Introduction to RF
Filter Design
RF Electronics
Spring, 2016
Robert R. Krchnavek
Rowan University
Objectives
•
Understand the fundamental concepts and
definitions for filters.
•
Know how to design filters using tabulated
parameters for common filter types.
•
Know how to convert lumped-element filter
designs into distributed-element filters.
Filter Configurations
ω
Ω=
ωc
where ωc is defined as the cutoff frequency for
low-pass and high-pass filters and the center
frequency for bandpass and bandstop filters.
Low-Pass Filters
Profiles for Three Common Types
•
•
•
Binomial or Butterworth - easy to implement; monotonic profile; requires
numerous elements to get step profile.
Chebyshev - equal amplitude variations; steeper profile than Butterworth.
Elliptic or Cauer - amplitude variations in both stopband and passband;
steepest profile; complicated to design.
Bandpass Filter - Profile
Filter Definitions
•
•
•
•
•
•
Pin
Insertion loss - how much power
IL = 10 log
is lost in going through the filter.
PL
Ripple - the flatness of the signal
in the passband.
Bandwidth - the width of the
BW 3dB = fU3dB − fL3dB
passband.
Shape factor - the sharpness of
60dB
BW
the filter response.
SF =
BW 3dB
Rejection - the attenuation of
the undesired signals.
fU60dB − fL60dB
= 3dB
Quality factor - see next slide.
3dB
fU
− fL
Q - Quality Factor
The quality factor, or Q, is a parameter that is used to
describe the selectivity of the filter.
The unloaded Q is defined as
Q = 2πfC
!
maximum energy stored in the filter at fC
power lost in the filter
"
The loaded Q is defined as
QLD = 2πfC
!
maximum energy stored in the filter at fC
power lost in the filter and to the external circuit
A higher Q indicates a more selective filter. Details to follow.
"
Series RLC Bandpass Filter
Find VR1 for this circuit.
Series RLC Bandpass Filter
Q = 2πfC
!
maximum energy stored in the filter at fC
power lost in the filter
"
1 2
LIp = maximum energy stored in the filter at fC
2
2
Irms
R = power lost in the filter
2
Irms
1 2
= Ip
2
L
L
Q = 2πfC = ωC
R
R
The resonant frequency is the
frequency where the imaginary
component of the impedance is
1
equal to 0:
ȷωC L +
ȷωC C
=0
1
ωC = √
LC
Series RLC Bandpass Filter
Solving for the frequencies at which VR1 is down 3 dB yields
R
ωU =
+
2L
!"
R
2L
#2
1
+
LC
and
−R
ωL =
+
2L
!"
R
2L
#2
The bandwidth is given by
R
BW = ωU − ωL =
L
and
1 R
BW = fU − fL =
2π L
And, using our previous result for unloaded Q, we see the
relationship between Q and BW is given by
fC
L
Q = 2πfC =
R
BW
or
fC
BW =
Q
1
+
LC
Series and Parallel Resonators
Quantitative Analysis of a Series RLC
Bandpass Filter
VL
H(ω) =
VG
VL
ZL
H(ω) =
=
VG
(ZG + ZL ) + R + ȷ [ωL − 1/(ωC)]
This filter is different from the previous one
because of the addition of ZG and ZL.
Series RLC Bandpass Filter
where RE = ZG + ZL
Define three different Q factors:
Unloaded, internal, or filter Q
External Q
Loaded Q
ωC L
QF =
R
ωC L
QE =
RE
QLD
ωC L
=
R + RE
Series RLC Bandpass Filter
Note: ZL=ZG=50Ω, R=20Ω, L=5 nH, and C=2 pF.
Series RLC Bandpass Filter
•
•
•
Series RLC bandpass filter is easy to analyze.
Minimum attenuation at the resonance point.
HOWEVER, the transition from passband to stop
band is not very sharp (large shape factor.)
Butterworth Filter
•
One of a series of special filter designs that
consist of more elements than a simple RLC and
give better control over the filter parameters.
•
•
Also known as a maximally flat filter - no ripple.
Strategy
•
•
First, do the normalized, low-pass filter.
•
Third, if necessary, create distributed
elements.
Second, implement the desired form through a
frequency scaling.
Butterworth Filter
gm
gN +1 =
!
⎧
⎫
⎨ inductance for series inductor ⎬
capacitance for shunt capacitor
=
⎩
⎭
(m ≡ 1, . . . , N )
load resistance if the last element is a shunt capacitor
load conductance if the last element is a series inductor
Two different networks that are used to implement the
Butterworth filter.
"
Butterworth Filter
!
"
Pin
2 2N
IL = 10 log
= 10 log 1 + a Ω
PL
The coefficient “a” is usually taken to be 1 so that the IL is 3
dB at the cutoff frequency.
Butterworth Filter
The attenuation vs frequency as a function of the number
of stages. Note: this design does NOT result in a linear
phase relationship.
Butterworth Filter
Coefficients for a maximally flat response.
Butterworth Filter
Coefficients for a linear phase response.
Comments
•
Coefficients for a 3 dB Chebyshev filter design
are in Table 5.4 (a).
•
Coefficients for a 0.5 dB Chebyshev filter design
are in Table 5.4 (b).
•
The generic, multisection, normalized element
circuits designs are the same for Butterworth
and Chebyshev filters.
Butterworth Example
Design a 4th-order, low-pass, standard (maximally flat),
3 dB Butterworth filter.
Frequency and Impedance
Transformations
•
The normalized values need to be modified to
produce
•
The desired response (low-pass, high-pass,
etc.)
•
•
At the desired center frequency.
With an impedance that is realistic.
Frequency Transformation
•
All of the different filter types are derived
from the low-pass filter.
•
The key is to determine a transformation
function that maps the normalized, low-pass
design into the appropriate (low-pass, high-pass,
etc.) at the desired frequency.
•
New values for L and C are obtained by
maintaining the same Z through the
transformation.
Frequency Transformation – Low Pass
ωc is the new cutoff frequency
! = ⌦!c
The impedances should remain the same:
!
ZL = |⌦L = | L = |!Lnew
!c
L
Lnew =
!c
1
!c
1
ZC =
=
=
|⌦C
|!C
|!Cnew
Cnew
C
=
!c
Normalized Low-Pass
Low-Pass
ω = Ωωc
ω
ZL = ȷΩL = ȷ L = ȷωLnew
ωc
L
Lnew =
ωc
ωc
1
1
=
=
ZC =
ȷΩC
ȷωC
ȷωCnew
C
Cnew =
ωc
Frequency Transformation – High Pass
ωc is the new cutoff frequency
!c
!=
⌦
Again, the impedances should
remain the same giving . . . .
Normalized Low-Pass
−ωc
ω=
Ω
Cnew
1
=
ωc L
Lnew
1
=
ωc C
High-Pass
Frequency Transformations
Bandpass and Bandstop
•
These transformations are more complex. See
the textbook for both.
Frequency Transformations
Summary
Impedance Transformation
•
For the Butterworth designs, the source and load
resistances have a value of 1.
•
For the Chebyshev designs, even-number ordered
designs have a non-unity load.
•
Impedance transformation is the process of
adjusting all the elements to account for
different source and load impedances.
Impedance Transformation
Assume the source impedance, RG, is scaled from 1 in the
original design to RG,new. Then, the new values are:
RG, new = 1RG, new
Lnew = LRG, new
C
Cnew =
RG, new
RL, new = RL RG, new
Distributed-Element Filters
•
Above approximately 1 GHz, lumped-element
filter design is problematic because the elements
are approaching a significant fraction of λ.
•
•
Distributed-element filters are common.
One approach is to design the lumped-element
filter and then convert it to a distributedelement realization.
Distributed-Element Filters
•
Assume you have a lumped-element filter design
that you want to build as a distributed-element
filter.
•
Recall our expression for the impedance of a
terminated (lossless) transmission line:
ZL + ȷZ0 tan βd
Z(d) = Z0
Z0 + ȷZL tan βd
•
If ZL = 0, then
Z(d) = ȷZ0 tan βd
•
If ZL = ∞, then
Z(d) = −ȷZ0 cot βd
•
The electrical length, βd, can be put in the
following form
2π
2πf
2π
d=
d=
d
βd =
λ
vp /f
vp
•
Assume we chose a line that is 1/8 of a
wavelength
d = λ/8
•
The expression for impedance explicitly in terms of
frequency is then (ZL = 0)
π f
2πf vp
= ȷZ0 tan
Z(d) = ȷZ0 tan
vp f0 8
4 f0
•
The impedance of the stub must equal the impedance
of the lumped element
π f
ȷωL = ȷZ0 tan
4 f0
•
For a capacitive element, you could use the opencircuited transmission line.
•
One significant difference is the frequency range is
shortened because the tan function is periodic.
•
For a d =
short-circuited line, we have the
following for the Richard’s transform:
•
For a d=
open-circuited line, we the
following for the Richard’s transform:
Distributed-Element Filters - Physical
Realization
•
Using transmission line sections to build the filter
may require sections of line that separate
elements from each other. These are called unit
elements.
•
The unit elements have an electrical length of
π f
βd =
4 f0
•
We also need to be able to create distributed
element sections for difficult-to-replace lumped
elements such as series inductors. We use
Kuroda’s identities for this.
Distributed Filter Implementation
Design a 4th-order, low-pass, standard
(maximally flat), 3 dB Butterworth filter. It
should have a cutoff frequency of 1 GHz.
1. Select the normalized filter order and parameters
to meet the design criteria.
2. Replace inductances and capacitances with
equivalent λ/8 transmission lines.
3. Convert series stub lines to shunt stub lines
through Kuroda’s identities.
4. Denormalize and select equivalent microstrip lines.
Distributed Filter Implementation
1.8478 H
0.7654 H
G=1
0.7654 F
1.8478 F
1. Select the normalized filter
order and parameters to meet
the design criteria.
Ω
R=1Ω
Distributed Filter Implementation
Z0=1.8478
Z0=0.7654
R=1Ω
Y0=0.7654
Y0= 1.8478
2. Replace inductances and
capacitances with equivalent
λ/8 transmission lines.
Ω
G=1
Distributed Filter Implementation
R=1Ω
Z0=0.7654
ZUE=1Ω
ZUE=1Ω
UE
UE
Y0=0.7654
Y0=1.8478
3. Convert series stub lines to shunt stub
lines through Kuroda’s identities.
G=1
Ω
Z0=1.8478
Distributed Filter Implementation
Z0=1.8478
Z0=0.7654
ZUE=1Ω
UE
UE
Y0=1.8478
3. Convert series stub lines to shunt stub
lines through Kuroda’s identities.
G=1
Ω
R=1Ω
Distributed Filter Implementation
Z0=1.8478
R=1Ω
UE
Y0=1.8478
3. Convert series stub lines to shunt stub
lines through Kuroda’s identities.
G=1
Ω
UE
Distributed Filter Implementation
Z0=1.8478
UE
UE
UE
Y0=1.8478
3. Convert series stub lines to shunt stub
lines through Kuroda’s identities.
G=1
Ω
R=1Ω
ZUE=1Ω
Distributed Filter Implementation
R=1Ω
UE
UE
Y0=1.8478
3. Convert series stub lines to shunt stub
lines through Kuroda’s identities.
G=1
Ω
UE