1. The Elegant Nature of the Tschebyscheff Impedance
Transformer and Its Utility for Broadband Radome Design
Dan Hillman
Abstract: The goal of this article is to demonstrate the utility and elegance of the Tschebyscheff
transformer for minimizing the input reflection coefficient of a multi-section radome over a
maximum possible bandwidth. The author also seeks to mitigate any fear or pain that might be
associated with Tschebyscheff analysis and design.
I. Introduction
The primary motivation of this study is to design a broadband radome with a maximum
transmission coefficient across a very wide bandwidth. A secondary motivation of this study is
to observe and appreciate a wonderful example of mathematical elegance and beauty.
It has been proved in [1] that the Tschebyscheff impedance transformer represents an optimum
design in that for a specified maximum tolerable input reflection coefficient magnitude, no other
design will yield a wider bandwidth. This maximum bandwidth does come at a cost; other
designs (such as the binomial design) perform much better over a relatively narrow bandwidth.
Thus, the design engineer needs to consider judiciously the trade-off between the desired
reflection coefficient and the desired bandwidth.
It is well known that Tschebyscheff impedance transformer is just that – an impedance
transformer. The goal of such transformers is to minimize reflections as energy propagates into a
medium of different constitutive parameters. In a radome situation, energy is ultimately
propagating from medium 0 (usually air) through the various layers of the radome (mediums 1
through 2N or 2N+1) and then back into air (medium 2N+1 or 2N+2). See Figure 1. In order to
provide a certain level of mechanical strength, part of the radome must have a dielectric constant
that is so high. Therefore, the design idea here is to break down the radome into two halves. In
both halves, the goal is to minimize reflections. The first half is an impedance transformer from
air to a denser dielectric layer, and the second half is an impedance transformer from the denser
dielectric layer to air. Since it is presumed that the material is isotropic, the two halves will be
symmetrical. Thus, the radome design problem really is an impedance transformer problem.
Details of Tschebyscheff analysis and design can be found in [2], [3], [4], and [5]. However, the
most clear explanation of the Tschebyscheff design process (in the opinion of this author) is to be
found in [1]. This article parallels the presentation therein.

2.
Figure 1: Radome Cross Section
II. First Order Approximation of a Multi-section Quarter-wave Transformer
Consider N layers of dielectric slabs between two semi-infinite media (Figure 2).
Figure 2: An N-layer Impedance Transformer

3. The electrical length θ of each layer is set to π/2 at the center frequency f0 corresponding to a
quarter wavelength. The intrinsic reflection coefficient at the nth
interface is given by
Equation 1
The relationship between the intrinsic reflection coefficient on either side of the nth
interface and
the intrinsic impedances of the material on either side of the interface is determined according to
the boundary conditions at the interface. Thus, Equation 1 constrains the optimization problem.
An exact expression for the input reflection coefficient takes into account the infinite number of
reflections and transmission in each layer. As long as the intrinsic reflection coefficients of each
layer are relatively small with respect to unity, a good approximation of the input reflection
coefficient is the sum of the first-order reflected waves only Equation 2.
| | | | | | ⋯ | | | |
Equation 2
Now, we assume (quite safely as design engineers) a symmetrical design. That is, |Γn| = | ΓN-n|
for all n. Equation 2 can then be written as
| | | | ⋯
Equation 3
If N is odd, then there are (N+1)/2 terms in Equation 3 and the last term is |Γ(N-1)/2|(ejθ
+ e-jθ
). If
N is even, then there are N/2+1 terms and the last term is |ΓN/2|. The distrusting or interested
reader is encouraged to work through several example scenarios in order to see this.
It should be immediately apparent that the motivation for writing Equation 3 is to utilize Euler’s
identity. See I.1 and [7] for a proof of Equation 4.
cos
Equation 4
Thus, Equation 3 reduces to
2 | | cos | | cos 2 ⋯ | | cos 2 ⋯
Equation 5
If N is odd, then the last term in the brackets is |Γ(N-1)/2|cos(θ ). If N is even, then the last term in
the brackets is ½ |ΓN/2|.

4. III. Correlating Fractional Bandwidth, Quarter-Wavelength, and Electrical Length
Collin notes in [1]:
“Since the series is a cosine series, the periodic function that it defines is periodic over the
interval π corresponding to the frequency range over which the length of each transformer
section changes by a half wavelength.”
It is worth taking some time to get straight the relationships between electrical length θ, relative
frequency f/f0, and quarter wavelength λ0/4. Recall some basic relationships (Figure 3):
Figure 3: Some Basic Equations Involving wavenumber, frequency, wavelength, phase velocity,
and electrical length
θ f/f0 λ0/4
0 0.000 0λ
π/50 0.040 0.01λ
π/20 0.100 0.025λ
π/10 0.200 0.05λ
π/8 0.250 0.0625λ
π/6 0.333 0.0833λ
π/4 0.500 0.125λ
π/3 0.667 0.1667λ
3π/8 0.750 0.1875λ
π/2 1.000 0.25λ
5π/8 1.250 0.3125λ
2π/3 1.333 0.3333λ
3π/4 1.500 0.375λ
5π/6 1.667 0.4167λ
7π/8 1.750 0.4375λ
9π/10 1.800 0.45λ
19π/20 1.900 0.475λ
49π/50 1.960 0.49λ
π 2.000 0.5λ
Table 1: Relating Electrical Length, Relative Frequency, and Quarter Wavelength

5. Thus, as θ varies from 0 to π, the relative frequency varies from 0 to 2 and a particular physical
length that is a quarter wavelength at the center frequency varies between 0 wavelengths (at dc)
and ½ wavelength at 2f0. Thus, if the fractional bandwidth (FBW = Δf/f0) is 1.0, then the
relative frequency f/f0 will vary between 0.5 and 1.5, the electrical length θ of each slab of the
transformer will vary between π/4 and 3π/4 radians.
IV. Tschebyscheff Polynomials
The first four Tschebyscheff polynomials (plus the Tschebyscheff polynomial of degree 0) are
given below along with the definition of the Tschebyscheff polynomial of degree n.
Equations 6(a-f): Tschebyscheff Polynomials
The first four Tschebyscheff polynomials (plus the zeroth order polynomial) are plotted in Figure
4 below:

6.
Figure 4: Tschebyscheff Polynomials
Note that for -1 < x < 1, the Tschebyscheff polynomials oscillate between -1 and 1. The reader
can see how Tschebyscheff polynomials just seem to be ideal to achieve equal ripple across a
desired band if the polynomials could be judiciously manipulated for any given design goals. It
is worth noting here that Tschebyscheff transformers are also called equal ripple transformers.
The equal ripple across a desired bandwidth property is even more apparent when we plot
absolute values of these polynomials (Figure 5).
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Tschebyscheff Polynomials
x
T
Zeroth
First
Second
Third
Fourth

7.
Figure 5: Absolute Values of Tschebyscheff Polynomials
V. The Derivation
If we let x = cos(θ), then as shown in [8] and I.2
cos cos cos cos
Equation 7
Some authors go through the trouble of introducing hyperbolic functions to avoid dealing with
imaginary arguments. In the opinion of this author, introducing hyperbolic functions
complicates everything, and it is unnecessary, because cos(jx) = cosh(x) (I.3 and [9]). In any
case, the passband is the band for which argument of Equation 7 is real. Further, by simply
ignoring x and only concentrating on the passband, the absolute values of the Tschebyscheff
polynomials are plotted as functions of θ.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Tschebyscheff Polynomials
x
T
Zeroth
First
Second
Third
Fourth

8.
Figure 6: Absolute Values of Tschebyscheff Polynomials in the Variable of the cos(θ)
Now the goal is to force |Γin| to have an equal-ripple characteristic from some θm to π – θm; that
is, from fm to 2f0-fm.
Therefore, a judicious choice is made to let x = cos(θ)sec(θm) yielding:
Equations 8(a-d): Tschebyscheff Polynomials varying in cos(θ)sec(θm)
If, for example, θm = π/4, the equal-ripple passband is set from π/4 < θ < 3π/4, which is, in terms
of relative frequency equivalently, 0.5 < f/f0 < 1.5.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Tschebyscheff Polynomials
theta (rad)
T Zeroth
First
Second
Third
Fourth

9.
Figure 7: Absolute Values of Tschebyscheff Polynomials in the Variable of cos(θ)sec(π/4)
Equations 8 is a series of the form of Equation 5. Therefore, let
2 | |cos | |cos 2 ⋯ | |cos 2 ⋯
cos sec
Equation 9
where A is not yet determined.
When θ=0,
sec
or equivalently,
1
sec
Equation 10
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Tschebyscheff Polynomials
theta (rad)
T
Zeroth
First
Second
Third
Fourth

10. Consequently,
2 | |cos | |cos 2 ⋯ | |cos 2 ⋯
cos sec
sec
Equation 11
In the passband, the maximum value for TN(cos(θ)sec(θm)) is unity. Therefore,
| |
1
| sec |
1
|cos cos sec |
Equation 12
represents the maximum tolerable value of the input reflection coefficient within the passband.
Note that if the passband (and thus θm) is specified, then |Γm| is fixed. Likewise, if |Γm| is
specified, then θm is fixed. Rearranging Equation 12,
sec cos
1
cos
1
| |
Equation 13
Thus, the trade-off of the Tschebyscheff design between a maximum tolerable input reflection
coefficient and bandwidth is seen in Equation 12 and Equation 13 and in Figure 8 below.

11.
Figure 8: Trade Off Between Maximum Tolerable Reflection Coefficient and θm for ηL=188.365 Ω
What remains to complete the Tschebyscheff design is to solve Equation 11 for the intrinsic
reflection coefficients at each interface. To do this, we need to use the binomial theorem found
in [6].
Equation 14
where
!
! !
Equation 15
Therefore as demonstrated in [1],
cos 2 1 2
2 cos cos 2 ⋯ cos 2 ⋯
Equation 16

12. The last term in Equation 16 is / for N = even and / cos for N = odd.
Using Equation 16 along with Equations 6,
sec cos sec cos (a)
sec cos 2 sec cos 1 sec 1 cos 2 1 (b)
sec cos sec cos 3 3 cos 3 sec cos (c)
sec cos
sec cos 4 4 cos 2 3 4 sec cos 2 1 1
(d)
Equations 17
Plugging the appropriate equation of Equations 17 into Equation 11, the coefficients are now
easily solved by matching terms and doing algebra. Finally, the intrinsic impedances, dielectric
constants, and thicknesses of each layer are found by simple algebra.
VI. Design Considerations and Procedures
The following is a useful checklist throughout the design process.
 Select N where N is the number of sections of the Tschebyscheff transformer. Keep in
mind that as N increases, bandwidth increases.
 Minimize the maximum tolerable reflection coefficient |Γm| of the passband.
 Maximize the fractional bandwidth by minimizing θm.
 Note the trade-off between bandwidth and reflection ripple level.
 Key Equations: Equation 1, Equation 11, Equation 12, Equation 13, Equation 15, Equations
17.
 Specify / find: fm/f0, 2-fm/f0, Δf/f0, θm, sec(θm), TN(sec(θm), |Γm|.
 Find the intrinsic reflection coefficients |Γn| at each interface by solving Equation 11 using
Equations 17 and lining up like terms.
 Plot |Γin| versus θ using both sides of Equation 11. This is a good way to troubleshoot,
since the two expressions in Equation 11 are equal.
 For each section find ηn, εnr, dn=λn0/4.
 Adjust |Γm| or θm to account for other design constraints and considerations.

13. VII. Examples, Solutions, and Design Tables
Using the right side of Equation 11 along with the appropriate equation of Equations 8, it is
possible to observe what the final performance of the design will be prior to completing the
design. Let |Γm| = 0.2 and ηL=188.365 Ω. Observing Figure 8, we see that for N = 2, θm 0.52,
which corresponds to a fractional bandwidth Δf/f0 1.67. For N = 3, obviously, θm is lower (θm
0.35) and the fractional bandwidth is greater (Δf/f0 1.78). See Figure 9. As N increases,
Δf/f0 also increases. Assuming that N is set, we can increase bandwidth by increasing the
maximum tolerable reflection coefficient, or we can achieve a lower maximum tolerable
reflection coefficient by decreasing the bandwidth.
N |Γm| θm Δf/f0
2 0.3 0.23 1.85
2 0.2 0.52 1.67
2 0.1 0.82 1.48
3 0.3 0.15 1.90
3 0.2 0.35 1.78
3 0.1 0.58 1.63
4 0.3 0.12 1.92
4 0.2 0.27 1.83
4 0.1 0.45 1.71
5 0.3 0.09 1.94
5 0.2 0.22 1.86
5 0.1 0.37 1.76
Table 2: Some approximate values of |Γm| and θm from Figure 8 (ηL=188.365 Ω)

14.
Figure 9: The performance of two Tschebyscheff Transformers (N=2 and N=3) each with |Γm|= 0.2
(ηL=188.365 Ω)
Figure 10: The performance of three 2-section Tschebyscheff Transformers
(|Γm|= 0.1, |Γm|= 0.2, |Γm|= 0.3) (ηL=188.365 Ω)
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient vs theta
theta (rad)
InputReflectionCoefficient
N=2
N=3
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient | IN
| vs 
 (rad)
InputReflectionCoefficient|IN
|
Wideband
Middleband
Narrowband

15.
Figure 11: The performance of three 3-section Tschebyscheff Transformers
(|Γm|= 0.1, |Γm|= 0.2, |Γm|= 0.3) (ηL=188.365 Ω)
If we let ηL=214 Ω, then we obtain the following results.
Figure 12: Trade Off Between Maximum Tolerable Reflection Coefficient and θm for ηL=214 Ω
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient | IN
| vs 
 (rad)
InputReflectionCoefficient|IN
|
Wideband
Middleband
Narrowband
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
m
MagnitudeofMaximumTolerableReflectionCoefficient|m
|
Magnitude of Maximum Tolerable Reflection Coefficient | m
| vs m
N=2
N=3
N=4
N=5

16. N |Γm| θm Δf/f0
2 0.25 0.22 1.86
2 0.15 0.57 1.64
2 0.1 0.75 1.52
3 0.25 0.15 1.90
3 0.15 0.39 1.75
3 0.1 0.53 1.66
4 0.25 0.11 1.93
4 0.15 0.30 1.81
4 0.1 0.41 1.74
5 0.25 0.09 1.94
5 0.15 0.24 1.85
5 0.1 0.33 1.79
Table 3: Design Table: Some approximate values from Figure 12 (ηL=214 Ω)
Figure 13: The performance of three 2-section Tschebyscheff Transformers
(|Γm|= 0.1, |Γm|= 0.15, |Γm|= 0.25) (ηL=214 Ω)
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient | IN
| vs 
 (rad)
InputReflectionCoefficient|IN
|
Wideband
Middleband
Narrowband

17.
Figure 14: The performance of three 3-section Tschebyscheff Transformers
(|Γm|= 0.1, |Γm|= 0.15, |Γm|= 0.25) (ηL=214 Ω)
Figure 15: The performance of three 4-section Tschebyscheff Transformers
(|Γm|= 0.1, |Γm|= 0.15, |Γm|= 0.25) (ηL=214 Ω)
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient | IN
| vs 
 (rad)
InputReflectionCoefficient|IN
|
Wideband
Middleband
Narrowband
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of Reflection Coefficient | IN
| vs 
 (rad)
InputReflectionCoefficient|IN
|
Wideband
Middleband
Narrowband

18. To realize these or other Tschebyscheff designs, we must solve for the intrinsic reflection
coefficients of Equation 11 using the appropriate equation of Equations 17 and lining up like
terms.
Let N = 2.
2| | cos 2 | | | | sec 1 cos 2 1 (a)
| | | | sec | | (b)
| | | | sec 1 (c)
Equations 18(a-c)
Let N = 3.
2| | cos 3 2| | cos | | sec cos 3 3cos 3 sec cos
(a)
| | | | sec | | (b)
| | | | sec sec 1 | | (c)
Equations 19(a-c)
Let N = 4.
2| | cos 4 2| | cos 2 | |
| | sec cos 4 4 cos 2 3 4 sec cos 2 1
1
(a)
| | | | sec | | (b)
| | | | sec sec | | (c)
| | | | 3 sec 4 sec 1 (d)
Equations 20(a-d)

19. Important Note on Signs: As the equations in this article are now, all the intrinsic reflection
coefficients come out to be positive. If the transformer is stepping up in impedance, this is fine.
If the transformer is stepping down in impedance, then the intrinsic reflection coefficients found
in Equations 18), Equations 19, and Equations 20 as well as in Table 4 need to be replaced by
their negatives.
When computing the intrinsic impedances, one should note that this design is a first-order
approximation. Therefore, by recursively applying Equation 1, starting from free space and
working toward the semi-infinite load, the astute reader will observe that the computed value of
the load impedance is slightly higher than the given value. Likewise, by starting at the load and
applying Equation 1 recursively, the computed value of the impedance of free space is slightly
less than η0 = 376.73 Ω. To obtain the best possible results, I have worked these values out both
ways and then for each section, I have assigned the geometric mean of the two computed
intrinsic impedance values as the intrinsic impedance of that section.
Table 4: Tschebyscheff Design Tables

20. VIII. Application to Radome Design
A MATLAB m-file (Fpanel_frq.m) has been generated by Rich Matyskiela which computes for
a given radome, the transmission coefficient, the reflection coefficient, and the power dissipated
in the radome as functions of frequency for several given angles of incidence. If a single section
of Astroquartz (εr=3.1, η = 214 Ω, tan(δ) = 0.004) with a thickness of 12 mils met the mechanical
specifications, it would be a nearly ideal radome (see Figure 16 and Figure 17).
Figure 16: No Design, Astroquartz Slab (εr=3.1, η = 214 Ω, tan(δ) = 0.004, T = 0.012 inches),
Parallel Polarization
Figure 17: No Design, Astroquartz Slab (εr=3.1, η = 214 Ω, tan(δ) = 0.004, T = 0.012 inches),
Perpendicular Polarization
0 2 4 6 8 10121416 18202224 26283032 34363840 42
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
FREQUENCY, GHz. ( One Section of Astroquartz, d = 12 mils )
dB
TX COEFFICIENT PARALLEL POL
0°
15°
30°
45°
60°
70°
0 2 4 6 8 10121416 18202224 26283032 34363840 42
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
FREQUENCY, GHz. ( One Section of Astroquartz, d = 12 mils )
dB
TX COEFFICIENT CROSS POL
0°
15°
30°
45°
60°
70°

21. It is assumed that a fundamental design requirement for the radome is that it must include a
section of Astroquartz section that is at least 0.027 inches thick along with at least one other
layer of dielectric needed for mechanical strength. Before attempting any design, let us observe
the performance of such a single Astroquartz slab that is 27 mils thick. Since the most important
performance parameter is the transmission coefficient, we will limit our analysis to the
transmission coefficient plots generated by the m-file.
Figure 18: No Design, Astroquartz Slab (εr=3.1, η = 214 Ω, tan(δ) = 0.004, T = 0.030 inches),
Parallel Polarization
Figure 19: No Design, Astroquartz Slab (εr=3.1, η = 214 Ω, tan(δ) = 0.004, T = 0.030 inches),
Perpendicular Polarization

22. It is readily apparent from Figure 18 and Figure 19 that the design challenge is to maximize the
transmission coefficient across the band for the perpendicular polarization without losing much
performance in the parallel polarization.
Previous design efforts have been performed which yields the current default design. The
current radome was designed to operate between 2 and 18 GHz. The design parameters are
given in Table 5, the associated transmission coefficient plots generated by the m-file are
displayed in Figure 20 and Figure 21.
Section d (inches) εr tan(δ) Material
1 0.027 3.100 0.004 Astroquartz /Epon-828
2 0.180 1.140 0.0071 Rohacell
3 0.027 3.100 0.004 Astroquartz /Epon-828
4 0.180 1.140 0.0071 Rohacell
5 0.027 3.100 0.004 Astroquartz/Epon-828
Table 5: Current Radome Default Design
Figure 20: Current Default Design, Parallel Polarization.

23. Figure 21: Current Default Design, Perpendicular Polarization.
By sandwiching two 2-section Tschebyscheff transformers between a center section of
Astroquartz, a much wider bandwidth radome is achieved at the cost of optimal performance in
the lower band. See Figure 22 and Table 6. If only a single design is permitted for the entire
bandwidth (2-40 GHz), this radome would perform reasonably well, but not as well as is desired.

24.
Figure 22: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 300 Ω, f0 = 25 GHz
Section d (inches) εr tan(δ)
1 0.101 1.360 0.007
2 0.092 1.630 0.007
3 0.027 3.100 0.004
4 0.092 1.630 0.007
5 0.101 1.360 0.007
Table 6: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 300 Ω, f0 = 25 GHz
To improve on this design, we break down the radome into two bands: 2-18 GHz and 18-40
GHz. After a plethora of iterations, we obtain the following for the lower band. See Figure 23:
Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 300 Ω, f0 = 10 GHz and Table 7.

25.
Figure 23: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 300 Ω, f0 = 10 GHz
Section d (inches) εr tan(δ)
1 0.253 1.360 0.007
2 0.231 1.630 0.007
3 0.027 3.100 0.004
4 0.231 1.630 0.007
5 0.253 1.360 0.007
Table 7: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 300 Ω, f0 = 10 GHz
This design performs well from 2-18 GHz, but the current default design performs better over
that band. See Figure 21.
Finally, after more design iterations, we obtain the following for the upper band. See Figure 24
and Table 8.

26.
Figure 24: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 275 Ω, f0 = 30 GHz
Section d (inches) εr tan(δ)
1 0.086 1.310 0.007
2 0.076 1.700 0.007
3 0.027 3.100 0.004
4 0.076 1.700 0.007
5 0.086 1.310 0.007
Table 8: Tschebyscheff Radome, N = 2, θm = 1.0, Zw = 275 Ω, f0 = 30 GHz

27.
IX. Conclusions and Future Work
The Tschebyscheff design process has been rigorously derived in detail. Design procedures,
examples, and tables have been documented. The design procedures have been utilized to
achieve broadband radome designs. Time permitting, further investigations may be made to
optimize performance for all incidence angles. Also, it may be worth modeling and simulating
the radome in HFSS and/or FEKO since physical optics is never quite valid, and this would give
further insight into the physics of the problem. It will probably be necessary to work closely
with mechanical and/or material engineers to formulate mechanical and dielectric specifications
constraining the design. It would be interesting also to determine what the optimal theoretical
performance would be for any given radome – taking into consideration all angles of incidence
and both polarizations. The difficulty with this is that each angle of incidence and each
polarization has its own directional impedance. This makes it challenging to decide which load
impedance to choose for the Tschebyscheff design.
It is also worth noting that the dip in performance in the low end of the band is due to the
presence of extra slabs (with extra thickness), and this dip lies outside of the Tschebyscheff
bandwidth. It may be worthwhile to explore how to either push this dip further to the left
(without losing performance in the high end of the band), or how to further mitigate its effect in
the operating bandwidth.

28. X. Appendices
1. Proof of Euler’s Identity [7]
i. Exponent Fundamentals
≜ …
…
1
1
ii. Taylor Series
!
0 0
0
2!
0
3!
⋯
iii. Definition of the Derivative
≜ lim
→
lim
→
1
iv. Defining e
lim
→
1
≜ 1
≜ lim
→
1
2.71828
v. Taylor Series of ex
and ejθ
!
1
2! 3!
⋯

29. !
1
2! 3! 4! 5!
⋯
1
2! 4!
⋯
3! 5!
⋯
vi. Derivatives of sin(θ) and cos(θ)
cos sin
sin cos
cos | 1 ,
0,
sin |
1 /
,
0,
vii. Taylor Series of sin(θ) and cos(θ)
cos
0
!
1
!
,
1
2! 4!
⋯
sin
0
!
1
!
,
3! 5!
⋯
cos sin

30. 2. Trigonometric Double, Triple, Quadruple, and n-tuple Angle Formulas [8]
3. Hyperbolic Functions [9]
sinh
1
2
sin
cosh
1
2
cos
1
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3
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4
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5
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6
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Calculus, Second Edition, John Wiley & Sons, New York, 1998.
7
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https://ccrma.stanford.edu/~jos/st/Proof_Euler_s_Identity.html, 2007, Online Book, Accessed January 25, 2011.
8
“List of Trigonometric Identities,” http://en.wikipedia.org/wiki/List_of_trigonometric_identities, Website,
Accessed January 24, 2011.
9
“Hyperbolic function,” http://en.wikipedia.org/wiki/Hyperbolic_function, Website, Accessed January 25, 2011.