrical cases where the termination is $R$. Simpler
T-coils can be derived by removing some resistor terms. A design process is
suggested based on selecting complex-pole angles.

###  IV-A T-coil History at Tektronix and Afterwards

The application of T-coils in products is documented in personal narratives by
C. R. Battjes [28] and J. L. Addis [27]. The first author adds T-coil
derivations done while at Tektronix and after leaving Tektronix. Around 1948,
William Hewlett had lunch in Oregon with Tektronix president Howard Vollum and
a key engineer Logan Belleville. Hewlett penciled out the distributed
amplifier circuit [22] on a paper napkin. Tektronix adopted distributed
amplifiers or distributed deflection circuits in cathode ray tubes in
oscilloscopes. The input for the Tektronix 519 oscilloscope (1 GHz) drove the
distributed vertical deflection plates directly. Other products included the
517 (50 MHz with distributed amplifier), 545 main frame with plugins (30 MHz),
545A (30 MHz), and 585A (100 MHz with distributed deflection plates). T-coils
for inter-stage peaking were used in Type K and L plugins and in the 3A6
plugin that connected directly to a cathode ray tube deflection structure.
This is just a short list of some early implementations. More information is
available by searching the product names. Note, some schematics assume, but do
not show the bridging capacitor $C_{B}$ (see Fig. 3).

The portable oscilloscopes, Tektronix 454 and 454A (150 MHz) used distributed
deflection elements and an output amplifier with T-coil peaking. It also used
T-coils for input delay line differential phase compensation. T-coils on
individual circuit boards were fabricated for transistor interstate peaking.
The transistor input was still approximated as capacitor. A leading edge (at
that time) oscilloscope the Tektronix 7904 (500 MHz) had minimum VSWR T-coil
that compensated for some additional parasitic degradations. The follow-on
7104 (1 GHz) oscilloscope used faster transistors, thin film conductors on
substrates, and a transmission line package design.

The so-called “Ross T-coil” is noted in [27, 28] and was implemented by thin
film deposition to peak and terminate the inter-chip transmission lines. Other
T-coils boosted the frequency response and minimized reflections in a
$50\Omega$ input for the 11A72 dual channel plugin preamplifier hybrid.

C. R. Battjes [28] (a Stanford University graduate) joined Tektronix to design
bipolar transistor amplifiers. He created and conducted internally the
amplifier frequency and time response (AFTR) class for new engineers. The
T-coil derivations were those of [22] or variations thereof. Rather than using
a full bipolar transistor hybrid-pi load input model used in Fig. 4(d) and
Figs. 5(c,d); Battjes favored the simpler $R_{S}-C$ load model in Fig. 4(b)
and Fig. 5(a). It offered better time-constant design intuition and the
$R_{P}$ contribution was considered large and negligible. Besides, the full
design equation mathematics shown in this paper were not known at that time.

Sitting next to Battjes, the first author became interested in working out the
mathematics for the $R_{S}-C$ model. He applied Wang algebra, but never
revealed the derivation details outside of Tektronix. Various forms of the
result were later published as the “Ross T-coil” [27, 28].

The Wang algebra approach differed from earlier known derivations. The coupled
inductor was split into three inductors, as shown in Fig. 3, to avoid any pre-
defined coupling assumptions. A constant resistance constraint was applied
initially to calculate some fixed element values (shown on and above the Gain
line in Table I). The equations were helpful in making later simplifications.
The parameter of interest was selected as $C_{B}$ because it could be varied
from 0 to nearly infinity. $C_{B}$ was easily related to the inductance $M$,
the coupling coefficient $k$, and a second order equation damping variable
$\delta$. This derivation did not require taking an absolute value of $k$, as
was done in some other derivations. Nor was it limited to symmetrical T-coils.
(For example, the T. T. True patent [31] for asymmetrical T-coils was known
internally, but never used.)

After leaving Tektronix, the first author investigated deriving the equations
for the full bipolar transistor hybrid-pi input model in Fig. 4(d) and Figs.
5(c,d), as discussed in this paper. As technology evolved, digital displays
replaced distributed deflection structures. Also, hybrid integrated circuits
and integrated circuits were replacing discrete transistor circuits. But as
the references show, T-coils are still being implemented in integrated
circuits and for ESD protection.

###  IV-B Constant-R Bridged T-coils

Figure 3: Standard constant-R bridged T-coil driving $C$, terminated by $R$,
and with a bridging capacitor $C_{B}$, (a) Coupled transformer model. (b)
Equivalent model with three inductors.

The standard T-coil is shown in Fig. 3(a). The structure contains a bridging
capacitor $C_{B}$ and is terminated by a resistor $R$. Of interest is the
transfer function to a load capacitor $C$. The three-terminal coupled
transformer in Fig. 1(a) can be modeled as shown in Fig. 3(b) by inductors
$L_{1}$, $L_{2}$, and $L_{3}$ with well-documented conversions between the two
forms. This allows for a wider range of solutions including cases (for some
extended configuration values) that make $L_{1}$ negative. In the standard
T-coil, $L_{1}=L_{2}$.

Figure 4: Symmetrical T-coils where $L_{1}=L_{2}$ and $R_{1}=R_{2}$ and with
calculated $R_{B}$. (a) Standard configuration (and no $R_{B}$, $R_{1}$ and
$R_{2}$). (b) With $R_{S}$. (c) With $R_{P}$. (d) With $R_{S}$ and $R_{P}$.

Fig. 4 shows symmetrical constant-R configurations with Fig. 4(a), the
standard configuration. Figs. 4(b)-4(d) adds additional resistor combinations.
These extensions add a bridging resistor $R_{B}$ to allow symmetry with
$L_{1}=L_{2}$ and $R_{1}=R_{2}$. Symmetrical structures might be easier to
fabricate.

Figure 5: Asymmetrical T-coils with added $R_{2}$ where needed. (a) With
$R_{S}$. (b) With $R_{P}$. (c) With $R_{S}$ and $R_{P}$. (d) Generalized with
$R_{S}$, $R_{P}$, and selectable $R_{1}$ and calculated $R_{2}$ to balance the
losses in the $L_{1}$ and $L_{2}$ branches.

Similar to Fig. 4, Fig. 5 shows asymmetrical constant-R T-coils
configurations. Fig. 5(d) provides a general configuration with a selectable
$R_{1}$ over a limited range as long as $R_{2}\geq 0$. For example, resistors
can be entered proportionally in both inductor branches for physical losses.

The first author derived the formulas in the late 1960s for the series
configuration in Fig. 5(a) to include bipolar transistor base resistance.
Equivalent formulas have been published in later works [26, 27, 28, 29, 30].
The constant-R component values for Fig. 5(b) were revealed in [31], without
the associated transfer function. Later, the first author published equations
for generalized configurations including both Fig. 4(d) and Fig. 5(d) [7]. The
asymmetrical configuration did not need a bridging resistor.

###  IV-C Constant-Z Bridged-T Networks

####  IV-C1 Driving Point Impedance

A general bridged-T network driving point impedance in Fig. 2 can be derived
in symbolic form from loop or nodal equations. Wang algebra simplifies the
process with the aforementioned rules. The resulting driving point admittance
is set equal to the terminating admittance ${1}/{e}$:

$\begin{split}&\frac{1}{Z}=\frac{1}{e}=\frac{N}{D}=\frac{(a+b+c)(b+d+e)}{(a+d)N}\\\
&=\frac{ab+ad+ae+bc+bd+be+cd+ce}{abc+abe+acd+ace+ade+bcd+bde+cde}.\end{split}$
(19)

In (19) the driving point admittance numerator $N$ is formed by adding the
impedances of a set of independent loops except the input loop, and then by
multiplying these sums with Wang algebra rules. The denominator $D$ is formed
by multiplying $N$ and the input loop sums and applying Wang algebra rules.

With regular algebra, (19) is simplified to produce symbolic, constant-Z
constraints for asymmetrical (20) and symmetrical (21) networks as arranged as

$\displaystyle\left(a+b\right)\left(d-e^{2}/c\right)+ab+\left(a-b\right)e-e^{2}=0,$
(20) $\displaystyle\ \ \ a=b,\ \ \ \ 2a\left(d-e^{2}/c\right)+a^{2}-e^{2}=0.$
(21)

The component relationships are formed by equating the factors of each power
of the Laplace variable $s$ to zero after inserting the actual impedances into
(20) or (21). This process yields a set of independent constraints. In some
cases the relationships show that the circuit is not suitable due to a
mathematical contradiction or to non-realizable (negative) components.

Figure 6: Transfer function to $V_{d}$. (a) Constant-$Z$ bridged-T network and
$V_{d}$. (b) Equivalent $V_{d}$ with added $f$ and $V_{f}$.
Figure 7: Thévenin equivalent circuit development to $V_{d}$. (a) Impedance
calculation. (b) Voltage calculation with scaled-mirror symmetry about
$V_{Th}$. (c) Thévenin equivalent circuit.

####  IV-C2 Transfer Function

A method is presented for calculating the transfer function to node $V_{d}$ in
Fig. 6(a), if the constant-Z constraint is satisfied. The configuration in
Fig. 6(b) produces an equivalent result with impedance $f$ and
$V_{f}=V(e+f)/e$. A Thévenin equivalent circuit can be derived at node $V_{d}$
as shown in Fig. 6. Without any loss in generality, the bridge in Fig. 7(a)
can be balanced by choosing $f=e/(ab)$ to null out the $c$ term. The Thévenin
impedance, $Z_{Th}$, becomes $(b+e)\|(a+f)$. As shown in Fig. 7(b) this also
produces a scaled-mirror symmetrical divider about $V_{Th}$ (which can be
shown by splitting $c$ into two impedances, $ca/(a+b)$ and $cb/(a+b)$, and
joining them at the $V_{Th}$ node. The values shown in Fig. 7(c) are:

$V_{Th}=V,\ \ Z_{Th}=b^{\prime}+e^{\prime}=\left(b+e\right)\frac{a}{a+b}.$
(22)

This equivalent circuit applies for any constant-Z bridged-T network. For
constant-R T-coils, the transfer function of interest is to the capacitor $C$
(usually part of a transistor model) within branch $d$. For symmetrical
networks the Thévenin impedance reduces to one-half of the impedances of the
$b+e$ branches. For asymmetrical networks, the Thévenin impedance also
produces reduced order transfer functions where applicable (sometimes after
applying some constant-Z identities). Fig. 7(a) suggests that poles associated
with branch $c$ do not appear in the transfer function. The conductance and
capacitance elements within branches $a+b$ in (22) can be cancelled, as shown
later.

###  IV-D General T-coil Equations

####  IV-D1 Root Locus

The Thévenin equivalent circuit approach produces second order transfer
functions for all T-coils in Fig. 4 and Fig. 5 and can be parameterized in
terms of the bridging capacitor $C_{B}$ for the complete range of solutions.
Also, $G_{P}=1/R_{P}$.

$\displaystyle\frac{V_{C}}{V}$
$\displaystyle=\frac{{1}/{\left(G_{P}+sC\right)}}{Z_{Th}+R_{S}+sL_{3}+\frac{1}{\left(G_{P}+sC\right)}}$
(23a) $\displaystyle\frac{V_{C}}{V}$
$\displaystyle=\frac{1}{B_{0}+B_{1}s+B_{2}s^{2}}=\frac{1}{B_{0}+\left(D_{0}+D_{1}C_{B}\right)s+D_{2}C_{B}s^{2}}.$
(23b)

Figure 8: Root locus on complex $s=\sigma+j\omega$ plane showing increasing
$C_{B}$. (a) Without $R_{P}$ ($G_{P}=0$). (b) With $R_{P}$.

In Fig. 8, the complex poles of interest are shown for increasing $C_{B}$ and
lie on a circle with

$\mathrm{center}\ =-\frac{B_{0}}{D_{0}},\ \ \ \mathrm{radius}\
=\frac{B_{0}}{D_{0}}\sqrt{1-\frac{D_{0}D_{1}}{B_{0}D_{2}}}.$ (24)

Fig. 7(b) shows the case where the circle defined by (24) is to the left of
the origin because $D_{1}>0$ when $R_{P}$ exists.

A design process is to solve for the poles of (23) and then find $C_{B}$ for a
specified pole-angle measured from the negative real axis. Both poles are
found by solving quadratic equations. Elements and terms for symmetrical and
asymmetrical transfer functions in (23) are given next. The $L_{3}$ value
varies with $C_{B}$ (and the pole-angle) as shown in (28) and (37) below. A
final step is to calculate physical inductor values and a coupling
coefficient.

####  IV-D2 Symmetrical T-coils

The T-coil equations for the general case in Fig. 4(d) are expressed using
$G_{P}=1/R_{P}$ and $G_{B}=1/R_{B}$ to support omitted resistor
simplifications with zero-valued $G_{P}$ and/or $G_{B}$ elements.

##### Constant-R Elements

$\displaystyle R_{1}$ $\displaystyle=R_{2}=R^{2}G_{p}/2$ (25) $\displaystyle
G_{B}$ $\displaystyle=R_{S}/R^{2}+G_{P}/4$ (26) $\displaystyle L_{1}$
$\displaystyle=L_{2}=R^{2}C/2$ (27) $\displaystyle L_{3}$
$\displaystyle=R^{2}C_{B}-L_{1}/2.$ (28)

##### Transfer Function Terms

$\displaystyle B_{0}$ $\displaystyle=1+G_{P}(R/2+R^{2}G_{P}/4+R_{S})$ (29)
$\displaystyle D_{0}$ $\displaystyle=\left(2R+R^{2}G_{P}+{4R}_{S}\right)C/4$
(30) $\displaystyle D_{1}C_{B}$ $\displaystyle=R^{2}G_{p}C_{B}$ (31)
$\displaystyle D_{2}C_{B}$ $\displaystyle=R^{2}CC_{B}.$ (32)

####  IV-D3 Asymmetrical T-coils

The T-coil equations for the general case in Fig. 5(d) are presented.

##### Constant-R Elements

$\displaystyle R_{2}$
$\displaystyle=\frac{R^{2}G_{P}-R_{1}\left[1+\left(R_{S}+R\right)G_{P}\right]}{1+(R_{S}+{R_{1}-R)G}_{P}}$
(33a) $\displaystyle R_{1}$
$\displaystyle=\frac{R^{2}G_{P}-R_{2}\left[1+\left(R_{S}-R\right)G_{P}\right]}{1+(R_{S}+{R_{2}+R)G}_{P}}$
(33b) $\displaystyle R_{1}+R_{2}$
$\displaystyle=\frac{\left(R-R_{1}\right)^{2}G_{P}}{1+\left(R_{S}+R_{1}-R\right)G_{P}}$
(33c) $\displaystyle R_{1}+R_{2}$
$\displaystyle=\frac{{(R+R_{2})}^{2}G_{P}}{1+(R_{S}+R_{2}+R)G_{P}}$ (33d)

$\displaystyle L_{T}G_{P}$
$\displaystyle=\left(L_{1}+L_{2}\right)G_{P}=\left(R_{1}+R_{2}\right)C$ (34a)
$\displaystyle L_{T}$
$\displaystyle=\frac{\left(R-R_{1}\right)^{2}C}{1+\left(R_{S}+R_{1}-R\right)G_{P}}$
(34b) $\displaystyle L_{T}$
$\displaystyle=\frac{{(R+R_{2})}^{2}C}{1+(R_{S}+R_{2}+R)G_{P}}$ (34c)

$\displaystyle L_{1}$
$\displaystyle=\frac{(R_{1}-R)(R_{1}+R_{S}-R)C}{1+\left(R_{1}+R_{S}-R\right)G_{P}+\sqrt{1+\left(R_{1}+R_{S}-R\right)G_{P}}}$
(35) $\displaystyle L_{2}$
$\displaystyle=\frac{(R_{2}+R)(R_{2}+R_{S}+R)C}{1+\left(R_{2}+R_{S}+R\right)G_{P}+\sqrt{1+\left(R_{2}+R_{S}+R\right)G_{P}}}$
(36) $\displaystyle L_{3}$
$\displaystyle={R^{2}C_{B}-L}_{1}L_{2}/(L_{1}+L_{2}).$ (37)

##### Transfer Function Terms

$\displaystyle B_{0}$
$\displaystyle=1+R_{S}G_{P}+R_{1}[1+R+R_{S}+R_{2}G_{P}]R+R_{2}$ (38)
$\displaystyle D_{0}$ $\displaystyle=L_{2}R[1+(R_{1}+R_{S}-R)G_{P}](R-R_{1})2$
(39) $\displaystyle D_{1}C_{B}$ $\displaystyle=R^{2}G_{p}C_{B}$ (40)
$\displaystyle D_{2}C_{B}$ $\displaystyle=R^{2}CC_{B}.$ (41)

The resistance constraint is implemented in (33a) after selecting $R_{1}$;
(33b) is an equivalent arrangement. Summations (33c) and (33d) are derived
from (33a) and (33b) respectively. The summations (33c) and (33d) are used in
(34a) to derive (34b) and (34c) such that zero-valued elements (besides $R$)
are allowed. These forms are useful to derive (35) and (36). Also, from (33a),
$R_{1}$ is limited to a range where $R_{2}\geq 0$. If $G_{P}=0$,
$R_{1}=R_{2}=0$.

From (22), (23a), and (34a), the transfer function $V_{C}/V$ in (23) remains
second order, as shown in (23b). The $sC+G_{P}$ term in the product
$(sC+G_{P})Z_{Th}$ drops out because in $Z_{Th}$,
$a+b=s(L_{1}+L_{2})+(R_{1}+R_{2})=(R_{1}+R_{2})(sC+G_{P}))/G_{P}$. If
$G_{P}=0$, then from (33c) or (33d), $(R_{1}+R_{2})/G_{P}=R^{2}$.

###  IV-E Example

Table I shows a set of load values and the designs for all T-coils. Elements
at and above the Gain line remain unchanged for each Fig. 4 or Fig. 5
configuration. Blank entries represent omitted elements for simpler
structures. Element values (negative values in parenthesis) for stated pole
angles are shown below single solid lines along with corresponding parameters
include bandwidth (BW) and bandwidth extension ratio (BWER) relative to a
$1+RCs$ single-pole bandwidth.

TABLE I: T-coil example for all configurations. $R=50\ \Omega$, $C=4$ pF,
$R_{S}=10\ \Omega$, $P_{P}=500\ \Omega$, Pole Angle $=30^{\circ}\ \&\
45^{\circ}$, $R_{1}$ is selected for Fig. 5(d).

##  V Conclusions

In this paper, we have presented a survey of Wang algebra and its applications
to T-coils. A Thévenin equivalent circuit method simplifies transfer function
extraction of constant-Z bridged-T networks. Because of a balanced-bridge
process, the Thévenin equivalent circuit produces a final result with common
poles and zeros already cancelled. Two sets of general constant-R bridged
T-coil equations are derived, and a design process is suggested.

Even greater bandwidth extensions have been reported by trading off the
constant-R property and controlling other parameters [39, 40]. These
extensions are outside of the scope of this paper.

## Acknowledgment

The first author acknowledges C. R. Battjes for his original work in designing
T-coils on printed circuits, on hybrid IC substrates, and in ICs; for teaching
the T-coil design methods within Tektronix, Inc.; and for inspiring others to
advance the technology. The first author also thanks Dr. Ivan Frisch,
Professor Emeritus at NYU, for teaching Wang algebra in a course at the
University of California, Berkeley. The second author would like to thank Dr.
Jin-Hai Guo for providing Ki-Tung Wang’s photo and several references.

##  Appendix A Symmetrical Constant-Z Derivations

The symmetrical network equations (25)-(28) can be derived for the Fig. 4(d)
elements with the substitutions in (42) entered in (21):

$\begin{split}a&=b=sL_{1}+R_{1},\\\ \frac{1}{c}&=\ sC_{B}+G_{B},\\\
d&=sL_{3}+R_{S}+\frac{1}{\left(sC+G_{P}\right)},\\\ e&=R.\end{split}$ (42)

This substitution expands to the product terms in (43):

$\begin{split}&s^{3}\left(-2CC_{B}L_{1}R^{2}+CL_{1}^{2}+2{CL_{1}L}_{3}\right)+\\\
&s^{2}(-2CG_{B}L_{1}R^{2}-2C_{B}{G_{P}L}_{1}R^{2}-2CC_{B}R^{2}R_{1}\\\
&+G_{P}L_{1}^{2}+2G_{P}L_{1}L_{3}+2CL_{1}R_{1}+2CL_{3}R_{1}+2CL_{1}R_{S}\\\
&s^{1}(-2G_{B}G_{P}L_{1}R^{2}-2CG_{B}R^{2}R_{1}-2C_{B}G_{P}R^{2}R_{1}\\\
&-CR^{2}+2G_{P}L_{1}R_{1}+2G_{P}L_{3}R_{1}+CR_{1}^{2}\\\
&+2G_{P}L_{1}R_{S}+2CR_{1}R_{S}+2L_{1})+\\\
&s^{0}(-2G_{B}G_{P}R^{2}R_{1}-G_{P}R^{2}+G_{P}R_{1}^{2}+2G_{P}R_{1}R_{S}+2R_{1})\\\
&=0.\end{split}$ (43)

One set of steps is outlined. Commercial or free symbolic mathematical tools
can automate the reductions by doing substitutions and cancellations. SageMath
(a free, open-source tool created by academia and available in the cloud or
for downloading at www.sagemath.org) is used in these steps.

The $s^{3}$ multiplier terms are set to zero to produce (28). After
substituting for $L_{3}$, the $s^{2}$ multiplier terms are simplified to yield
a preliminary expression for $R_{1}$ that includes $G_{B}$ and also to yield
(26) and (27) directly from the simplified $s^{1}$ and $s^{0}$ multiplier
terms that are set to zero. Finally, (26) is used to simplify the previous
$R_{1}$ result to produce (25).

To form the transfer function as a function of $C_{B}$, (27) is also used so
that $C_{B}$ can range from zero to infinity. This provides a complete
solution space covering real and complex poles. Values of $C_{B}$ (and
$L_{3}$) can be determined for complex poles and selected pole angles, as
shown in Table I.

##  Appendix B Asymmetrical Constant-Z Derivations

The asymmetrical network equations (33)-(37) can be derived for the Fig. 5(d)
elements with the substitutions in (44) entered in (20):

$\begin{split}a&=sL_{1}+R_{1},\\\ b&=sL_{2}+R_{2},\\\ \frac{1}{c}&=\
sC_{B},\\\ d&=sL_{3}+R_{S}+\frac{1}{\left(sC+G_{P}\right)},\\\
e&=R.\end{split}$ (44)

This substitution expands to the product terms in (45):

$\begin{split}&s^{3}(-CC_{B}L_{1}R^{2}-CC_{B}L_{2}R^{2}+CL_{1}L_{2}+{CL_{1}L}_{3})\\\
&+{CL_{2}L}_{3})+\\\
&s^{2}(-{C_{B}G}_{P}L_{1}R^{2}-{C_{B}G}_{P}L_{2}R^{2}-CC_{B}R^{2}R_{1}\\\
&-CC_{B}R^{2}R_{2}+G_{P}L_{1}L_{2}+G_{P}L_{1}L_{3}\ {+\ G}_{P}L_{2}L_{3}\\\
&+CL_{1}R+CL_{2}R+CL_{2}R_{1}+CL_{3}R_{1}+\ CL_{1}R_{2}\\\
&+CL_{3}R_{2}+CL_{1}R_{S}+CL_{2}R_{S})+\\\
&s^{1}(-C_{B}G_{P}{R^{2}R}_{1}-C_{B}G_{P}{R^{2}R}_{2}+G_{P}L_{1}R\\\
&+G_{P}L_{2}R+CR^{2}+G_{P}L_{2}R_{1}+G_{P}L_{3}R_{1}+CRR_{1}\\\
&+G_{P}L_{1}R_{2}+G_{P}L_{3}R_{2}-CRR_{2}\\\
&+CR_{1}R_{2}+G_{P}L_{1}R_{S}+G_{P}L_{2}R_{S}\\\
&+CR_{1}R_{S}+CR_{2}R_{S}+L_{1}+L_{2})+\\\ &s^{0}({-\
G}_{P}R^{2}+G_{P}RR_{1}-G_{P}RR_{2}+G_{P}R_{1}R_{2}\\\
&+G_{P}R_{1}R_{S}+G_{P}R_{2}R_{S}+R_{1}+R_{2})=0.\end{split}$ (45)

One set of steps is briefly outlined. Some of the later steps are done
manually. The $s^{3}$ multiplier terms are set to zero to produce (37). The
$s^{0}$ multiplier terms are set to zero to produce (33a) and (33b). After
replacing $L_{3}$ with (37), all terms with $L_{3}$ and $C_{B}$ elements are
eliminated in a modified expansion.

In this expansion, the $s^{2}$ multiplier terms yield (46) after they are set
to zero and divided by $C$:

$L_{2}^{2}\left(R_{S}+R_{1}-R\right)+2L_{1}L_{2}R_{S}+L_{1}^{2}\left(R_{S}+R_{2}+R\right)=0.$
(46)

The $s^{1}$ multipliers are simplified with two substitutions. The terms that
are multiplied by $G_{P}$ are identical to the terms in the left side of (46)
and are removed. The remaining terms that are multiplied by $C(L_{1}+L_{2})$
are replaced because they are identical to the first six terms of the $s^{0}$
multiplier in (45) after these six terms are divided by $-G_{P}$. The
mathematical process produces $(L_{1}+L_{2})(R_{1}+R_{2})/G_{P}$, which is set
equal to a remaining $(L_{1}+L_{2})^{2}$. Equation (34a) follows after
dividing both sides by $(L_{1}+L_{2})$.

A quadratic equation formula is used to solve (46) for either $L_{2}$ in (47)
or $L_{1}$ in (48) in terms of the other inductor. Each solution has the same
six product terms under the radical as the first six $s^{0}$ multiplier terms
divided by $-G_{P}$ in (45). These terms can be simplified as described above
to form $\sqrt{(R_{1}+R_{2})/G_{P}}$.

$\displaystyle L_{2}$
$\displaystyle=L_{T}-L_{1}=L_{1}\frac{-R_{S}+\sqrt{(R_{1}+R_{2}){/G}_{P}}}{R_{S}+R_{1}-R},$
(47) $\displaystyle L_{1}$
$\displaystyle=L_{T}-L_{2}=L_{2}\frac{-R_{S}+\sqrt{(R_{1}+R_{2}){/G}_{P}}}{R_{S}+R_{2}+R}.$
(48)

After doing some mathematical simplification, (35) follows from (47), (33c),
and (34b). Equation (36) follows from (48), (33d), and (34c).

Equations (35) and (36) are formed with positive radicals in (47) and (48).
The negative radical solutions produce divide by zero cases if $G_{P}=0$.

As done in the symmetrical case, (37) is substituted to express the transfer
denominator (38)-(41) as a function of $C_{B}$.

## References

  * [1] K. T. Wang, “On a new method of analysis of electrical networks,” _Memoir of National Research Institute of Engineering, Academia Sinica_ , vol. 2, pp. 1–11, 1934.
  * [2] S.-L. Ting, “On the general properties of electrical network determinants,” _Chinese Journal of Physics_ , vol. 1, pp. 18–40, 1935.
  * [3] C.-T. Tsai, “Short cut methods for expanding the determinants involved in network problems,” _Chinese Journal of Physics_ , vol. 3, pp. 148–181, 1939\.
  * [4] W.-L. Chow, “On electric networks,” _J. Chinese Math. Soc._ , vol. 2, pp. 321–339, 1940.
  * [5] Y.-H. Ku, “Résumé of Maxwell’s and Kirchhoff’s rules for network analysis,” _Journal of the Franklin Institute_ , vol. 253, no. 3, pp. 211–224, 1952.
  * [6] S. Bellert, “Topological analysis and synthesis of linear systems,” _Journal of the Franklin Institute_ , vol. 274, no. 6, pp. 425–443, 1962\.
  * [7] B. Ross, “Generalization of T-coil equations,” in _3rd Electrotechnical and Comput. Sci. Conf._ , Portoroz, Slovenia, 1994, pp. 39–43.
  * [8] ——, “Wang algebra and interconnects,” in _Asian IBIS Summit_ , Beijing, China, 2007. [Online]. Available: https://ibis.org/summits/sep07a/ross.pdf
  * [9] W.-K. Chen, _Graph Theory and Its Engineering Applications_. Singapore: World Scientific, 1997.
  * [10] R. J. Duffin, “An analysis of the Wang algebra of networks,” _Trans. Amer. Math. Soc._ , vol. 93, pp. 114–131, 1959.
  * [11] R. Duffin and T. Morley, “Wang algebra and matroids,” _IEEE Transactions on Circuits and Systems_ , vol. 25, no. 9, pp. 755–762, 1978.
  * [12] R. J. Duffin, “Some problems of mathematics and science,” _Bulletin of the American Mathematical Society_ , vol. 80, pp. 1053–1070, 1974.
  * [13] J.-H. Guo, “K. T. Wang: First Chinese scholar to publish a mathematical paper in international journals,” 2015, (in Chinese), Institute for the History of Natural Sciences, Chinese Academy of Sciences. [Online]. Available: http://www.ihns.cas.cn/kxcb_new/kpwz_new/201602/t20160229_4538251.html
  * [14] ——, “From the Nine Chapters to ancient Chinese mathematics: An interpretation of Ki-Tung Wang’s letter to Yan Li,” _Journal of Gaungxi University of Nationalities (Natural Science Edition)_ , vol. 21, no. 1, pp. 14–18, 2015, (in Chinese).
  * [15] K. T. Wang, “The differentiation of quaternion function,” _Proceedings of the Royal Irish Academy_ , vol. 29, pp. 73–80, 1911/1912.
  * [16] J.-H. Guo, “K. T. Wang and the differentiation of quaternion functions,” _China Historical Materials of Science and Technology_ , vol. 23, no. 1, pp. 65–70, 2002, (in Chinese).
  * [17] ——, “Ki-Tung Wang’s new method for the analysis of electric network and its scientific influence,” _China Historical Materials of Science and Technology_ , vol. 24, no. 4, pp. 312–319, 2002, (in Chinese).
  * [18] ——, “An inquiry of mathematics teaching in the Imperial Tungwen College,” _Journal of Gaungxi University of Nationalities (Natural Science Edition)_ , vol. 22, no. Supplementary Issue, pp. 47–60, 2003, (in Chinese).
  * [19] K. T. Wang, _Comparative Study of Buddhism and Sciences_. Shanghai Buddhism Press, 1933; reprinted by Shanxi People’s Press, 2014, (in Chinese). [Online]. Available: http://www.nnycjd.com/jsrw/wjt/8000.html
  * [20] T. H. Lee, _Planar Microwave Engineering: A Practical Guide to Theory, Measurements, and Circuits_. New York, NY: Cambridge University Press, 2004.
  * [21] E. Ginzton, W. Hewlett, J. Jasberg, and J. Noe, “Distributed amplification,” _Proceedings of the IRE_ , vol. 36, no. 8, pp. 956–969, 1948.
  * [22] N. B. Schrock, “A new amplifier for milli-microsecond pulses,” _HP Journal_ , vol. 1, no. 1, pp. 9–13, Sept. 1949.
  * [23] P. Starič and E. Margan, _Wideband Amplifiers_ , 4th ed. Dordrecht, The Netherlands: Springer, 2015, pp. 2.35-2.50.
  * [24] J. Paramesh and D. J. Allstot, “Analysis of the bridged T-Coil circuit using the extra-element theorem,” _IEEE Transactions on Circuits and Systems II: Express Briefs_ , vol. 53, no. 12, pp. 1408–1412, 2006.
  * [25] S. C. D. Roy, “Comments on “Analysis of the bridged T-coil circuit using the extra-element theorem”,” _IEEE Transactions on Circuits and Systems II: Express Briefs_ , vol. 54, no. 8, pp. 673–674, 2007.
  * [26] D. L. Feucht, _Handbook of Analog Circuit Design_. San Diego, CA: Academic Press, 1990.
  * [27] J. L. Addis, “Good engineering and fast vertical amplifiers,” in _Analog Circuit Design: Art, Science, and Personalities_ , J. Williams, Ed. Boston, MA, Butterworth-Heinemann, 1991, pp. 107–122.
  * [28] C. R. Battjes, “Who wakes the bugler?” in _The Art and Science of Analog Circuit Design_ , J. Williams, Ed. Boston, MA, Butterworth-Heinemann, 1995, pp. 121–138.
  * [29] A. L. Hollister, _Wideband Amplifier Design_. Raleigh, NC, SciTech Publishing, 2007, pp. 20-46.
  * [30] L. Selmi, D. Estreich, and B. Ricco, “Small-signal MMIC amplifiers with bridged T-coil matching networks,” _IEEE Journal of Solid-State Circuits_ , vol. 27, no. 7, pp. 1093–1096, 1992.
  * [31] T. T. True, “Bridged T termination network,” U.S. Patent 3 155 927, Nov. 1964\.
  * [32] S. Galal and B. Razavi, “Broadband ESD protection circuits in CMOS technology,” _IEEE Journal of Solid-State Circuits_ , vol. 38, no. 12, pp. 2334–2340, 2003.
  * [33] E. Pillai and J. Weiss, “Novel T-coil structure and implementation in a 6.4-Gbs CMOS receiver to meet return loss specifications,” in _Electron. Components and Tech. Conf._ , 2007, pp. 147–153.
  * [34] E. R. Pillai, L. L. Hsu, W. Sauter, and D. W. Storaska, “On-pad broadbanding matching network,” U.S. Patent 7 265 433 B2, Sept. 2007.
  * [35] B. Razavi, “The bridged T-coil [a circuit for all seasons],” _IEEE Solid-State Circuits Magazine_ , vol. 7, no. 4, pp. 9–13, 2015.
  * [36] P. Rako, “What’s all this T-coil stuff, anyhow?” Electronic Design, April 2019\. [Online]. Available: https://www.electronicdesign.com/technologies/analog/article/21807815/whats-all-this-tcoil-stuff-anyhow
  * [37] G.-Z. Hu, _Applied Modern Algebra_. Tsinghua University Press, 1993, (in Chinese).
  * [38] K. Shen, J. N. Crossley, A. W.-C. Lun, and H. Liu, _The Nine Chapters on the Mathematical Art: Companion and Commentary_. Oxford University Press, 1999.
  * [39] S. Shekhar, J. Walling, and D. Allstot, “Bandwidth extension techniques for CMOS amplifiers,” _IEEE Journal of Solid-State Circuits_ , vol. 41, no. 11, pp. 2424–2439, 2006.
  * [40] J. S. Walling, S. Shekhar, and D. J. Allstot, “Wideband CMOS amplifier design: Time-domain considerations,” _IEEE Transactions on Circuits and Systems I: Regular Papers_ , vol. 55, no. 7, pp. 1781–1793, 2008.