ECE 4990/5990 Design of RF Filters:

1

• ECE 4990/5990 Design of RF
  Filters
• Why we need filters
• To filter out unwanted signals.
• Low pass, high pass, band pass and band stop
  filters.
• Transmission Based Line Filters

Z
Zin
Y
Zin Z 1/Y Zin Zin Z Z2
4(Z/Y)0.5/2 Z/21 14/(ZY)0.5 ZjwL, YsC
Zin (jwL/2) 1-1-4/w2LC0.5 At low
frequencies, Zin (L/C) 0.5, the characteristics
impedance Zin k, and this is called constant k
filters.
2
As the frequency increases, the characteristic
impedance becomes negative resulting in no energy
transfer. The frequency at which 1-4/w2LC0 is
called the cut off frequency, Wh 2/LC0.5 This
assumes infinite number of sections. But filters
have finite sections. Difficult to control roll
off and control the characteristics of the
line. More sections, the better it is.
L/2
L/2
L/2
L/2
L/2
L/2
C/2
C
C
C/2
Two Cascaded T sections
3
L/2
L/2
L/2
L/2
L/2
L/2
C
C
C
Two Cascaded p sections
Design Equations Wh, cut off frequency, R is the
characteristic impedance, C is the
capacitance C(2/wh)1/R , L(2/wh)R R50 ohms,
wh 500 MHz x2p 3000 Mradians/sec C
(2/3000)1/50 2/150000 13pf L (2/3000)50
1/30 33 nH.
4

• Low-pass m-derived filter using two
  cascaded T-sections
• m-derived filter has a response generally
  attenuates more strongly as the cut off frequency
  w1 is approached.
• The effect of Capacitance is reduced by adding
  a series inductor.

L1/2
L1/2
L1/2
L1/2
L1/2
L1/2
2L2
L2
2L2
L2
C1/2
C1
C1
C1/2
Two Cascaded T sections
5
The cut-off frequency for a m derived section is
given by w1 2(R/L1) /4(L2/L1) 1 w1
2m/L1C10.5 R L1/C10.5 L2 (1-m2)R/2mw1 C1
(2m/w1)(1/R) L1 (2m/w1)R
6
C2
C2
2C2
2C2
L1/2
L1
L1
L1/2
C1
C1
C1
C1(2m/w1)1/R L1(2m/w1)R C2 (1-m2)/2mw1R
7
High-Pass
Filters
8
Band Pass
Filters
C1
C1
L1
L1
L1
C1
2L2
2L2
L2
C2
C2/2
C2/2
L2
C2
L1 2R/(w2-w1) C1 (w2-w1)/2wo2R C2
2/R(w2-w1) L2 (w2-w1)R/2wo2
9

• Filter Classifications
  and Specifications
• Low Pass b) Band pass c) High Pass d) Band Stop
  (Band Eject)
• Parameters a) band width b) shape factor (skirt,
  selectivity) c) ripple.

S6/60 Dw60/Dw6
-6dB
Dw6
-60dB
Dw60
1
H(jw)2
Transition band
1/(1e2)
Passband
Stopband
1/As2
wp
ws
10
Butterworth Filters The Butterworths response
magnitude as a function of frequency H(jw)2
1/1e2(w/wp)2n Where the band edge wp is the
frequency at which the power attenuation is
(1e2) The parameter n is the order(or degree)
of the filter and it is equal to the number of
independent energy storage elements as well as to
the power of w with which the response magnitude
finally rolls off. The 3-dB frequency wc is
1/1e2(wc/wp)2n 1/2 wpewc The filter
order can be determined depending by the
attenuation in the stop band edge, 1/As2
1/1e2(ws/wp)2n n ln(As2-1)0.5/e/lnws/wp ln
(As/e)/ln(ws/wp)
11

• Design of
  Filter
• We need to design a Butterworths filter with 1 db
  loss (0.794 gain) at the pass band edge of 1 GHz
  and require a 30 db attenuation at a 3 GHz stop
  band edge.
• 10 1dB/10 - 1 0.5088
• For the stop band specification,A2 1000, n
  ln(9990.5/0.5088)/ln 3 3.76
• n4
• bk 2(e1/n)sin (2k-1)p/2n where k ranges from
  0 to 1.
• k ranges from 1 to n, for k1
• b1 2(0.50881/4) sin (2-1)p/2x4 2x0.844x0.3826
  0.645.
• Lk (R/wp) bk , L1 R.b1/ 2pf
  50x0.645/(2x3.14x109) 5.135nH.
• Ck bk/2pfR 1x0.645/(2x3.14x109x50) 2.0576 pF.
• L first C first
• L1 5.1441nH C1 2.0576pF
• C24.9675pF L2 12.419nH
• L3 12.419nH C3 4.9675 pF
• C42.0576pF L4 5.1441 nH

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Low Pass Filter
L first
L1
L3
C4
C2
13
Band Pass Filters Butterworth type
Bandstop
Bandpass
Low Pass
High Pass
(BW)L/wo2
1/(BW)L
1/woL
L/BW
L/wo
BW/Lwo2
1/(BW)C
BW/wo2C
C/BW
C/wo
1/woC
Design a Band Pass Filter with BWDf100 MHz,
fo1GHz. BW2p(f2-f1) fo(f1f2)0.5
L1
L3
C4
C2
LPF
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C3s
L1s
L1
L3
C2s
L3s
LPF
L2s
L4p
C4
C2
C2p
C4p
L1 5.1441nH C24.9675pF L3 12.419nH C42.0576pF
L1s Lwo/BW 5.1441x1e9/.1e9 51.44nH C2s
BW/Lwo2 0.1/5.1x6.2x1e9 e-10/5.1x6.2x6.23pf/6.
20.5pf L3s Lwo/BW 12.419/0.1 124.1nH C3s
BW/Lwo2 0.1/12.41x(6.2)6.2x1e9 0.21pf C2p
C/BW Cwo/BW 4.97/.1 49.71pF. L2S BW/wo2C
0.1/(6.2 e9)5e-12x6.2e9 520 pH C4p C/BW
Cwo/BW 20.57pF L4p BW/wo2C
0.1/(6.2e9)2e-12x6.2e9 1.3nH
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• 
  Chebyshev Filters
• Mathematically, the Chebyshev response is of the
  form
• H(jw)2 1/1 e2Cn2((w/wp)
• At wwp, the H(jw)2 1/(1e2)
• is known as the ripple parameter and specified
  by the designer.
• The Chebyshev polynomial
• Cn(x) 2x Cn-1(x) Cn-2(x)
• Co1 and C1 x
• C2 2xC1-C0 2x2-1, C3 2x(2x2-1)-x4x3-3x
• Another method of generating Chebeshev polynomial
  is in terms of some trignometric functions from
  which the oscillation between -1 and 1 (FOR
  X,1)
• Cn(x) cos(ncos-1x) for xlt1
• For arguments larger than unity, Cn(x)
  cosh(ncosh-1x) for xgt1
• x cost, tcos-1x
• y cosnt , y cos (ncos-1x)

16
The relationship between the stop band frequency
ws, gainA(s)and order n is given by n
cosh-1(as2-1/e)/cosh-1(ws/wp)
Cosh-1(As/e)/cosh-1(ws/wp) The Chebyshev is
superior to Butterworth in the stop band. At
high frequencies, a Butterworth with e1 provides
an attenuation A(jw/wp)2 (w/wp)2n For
Chebyshev, A(jw/wp)2 22n-2(w/wp)2n Computation
of filter elements b sinh(tanh-1(1/(1e2)0.5/n)
The element values (normalized to 1 rps and 1
ohm) are then, c1b/b Ck bkbk-1/ck-1 (b2
sin(k-1)p/n2) bk 2sin(2n-1)p/2n
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Design of Chebyshev Low Pass Filters Normalized
element values for 1.0dB ripple Chebyshev Low
Pass filter C1(L1) 2.024 L2(C2) 0.994
C3(L3) 2.024 Low pass filter design Design a
low pass filter with 1 dB pass band response,
pass band edge at 1 GHz, stop band edge at 3 GHz,
gain 30 dB. n2.73 3 order. Lk (R/wp) Ck , L1
50x2.024/(6.2x1e9) 16e-9 H 16nH. Ck Ck/6.2fR,
C2 0.994/6.2x1e9x50 3pf L3 50x2.024/6.2x1e9
16nH.
L316n
L116n
LPF
C23pF
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Design of Bandpass Chebyshev Filters
173fF
L1160 n
L3160n
L316n
173fF
L116n
C230 pF
LPF
BPF
L20.8nH
C23pf
C2 Cwo/BW 3pf/0.1 30pF L2 BW/wo2C
0.1/(6.2Xe9)2x3e-12 0.8nH.
L/BW
L/wo
BW/Lwo2
C/wo
BW/wo2C
C/BW
19
Microstrip, stripline and Planar
Passive Components General Characteristics of PC
Boards Metal thickness I ounce of copper
corresponds to 1.34 mil or (35 micron in
thickness). 2 Oz(70 micron) and 0.5 Oz(17.5
micron) are also common thicknesses. The
resistivity of bulk copper 1.8 mW- cm and the
corresponding sheet resistance is 0.5mW per
square. The skin depth of copper 2.1 mm at 1
GHz. And sheet resistance is about 8mohm per
sq. Usually, the resistivity of film copper is
higher than bulk copper by a factor of 2. The
resistance depends on surface roughness, Fsr 1
2/ptan-11.4(D/d) where D is the surface
roughness , d is the skin depth. Common
Dielectric thicknesses are 1/32(0.8mm),
1/16(1.6mm)or 1/8 (3.2mm) In multilayer boards,
the thickness of dielectric may be 1/64. Most
common PC board used at low frequencies is FR-4.
The typical loss is 0.03dB/cm/GHz for 1/16. The
dielectric constant of FR- varies from 4.2 to 4.7
at 1 GHz. Suitable upto 5GHz.
20
Characteristics of PCBs The dielectric
constant of PCB increases with increase in
frequency dispersion. The dielectric constant in
FR-4 increases by 5 from 100 MHz to 5 GHz.
Other materials are from Rogers Corporation. In
addition, for high performance one can use
saphire, allumina, beryllia and quartz.
Transmission lines on PC Boards
W
W
T
T
H
H
Microstrip
Stripline
The charactersitics of strip line 1) Stripline
is nearly self shielding minimizes couling and
radiation losses. 2) TEM propogation
Disadvantage Making connections to center
conductor. The Charactersitic Impedance of
stripline, Zo( 60/er )ln6H/pW(0.8T/W)
21

• The above formula is valid for narrow lines i.e.
  lines defined by
• W/(H-T),0.35 for a total dielectric thickness of
  1/8 using 1 Oz copper, a 50 ohm line requires a
  conductor of approximately 1.25mm.
• For more accurate Zo is given by Cohns
  expression
• Zo hK(k)/4(er)0.5 K(k) where k
  cosh(pW/2H) -1
• is the impedance of free space, (mo/eo)0.5120p
• K(k) /K(k) 1/p ln(2(1k0.5)/(1-k0.5) for
  0ltklt0.7070
• 1/p ln(2(1k0.5)/(1-k0.5)
  for 0.7070ltklt1
• and k (1-k2)0.5
• Equations for Microstrips
• For microstrips, Zo (mr mo/ereo)0.5 (H/W)
  11.735er-0.0724(W/H) -.836 -1
• If the thicknesses of the metals are
  included,replace W by Weff WT/pln(2H/T) 1
• Disadvantages
• Energy is delivered to substrates leading to loss
  if the substrate is dissipative.
• Ground plane is not easily accessible.

22
Line-To-Line Discontinuities
Discontinues in the transmission Line can result
higher excitation Modes.
H/2
W
H/2
Ceq e(WH/2 H2/4)/H e (W/2 H/4)
Zo
The effective length extension Dl is given by,
Dl/H 0.412(ere 0.3)(W/H 0.262)/(ere-0.258)(W
/H0.813) ere, effective realtive dielectric
constant (er 1)/2 (er-1)(1/2)(110H/W) -0.5
Excess capacitance due to sharp bend.
Use Chamfered bend. For a right angle bend 1.8W
area.
23
Coplanar Waveguide (CPW) and Coplanar
Strip
W
W
w
S
S
Advantages of Coplanar Structures a) Both ground
and signal lines are accessible from the top
surface. Probing is easy. b) Less loss. The
characteristic impedance impedance Zo
hK(k)/4(ee)0.5K(k) for CPW

hK(k)/(ee)0.5K(k) for CPS k S/2/S/2
W ee 1 (er-1)/2 K(k) K(k1)/K(k)K(k1) In
turn, k1 sinh(pS/4H)/sinh(pS/2 W/2H) K(k)
/K(k) 1/p ln(2(1k0.5)/(1-k0.5) for
0ltklt0.7070 1/p
ln(2(1k0.5)/(1-k0.5) for 0.7070ltklt1 and k
(1-k2)0.
24
Line
Discontinuities
W2
b 0.4W12W220.5
b
W1
q tan-1(W1/w2)
Circular bend with radius greater than 3 times
the width.
The capacitance discontinuity (W2-W1)l
l
W1
W1
W2
l
Tapered transition can be used to reduce the
energy stored due to higher order modes.
Inductive notched line.
25

Line Discontinuities Effects The inductance due
to discontiuity, L lZo,n(er,eff,n)0.51-(Zo,w/Z0,
n)2/c Where the subscripts n,w stand and wide,
respectively. T Junction Shortening The
shortening (due to capacitive coupling)occurs at
the junction of a wide and narrow line.
Dl/H 120p/Zl(ere,ser)0.5-0.16(Z1/Z2)(1-2lnZ1/Z
2)
Z1
Transitions between connectors and transmission
lines
Z2
26
Passives made from Transmission line
Segments Z(l)/Zo (ZLn jtanbl)/1jZLntanbl
where Zln is the normalized load impedance Where
b is an imaginary part of the propogation
constant, g bImg wLC0.5w/v er,eff
10.63(er-1)(W/H)0.1255 for W/Hgt0.6
10.63(er-1)(W/H)0.1255 for W/Hlt0.6 For a 50
ohm line on FR4, W/H2 and the effective
dielectric constant is about 3.5(if the actual
bulk er4.5). If the load impedance goes to
infinity, Z Zo/jbl Zo/jw(l/v) The capacitance
is therefore, Cl/vZo l(er,eff)0.5/cZo The time
constant Zoc is simply the one-way time of
flight. FR-4 offers a capacitance of 2.5pF/cm2. A
short segment terminated in a short circuit will
appear inductive since we have a Current
loop. ZZo bl lZo(er,eff)0.5/c The line
dimensions of components must be smaller than
wavelength.
27

Resonators Input Impedance of a line that
is l/4 in extent The impedance infinity. If the
length is greater than l/4, the impedance is
capacitive. If the length is less than l/4, the
impedance is inductive. The Q of the resonator
inductive or capacitive reactance/real part of
impedance
Z(l)/Zo (Zln tanhgl)/(1 Zln tanhgl)
1/tangl
1/tanh(ajb)l If alltlt1, input resistance, Reff
Zoal Zo(al/4) L Q woL/Reff
28
Combiners Combiners are used to combine
signals from multiple sources to create a single
output. They can also be used as power
splitters. Resistive Splitter/combiner
V1
V1
Zo/3
3Zo
3Zo
Zo/3
Vin
Vin
3Zo
Zo/3
2Zo
V2
V2
Distributed Combiners
2Zo
Zo
Zo
2Zo
2Zo
29
Zo
l/4
Z
Z0
R
l/4
Zo
Z
30
Hybrids and Baluns
Classic telephone Hybrid
Port3
Through Output
Input
line
A
C
Port1
Port2
Isolated output
R
D
Auxiliary(coupled) output
B
talk
listen
Through
Input
Coupled
Isolated
Symbols for hybrid couplers
31
C(null ouptput) Zo
B, Output1, Zo
Circumference of the circle is 3l/2.
S
Input signal at A, when enters the hybrid, at
point B, the clockwise signal traverses l/4and
anticlockwise, and conuter clockwise
5l/4(3l/2-l/4) Therefore signals add up.
Input A, Zo
D
Zo, D Output2
Ring (or rat race ) hybrid
32
Lumped 180 Splitter/combiner
If Zo is the source resistance driving the input
port, the input impedance of each filter is
2Zo. ZL is the load impedance from each output to
ground, The characteristic impedance of each arm
is given by (L/C)0.5 (2ZoZL)0.5 (L/C)
2ZoZL wo 1/(LC)0.5 C 1/(wo)(2ZoZL)0.5 L
(2ZoZL)0.5/wo
Vlag
L
2ZL
C
C
Vin
C
C
Vlead
L
For a 1 GHz hybrid driven by 50 ohm, and
terminated in 100 ohm, 50 ohm from each output to
ground, the component values are about 11.3nH and
2.25 pF.
33
Lumped Coupler l/4 sections are replaced by
low pass p sections and l/2 sections are replaced
by high pass T network.
B
C
L1
C1
C1
C1
C1
L1
L1
C1
C1
A
D
C1
C1
If all the inductors and capacitors are equal,
the ring impedance at the center frequency of
operation C1 1/(woZo20.5) and
L1Zo20.5/wo Lumped versions have the same narrow
bandwidth (10 to 15) as the distributed
versions.
34
Branch Line Hybrid
A(input)
B(output 1)
Zo
Zo
Zo/20.5
l/4
Zo
Zo
D(output 2)
C(isolated)
Zo/20.5
Zo
Zo
l/4
Branch line hybrids are used to generate(or
combine) two signals that are in quadrature
rather than in antiphase with each other.
Zs
Zs
Zo
Zo
Zo
Zo
B, (output 1)
A(input)
B, (output 1)
A(input)
Zp
Zp
Zp
Zp
Common mode
D,(output 2)
C(isolated)
Zo
Zs
Zo
35
Lumped implementation of 90 hybrid
C
L1
C1
C2
C2
C1
L2
L2
C2
C2
A
D
L1
C1
C1
C
L1
C1
C2
C2
C1
L2
L2
C2C1
A
D
L1
C1C2
36
Directional Couplers Directional couplers are
four port devices capable of splitting power by
prescribed amounts that generally differ
considerably from 11. They are used for feed
back in power amplifiers They are used to resolve
forward and reverse signals. Figures of merit for
a directional coupler include coupling factor,
isolation and directivity. The coupling factor is
defined as the ratio of input power to the power
delivered to the coupled (auxiliary) port. CF
PIn/PAuxforward Typical coupling factor is from
3dB to 20 dB. If the directional is operated in
reverse, with power now supplied to the through
output with input terminated, some power leaks
to the auxiliary port. A measure of how well the
reverse leakage is suppressed is the isolation
factor. I Pin/Paux reverse Isolation factors
of 30-60 dB are not uncommon for commercially
available units. The directivity D I/CF
Pauxforward/Pauxreverse
37
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