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INF5481 RF Circuit, Theory and DesignAssignment #4
Use your favorite mathematical “spreadsheet” program for
presenting thegraphs (Python, Matlab, ...).
I have written some rough python code to keep track of filter
coefficientsand do varies scaling operations, it is not very pretty
so I will not includethe code. All the filter responses are created
by multipling the appropriateABCD representation and plotting
abs(2/ABCD[0,0]).
Problem 1
Design a maximally flat high-pass filter with a cutoff frequency
of 1 GHz, theattenuation at 2 GHz should be at least 20 dB at 2
GHz.
(a)
Start with a low pass Butterworth prototype filter, what filter
order N isrequired for 20 dB at Ω = 2? Find the filter coefficients
and plot the attenu-ation in dB.
(b)
Transform your normalized low-pass filter to the desired high
pass filter. Plotthe attenuation in dB.
We start by creating a low-pass Butterworth prototype filter
with 20 dBattenuation at f = 2f3 dB.
The attenuation or insertion loss of a Butterworth filter of
order N canbe expressed as ([1] equation (5.41))
IL(Ω) = 10 log(1 + Ω2N
),
1
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INF5481: Assignment #4
where we have set a = 1. Our required response is
IL(Ω = 2) ≤ 20 dB10 log
(1 + 22N
)≤ 20 dB
1 + 22N ≤ 10022N ≤ 100− 1
N ≤ log2(100− 1)2
N ≤ 3.314so we use N = 4, which is the closest integer
fulfilling this requirement. Fromtable 5.2 in [1]
filt = [1, 0.7654, 1.8478, 1.8478, 0.7654, 1]
RG = 1 Ω
C1 = 0.765 F
L2 = 1.85 H
C3 = 1.85 F
L4 = 0.765 H
RL = 1 Ω
To confirm our results we plot the insertion loss for N = 4 and
N = 3,see figure 1. We note that N = 4 is indeed the lowest order
that satisfiesIL(Ω = 2) ≤ 20 dB.
We now convert our low pass filter into a high pass filter with
cutoff ωcusing the transformations
R← R
L← 1Cωc
C ← 1Lωc
.
This results in
RG = 1 Ω
L1 = 2.08× 10−10 HC2 = 8.61× 10−11 FL3 = 8.61× 10−11 HC4 = 2.08×
10−10 FRL = 1 Ω
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INF5481: Assignment #4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Ω [rad/s]
0
5
10
15
20
25
30
35
40
3
IL [dB]
N=4
N=3
Figure 1: Butterworth insertion loss for order 3 and 4. At Ω =
2, N = 4 isabove the required insertion loss of 20 dB.
And the response is shown in figure 2 and we note the 3 dB
cutoff at 1 GHz.
0.0 0.5 1.0 1.5 2.0
f [GHz]
0
10
20
30
40
50
60
70
80
90
3
IL [dB]
N=4
Figure 2: Insertion loss for high-pass Butterworth filter of
order 4.
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INF5481: Assignment #4
Problem 2
In the design of an amplifier for cellular phone applications,
it is discoveredthat the circuit exhibits excessive noise at 3
GHz.
Develop a bandstop filter with the minimum number of components
withthe following properties:
• center frequency of fc = 3 GHz
• 30 dB stopband width of 10 % (from 2.85 GHz to 3.15 GHz)
• 3 dB stopband width of 36.7 %
3 dB ripple in the passband is acceptable. Plot the attenuation
profile.We want to design a filter with a geometric center
frequency
fc = 2.9491× 109 Hz
and upper and lower frequencies
fL = 2.4495× 109 Hz fU = 3.5505× 109 HzΩL = −1.0000 ΩU =
1.0000
at the frequencies
f30 dB = 2.8016× 109 Hz f30 dB = 3.0965× 109 HzΩ30 dB = −3.6377
Ω30 dB = 3.8245
we require an attenuation greater than 30 dB. The solution is
visualized infigure 4.
Theory
We use the Chebyshev Polynomials Tn(Ω) to create a filter with
an insertionloss
IL(Ω) = 10 log(1 + a2T 2N(Ω)
)where TN(Ω) =
{cos (N arccos (Ω)) for |Ω| ≤ 1cosh (N arccosh (Ω)) for |Ω| ≥
1
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INF5481: Assignment #4
In the pass-band Ω ≤ 1 and T 2N is bound to the interval 0 to 1,
so in thepass-band we can write
IL(Ω ≤ 1) = 10 log(1 + a2
)10IL(Ω≤1)/10 =
(1 + a2
)⇒ a2 = 10IL(Ω≤1)/10 − 1, (1)
which allows us to control the amount of ripple.
Solution
Low-pass
We start by designing a low-pass prototype Chebyshev filter
normalized toa cutoff frequency of
ωc = 2.4495× 109 Hz
we require IL(Ω30 dB) ≥ 30 dB at
Ω30 dB = 3.8245
We start with the ripple, from (1)
a2 = 103/10 − 1 ≈ 1 .
We then find the order of the filter
IL(Ω30 dB) ≥ 30 dB10 log
(1 + a2T 2N(3.8245)
)≥ 30 dB
1 + T 2N(3.8245) ≥ 103
TN(3.8245) ≥√
103 − 1cosh (N arccosh (3.8245)) ≥
√103 − 1
N arccosh (3.8245) ≥ arccosh(√
103 − 1)
N · 2.0170 ≥ arccosh(√
103 − 1)
N ≥arccosh
(√103 − 1
)2.0170
N ≥ 2.0557
Which is satisfied with N = 3. Since a = 1 the filter
coefficients can be foundin table 5.4(a) in [1]
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INF5481: Assignment #4
filt = [1, 3.3487, 0.7117, 3.3487, 1]
RG = 1 Ω
C1 = 3.35 F
L2 = 0.712 H
C3 = 3.35 F
RL = 1 Ω
We verify our progress by creating figure 3.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.03.9
Ω [rad/s]
0
10
20
30
40
50
3
IL [dB]
N=3
Figure 3: Low pass prototype Chebyshev filter
Bandstop
We convert our filter with the following transformations
LP ←BWL
ω20
CP ←1
BWL
LS ←1
BWC
CS ←BWC
ω20,
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INF5481: Assignment #4
where
BW ≡ ωU − ωL= 6.9178× 109 rad/s
ω0 ≡√ωUωL
= 1.8529× 1010 rad/s.
RG = 1 Ω
CS1 = 6.75× 10−11 FLS1 = 4.32× 10−11 HCP2 = 2.03× 10−10 FLP2 =
1.43× 10−11 HLS3 = 4.32× 10−11 HCS3 = 6.75× 10−11 FRL = 1 Ω
The insertion loss is shown in figure 4, where figure 5 uses a
logaritmicx-axis and figure 6 displays the magnitude.
1.0 2.0 2.4 2.8 3.1 3.6 4.0 5.0
f [GHz]
0
10
20
30
40
50
60
70
80
3
IL [dB]
N=3
Figure 4: Bandstop Chebyshev filter.
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INF5481: Assignment #4
100 101
f [GHz]
0
10
20
30
40
50
60
70
80
3
IL [dB]
N=3
Figure 5: Bandstop Chebyshev filter, symmetric with logaritmic
x-axis.
100 101
f [GHz]
0.000
0.200
0.400
0.600
0.800
1.000
0.708
IL
N=3
Figure 6: Bandstop Chebyshev filter, magnitude (not db-y
scale).
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INF5481: Assignment #4
Problem 3
(a)
Based on the maximally flat low-pass filter from problem 1a),
convert it toa bandstop filter with a 3 dB bandstop at [2.22 GHz,
2.58 GHz]. Match yourfilter to 50 Ω on both ends. Plot the
attenuation of your bandstop filter.
(b)
Transform your filter to realizable transmission line
elements.
Theory
In this exercise we are going to replace the lumped elements L
and C withmicrostriplines, writing out the impedance for a
short-circuit transmissionline
Zshort = jZ0 tan(βl)
with l = λ0/4 and f0 ≡ vp/λ0 we can write
Zshort = jZ0 tan(2πf
vpλ0/4)
= jZ0 tan(2πf
f0).
This is inductive when
ZL = Zshort
jωL = jZ0 tan(πΩ/2)
jωL = jZ0 tan(π/2 ·ω/ω0)
this can not be fulfilled for all ω, but at ω = ω0, it reduces
to
jωL = jZ0 tan(π/2)
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INF5481: Assignment #4
which goes to infinity, so we do little trick. We multiply our
transmissionline with the so called “bandwidth factor”
1
tan(π/2 ·ωL/ω0)jωL = jZ0 tan(π/2 ·ω/ω0)
1
tan(π/2 ·ωL/ω0)1
tan(π/2 ·ωL/ω0)jωL = jZ0
tan(π/2 ·ωL/ω0)tan(π/2 ·ωL/ω0)
1
tan(π/2 ·ωL/ω0)jωLL = jZ0
ωLL
tan(π/2 ·ωL/ω0)= Z0
So we replace an inductor L with a transmission line with
impedance Z0 =(bf)L. In a similar manner we can derive that a
capacitor C should bereplaced with an open line of impedance Z0 =
1/((bf)C).
Solution
With the filter coefficents
filt = [1, 0.7654, 1.8478, 1.8478, 0.7654, 1]
and the bandwidth factor
bf =1
tan(π/2 ·ωL/ω0)
=1
tan(π/2 · 2.220/2.4)= 0.12
we obtain the transmission line elements
filt[1] = bf*filt[1] # series
filt[2] = 1/(bf*filt[2])# shunt
filt[3] = bf*filt[3] # series
filt[4] = 1/(bf*filt[4])# shunt
1, 0.0906, 4.57, 0.219, 11, 1
We now insert unit elements to each side of the filter
filt = [filt[0], 1, filt[1], filt[2], filt[3], filt[4], 1,
filt[5]]
we now move these unit elements “inwards”, first the
leftmost:
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INF5481: Assignment #4
Z_ue = filt[1]
Z_comp = filt[2]
N = 1 + Z_ue/Z_comp
filt[1] = N*Z_ue # shunt
filt[2] = N*Z_comp # ue
1, 12, 1.09, 4.57, 0.219, 11, 1, 1
then the rightmost:
Z_ue = filt[-2]
Z_comp = filt[-3]
N = 1 + Z_ue/Z_comp
filt[-3] = Z_ue/N # series
filt[-2] = Z_comp/N # ue
1, 12, 1.09, 4.57, 0.219, 0.917, 10.1, 1
and another move
Z_ue = filt[5]
Z_comp = filt[4]
N = 1 + Z_ue/Z_comp
filt[4] = Z_comp*N # ue
filt[5] = Z_ue*N # shunt
1, 12, 1.09, 4.57, 1.14, 4.76, 10.1, 1
We now insert another set of unit elements to each side of the
filter
filt = [filt[0], 1, filt[1], filt[2], filt[3],
filt[4], filt[5], filt[6], 1, filt[7]]
and move the rightmost
Z_ue = filt[-2]
Z_comp = filt[-3]
N = 1 + Z_ue/Z_comp
filt[-3] = Z_ue*N # shunt
filt[-2] = Z_comp*N # ue
1, 1, 12, 1.09, 4.57, 1.14, 4.76, 1.1, 11.1, 1
1, 1, 12, 1.09, 4.57, 1.14, 4.76, 1.1, 11.1, 1
Which, de-normalized
50, 50, 602, 54.5, 229, 56.8, 238, 54.9, 556, 50
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INF5481: Assignment #4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Frequency (Hz) 1e10−80
−70
−60
−50
−40
−30
−20
−10
0
Magnitude (dB)
S11 S21
Figure 7: Simulated filter response
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
Frequency (Hz) 1e9−4
−3
−2
−1
0
1
Magnitude (dB)
S11 S21
Figure 8: Zoomed in view of figure 7
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INF5481: Assignment #4
Problem 4
Following example 2.5 in the book, you designed a 50 Ω
transmission line.Your goal is to short out 2 GHz and pass 4 GHz.
Your reasoning leads youto the open λ/4 stub in figure 9.
Z0 = 50 Ωl = 20.3 mm Cfringe
h = 1 mm
Figure 9: Open 50 Ω transmission line on a 1 mm FR4 substrate
with anunkown fringe component at the end.
(a)
Express the input impedance of the ideal open transmission line
stub assum-ing vp = 1.61× 108 m/s. Vertify that |Zin(f = 2 GHz)| ≈
0 Ω.
(b)
Due to fringing fields some “leakage” will occur, model this as
a fringe ca-pacitor Cfringe. What is the input impedance expression
now?
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INF5481: Assignment #4
(c)
Since you find conformal mapping1 too difficult and do not know
how to useEM simulation software, you decide to guess that Cfringe
is between 0.1 pFand 1 pF.
Plot the magnitude of the input impedance |Zin| from DC to 8 GHz
forthe following cases: ideal, Cfringe = 0.1 pF and Cfringe = 1
pF.Hint: Please limit the y axis to [0 Ω, 500 Ω].
We use the normal formula for a terminated transmission line,
and alsostudy what happens with an open line
Zin = Z0ZL + jZ0 tan(βd)
Z0 + jZL tan(βd)
limZL→∞
Zin = limZL→∞
Z01 + jZ0/ZL tan(βd)
Z0/ZL + j tan(βd)= Z0
1
j tan(βd)
Writing these two equations in terms of frequency f we
obtain
Zin = −jZ01− (2πfC)Z0 tan
(2πvpfd)
Z0(2πfC) + tan(
2πvpfd)
limZL→∞
Zin = −jZ01
tan(
2πvpfd)
For small values of fC the first equation will simplify to the
last, for smallpositive values of f this goes rapidly from −∞ up
to∞. Taking the absolutevalue this is turned around and becomes
down to 0. For larger values we seethe periodicity of tan. The code
is presented bellow and the plot in figure10.
import numpy as np
import pylab as plt
def plot_tl_C(f, C=np.inf, **kwargs):
omega = 2*np.pi*f
Z_0 = 50.
Z_C = 1/(1j*omega*C)
d = 80.67e-3/4
v_p = 1.61e8
beta = omega/v_p
1see for instance
http://physics.usask.ca/~bradley/pll_plate_edge.pdf
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http://physics.usask.ca/~bradley/pll_plate_edge.pdf
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INF5481: Assignment #4
t = np.tan(beta*d)
if C == np.inf:
Z_in = Z_0/(1j*t)
else:
Z_in = Z_0*(Z_C + 1j*Z_0*t)/(Z_0 + 1j*Z_C*t)
plt.plot(f/1e9, np.abs(Z_in), **kwargs)
plt.ylim(0, 500)
plt.xlabel(’freq GHz’)
plt.ylabel(’$|Z_{in}|$ [$\Omega$]’)
plt.figure()
f = np.linspace(0.1e9, 8e9, 1024)
plot_tl_C(f, label=’ideal’)#, ls=’--’)
plot_tl_C(f, C=0.1e-12, label=’$C=0.1$ pF’)#, ls=’--’)
plot_tl_C(f, C=1e-12, label=’$C=1$ pF’)#, ls=’:’)
plt.legend(loc=’best’)
0 1 2 3 4 5 6 7 8
freq GHz
0
100
200
300
400
500
|Zin
| [Ω]
idealC=0.1 pF
C=1 pF
Figure 10: Input impedance of an open transmission line, with
load capaci-tance (shown in blue) and without (shown in green).
Lower plot is a zoomedin view of the upper with linear x axis.
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INF5481: Assignment #4
References
[1] R. Ludwig and G. Bogdanov, RF circuit design: theory and
applications,2nd ed. N.J.:Prentice-Hall, 2009.
Page 16 of 16
Problem 1(a)(b)
Problem 2TheorySolutionLow-passBandstop
Problem 3(a)(b)TheorySolution
Problem 4(a)(b)(c)