Filter
Prof. Tzong-Lin Wu
EMC Laboratory
Department of Electrical Engineering
National Taiwan University
2013/5/14
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Accept desired signal and reject signal outside the operating band
Low-pass, high-pass and bandpass filters.
Insertion loss method will be introduced
The insertion loss method is based on network synthesis techniques, and can be used
to design filters having a specific type of frequency response. The technique begins
with the design of a low-pass filter prototype that is normalized in terms of impedance
and cutoff frequency.
Impedance and frequency scaling and transformations are then used to convert the
normalized design to the one having the desired frequency response, cutoff frequency,
and impedance level.
Additional transformations, such as Richard's transformation, impedance/admittance
inverters, and the Kuroda identities, can be used to facilitate filter implementation in
terms of practical components such as transmission lines sections, stubs, and resonant
elements.
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5.1 Filter design by the insertion loss method
For minimum loss: binomial response,
For sharpest cutoff: Chebyshev response
Linear phase needs sacrifices attenuation rate
Power loss ratio:
Thus, for a filter to be physically realizable its power loss ratio must be of the form.
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Maximally flat or binominal or Butterworth
This characteristic is also called the binomial or Butterworth response, and is optimum in
the sense that it provides the flattest possible passband response for a given filter
complexity, or order.
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Maximally flat or binominal or Butterworth
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Chebyshev response or equal ripple
Chebyshev polynomial is used to specify the insertion loss of an N-order low-pass filter
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Linear Phase
The above filters specify the amplitude response, but in some applications
(such as multiplexing filters in frequency-division multiplexed communications system).
It is important to have a linear phase response in the passband to avoid signal distortion.
A linear phase characteristic can be achieved with the following phase response:
which shows that the group delay for a linear phase filter is a maximally flat function.
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We will next discuss the design of low-pass filter prototypes which are normalized in
terms of impedance and frequency;
This type of normalization simplifies the design of filters for arbitrary frequency,
impedance, and type (low-pass, high-pass, bandpass, or bandstop).
The low-pass prototypes are then scaled to the desired frequency and impedance,
and the lumped-element components replaced with distributed circuit elements for
implementation at microwave frequencies.
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Maximally flat low-pass filter prototype
Consider the two-element low-pass filter prototype circuit shown in Figure 5.3; we will
derive the normalized element values, L and C, for a maximally flat response.
The desired power loss ratio will be, for N = 2,
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Maximally flat low-pass filter prototype
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Maximally flat low-pass filter prototype
Comparing to the desired response
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Maximally flat low-pass filter prototype
In principle, this procedure can be extended to find the element values for filters with
an arbitrary number of elements, N, but clearly this is not practical for large N.
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Equal ripple low-pass filter prototype
Desired response
Chebyshev polynomials have the property that
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Equal ripple low-pass filter prototype
Chebyshev polynomial of order 2 is given as
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Equal ripple low-pass filter prototype
Note that (5.14) gives a value for R that is not unity, so there will be an impedance
mismatch if the load actually has a unity (normalized) impedance;
this can be corrected with a quarter-wave transformer, or by using an additional
filter element to make N odd. For odd N, it can be shown that R = 1.
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Linear Phase Low-Pass Filter Prototype
Filters having a maximally flat time delay, or a linear phase response, can be designed
in the same way, but things are somewhat more complicated because the phase of the
voltage transfer function is not as simply expressed as is its amplitude. Design values
have been derived for such filters, however, again for the ladder circuits of Figure 5.4,
and are given in Table 5.3 for a normalized source impedance and cutoff frequency
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5.2 Filter scaling and transformation
Impedance scaling (1 -> R0)
In the prototype design, the source and load resistances are unity.
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Frequency Scaling for Low Pass Filter
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Frequency Scaling for Low Pass Filter
Both impedance and frequency scaling
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Low-pass to high-pass transformation
Both impedance and frequency scaling
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Bandpass and bandstop transformation
Fractional bandwidth
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Bandpass and bandstop transformation
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Bandpass and bandstop transformation
The new filter elements are determined
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Band stop transformation
Then series inductors of the low-pass prototype are converted to parallel LC circuits
having element values given by
The shunt capacitor of the low-pass prototype is converted to series LC circuits having
element values given by
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5.3 Low pass and high pass filters using
transmission line stubs
The lumped-element filters discussed in the previous sections generally work well at low
frequencies, but two problems arise at higher RF and microwave frequencies.
First, lumped elements such as inductors and capacitors are generally available only for
a limited range of values, and are difficult to implement at high frequencies.
In addition, at microwave frequencies the electrical distance between filter components
is not negligible.
Richard's transformation can be used to convert lumped elements to transmission line
stubs.
Kuroda's identities can be used to separate filter elements by using transmission line
sections. Because such additional transmission line sections do not affect the filter
response, this type of design is called redundant filter synthesis.
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Richard’s Transformation
This transformation was introduced by P. Richard to synthesize an LC network using
open- and short-circuited transmission lines.
if we replace the frequency variable ω with Ω, the reactance of an inductor can be
written as
and the susceptance of a capacitor can be written as
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Richard’s Transformation
Cutoff occurs at unity frequency for a low-pass filter prototype; to obtain the same
cutoff frequency for the Richard's-transformed filter,
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Richard’s Transformation
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Kuroda Identities
The four Kuroda identities use redundant transmission line sections to achieve a more
practical microwave filter implementation by performing any of the following operations:
The additional transmission lines are called unit elements, and are λ /8 long at wc
The unit elements are thus commensurate with the stubs obtained by Richard's
transform from the prototype design.
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Kuroda Identities
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Kuroda Identities
Prove identity (a)
For Fig. 5.14(a)
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For Fig. 5.14(b)
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5.4 Stepped-impedance low-pass filter
A relatively easy way to implement low-pass filters in microstrip or stripline form is
to use alternating sections of very high and very low characteristic lines. Such filters are
usually referred to a stepped-impedance.
Popular because they are easy to design and take up less space than a similar low-pass
filter using stubs.
Because of the approximations involved, however, their electrical performance is often
not as good as that of stub filters, so the use of such filters is usually limited to
applications where a sharp cutoff is not required, such as for rejection of out-of-band
mixer products.
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Approximate Equivalent Circuits for Short Transmission
Line Sections
The open-circuit impedance matrix elements for a transmission line of length l and
characteristic impedance Zo can easily be found as
The series elements of a T-equivalent circuit for the transmission line section are then
given as Zll - Z12 for the series arms, and Z12 for the shunt arm.
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Approximate Equivalent Circuits for Short Transmission
Line Sections
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Large Z0
Small Z0
the actual values of Zh and Ze are usually set to the highest and lowest characteristic
impedances that can be practically fabricated (the thinnest and widest lines, respectively).
The lengths of the lines can then be determined from (5.39) and (5.40); to get the best
response near cutoff, these lengths should be evaluated at w = wc
Combining the results of (5.39) and (5.40) with the impedance scaling equations of (5.15)
allows the electrical lengths of the inductor sections to be calculated as
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5.5 BANDPASS FILTERS USING TRANSMlSSlON LINE RESONATORS
Bandpass filters perform a variety of critical functions in wireless systems, being used
to reject out-of-band and image signals in the front end of a receiver, to attenuate
undesired mixer products in transmitters and receivers, and to set the IF bandwidth of
the receiver system.
Impedance and Admittance Inverters
Bandpass filter prototypes require shunt elements consisting of parallel LC resonators
and series elements consisting of series LC resonators.
Such an arrangement is very difficult to implement using transmission line sections, for
which it is preferable to have either all shunt, or all series, elements.
While the Kuroda identities are useful for transforming capacitors or inductors to either
series or shunt transmission line stubs, they are not useful for transforming LC
resonators.
For this purpose, impedance (K) and admittance (J) inverters can be used. Such
techniques are especially useful for bandpass and bandstop filters having narrow (t
10%) bandwidths.
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Impedance and Admittance Inverters
An impedance inverter converts a load impedance to its inverse, while an admittance
inverter converts a load admittance to its inverse:
where K is the impedance inverter constant, and J is the admittance inverter constant.
They can be used to transform between series-connected and shunt-connected
elements. Thus, a series LC resonator can be transformed to a parallel LC resonator,
or vice versa.
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Impedance and Admittance Inverters
In its simplest form, a K or J inverter can be constructed using a quarter-wave transformer
of the appropriate characteristic impedance, Fig. (b)
Others: T network (K inverter), or a pi network (J inverter), of capacitors. Fig. (c)
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Practical Realization of Immittance Inverters
Z1
Z2
Z3
proof
+900
 Z1
1  Z
3

 1
 Z
 3
Z1Z 2 
 0
Z3  
 1
Z2
 
1
  j L
Z3

Z1  Z 2 
-900
 j L 

0 

-900
+900
Immittance inverters comprised of lumped and transmission line
elements
A circuit mixed with lumped and transmission line elements
Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
Since quarter-wave short-circuited transmission line stubs look like parallel resonant
circuits, they can be used as the shunt parallel LC resonators for bandpass filters.
Quarter wavelength connecting lines between the stubs will act as admittance
inverters, effectively converting alternate shunt stubs to series resonators.
For a narrow passband bandwidth (small Δ), the response of such a filter using N
stubs is essentially the same as that of a lumped element bandpass filter of order N.
The circuit topology of this filter is convenient in that only shunt stubs are used, but a
disadvantage in practice is that the required characteristic impedances of the stub lines
are often unrealistically low. A similar design employing open-circuited stubs can be
used for bandstop filters
How to design Z0n ?
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Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
Note that a given LC resonator has two degrees of freedom: L and C, or equivalently, ω0,
and the slope of the admittance at resonance.
For a stub resonator the corresponding degrees of freedom are the resonant length and
characteristic impedance of the transmission line Z0n .
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Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
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These two results are exactly equivalent for all frequencies if the following conditions are satisfied:
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Bandpass Filters Using Capacitively Coupled
Quarter-Wave Resonators
An Nth order filter will use N stubs, which are slightly shorter than λ/4 at the filter center frequency.
The short-circuited stub resonators can be made from sections of coaxial line using ceramic materials
having very high dielectric constant and low loss, resulting in a very compact design even at UHF
frequencies [6].
Such filters are often referred to as ceramic resonator filters, and are presently the most common
type of RF bandpass filter used in portable wireless systems.
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Admittance inverter constant J
– 2 order bandpass filter circuit
J01
-90o
Zo
L1
Zin
J01
-90o
Zin 
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L1
C1
Z2 
Zin
Zo
L2
J12
-90o
J23
-90o
C2
Y3 
Z2
J01
-90o
Zo
C1
J12
-90o
L2
C2
R=Zo
2
J 23
2
 J 23
Zo
Yo
Y3
YL

1 YL
1 
1
 2  2  jC2 
 Y3 
Y2 J12 J12 
j L2

L1
C1
Z2
1 YLoad
1  1
1 
 2  2 
 jC1  
Yin
J 01
J 01  j L1
Z2 
MW & RF Design / Prof. T. -L. Wu
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g1
J01
-90o
Zo
L1
g2
C1
Lowpass to
Bandpass
transformation
Y
1 
1
1 
Zin  Load
 2  jC1 
 
2
J 01
J 01 
j L1 Z 2 
C1 J 012
g1Z o
o 
J 012 L1
G
J 122
L2
J 012

g1Z oo
J 012 2
J 23Z o
J 122
R  g3 Z o
Zo 
g 2o
J 012
C2
J 122
Equivalent L/C of Short-circuited stub : L 
g3
4Zo
o
C
g2
Z oo

4o Z o
The two results are exactly equivalent for all frequencies :


4o Z o g1Z o
Z
J


o 01
4 g1
J 012
o 

J122 4 Zo Zo 
Z
J


o 12
4 g1 g2
J 012 o g2o
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1
J122

 2 2
 g3Z o
Z o J 23 
G J 01MW
J 23Z&o RF Design / Prof. T. -L. Wu 4 g2 g3
75
π-network for realizing BPF
Bandpass filter structure
J01
L1
J12
C1
C2
L2
CN JN, N+1
LN
π-network of J-inverter
C
-C
-C
J=ωC
J-inverter
Circuit implementation
C01
-C01
-C01
CN, N+1
C12
L1
C1 -C12
-C12
L2
C2
LN
-CN, N+1
CN
-CN, N+1
How to deal with the J-inverter at the two ends due
to the negative capacitors next to the terminations?
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The coupling capacitor values at
input/output end for matching (I)
BPF circuit with admittance inverter
BPF circuit with capacitor circuit
Cx
Zo
J01
L1
G
J 012
 J 012 Z o
YL
C1
Cy
Zo
Y  j0C y 
 j0C y 

1
1
 Zo
j0C x
 j0C y 
1   Z o0C x 
2
1   Z o0C x 
MW & RF Design / Prof. T. -L. Wu
C1
j0C x  Z o 0C x 
Z o 0C x 
Equating G to the real part of Y :
2
Z

C
2
2


J 012  0Cx  1  J x2  Zo  
J 012 Z o  o 0 x 2


1   Zo0Cx 
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L1
2

j0C x
1  jZ o0C x
2
2
j0C x
1   Z o0C x 
0Cx 
2
 j0C y  G  jB
J 01
1  J 012  Zo 
77
2
The coupling capacitor values at
input/output end for matching (II)
The imaginary part of Y should be eliminated:
2
Zo 0Cx 
j0Cx
Y

 j0C y  G  jB
2
2
1   Zo0Cx  1   Zo0Cx 
j0Cx
1   Zo0Cx 
Note:
2
 j0C y  0
Cy  
Cx
1   Zo0Cx 
2
In additional to the modified coupled capacitor Cx, the shunt capacitor Cy is also revised. The Cy is
not directly equal to the Cx unless ω0Cx << 1 and this is the assumption in the textbook.
C01
Textbook
-C01
L1
C1 -C12
C01
Accurate
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-C’01
'
C01

CN, N+1
C12
-C12
L2
C2
LN
CN
CN, N+1
C12
L1
C1 -C12
C01
1   ZoC01 
2
-C12
-CN, N+1
L2
C2
MW & RF Design / Prof. T. -L. Wu
LN
CN
-C’N, N+1
78
Trick for the final circuit for implementing
the BPF
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Third order BPF with 0.5 dB Chebyshev prototype
g1 = 1.5963, g2 = 1.0967, g3 = 1.5963, g4 = 1
f0 = 1 GHz, FBW = 5 %, Zo = 50
Difference on the two circuits
Ideal
J01
L
C
C01
Accurate
-C’01
J12
C
L
L
C
J34
C34
C23
-C12
C
L
C
L
-C23
-C’34
40
Ideal
Textbook
3.8
Ideal
Accurate
3.8
3.6
-10
f3/f1 (dB)
S-parameter
3.6
-10
f3/f1 (dB)
S-parameter
C
L
C12
40
3.4
3.2
-20
3
2.8
-30
2.6
3.4
3.2
-20
3
2.8
-30
2.6
2.4
2.4
-40
2.2
0.9
0
J23
0.1
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0.2 0.95
0.3
0.4
1
0.5
0.6
Freq(GHz)
u=
2/( 1+ 2)
0.7 1.05
0.8
0.9
1.1
1
-40
2.2
0.9
0
0.1
0.2 0.950.3
MW & RF Design / Prof. T. -L. Wu
0.4
1
0.5
0.6
Freq(GHz)
u=
2/( 1+ 2)
0.7 1.050.8
0.9
80
1.1
1
The transformation of the stub length to account for the change in capacitance is
illustrated in Figure 5.25d. A short-circuited length of line with a shunt capacitor at its
input has an input admittance of
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Dielectric material properties play a critical role in the performance of dielectric resonator
filters. Materials with high dielectric constants are required in order to provide miniaturization at the
frequencies typically used for wireless applications.
Losses must be low to provide resonators with high Q, leading to low passband insertion loss and
maximum attenuation in the stopbands.
And the dielectric constant must be stable with changes in temperature to avoid drifting of the filter
passband over normal operating conditions.
Most materials that are commonly used in dielectric resonator filters are ceramics such as Barium
tetratitanate, Zinc/Strontium titanate, and various titanium oxide compounds.
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