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Lecture 11
Passive Components(Capacitors, Inductors, Terminations, Attenuators,Power Dividers, Directional Couplers and Hybrids)
Acknowledgement: Some diagrams are from M. Steer’s book “Microwave and RF Design” and from D. Pozar’s book “Microwave Engineering”
Lumped Capacitors
ElecEng4FJ4 2L12: PASSIVE COMPONENTS
lumped capacitors are used in RF and microwave ICs• on-chip capacitors• standalone chip capacitors for HMICs
frequency range – up to several GHz types of chip capacitors
• metal-dielectric-metal• metal-dielectric-semiconductor (structure of MOS transistor)• semiconductor junction (reverse biased p-n or Schottky barrier)
Read about Walter Schottky in the Microwave Hall of Famehttp://www.microwaves101.com/encyclopedias/microwave-hall-of-fame-part-i#schottky
Note: Schottky barrier (aka Schottky contact) is a junction between a metal and a semi-conductor; the gate of a FET is a Schottky barrier
Schottky barrier is used to make Schottky diodes: very fast switching makes them ideal for microwave electronics
Chip Capacitors (Series Mount)
ElecEng4FJ4 3L12: PASSIVE COMPONENTS
parallel-plate type
gap type
interdigitated type
Terminations
ElecEng4FJ4 5L12: PASSIVE COMPONENTS
• terminations (loads) are 1-port devices designed to completely absorb (ideally without reflection) the incident power
• the overall resistance of the termination must match Z0 of the interconnect
• the overall geometry of the termination must conform to that of the interconnect to minimize reflections
coaxial microstrip
Attenuators
ElecEng4FJ4 6
• attenuators are 2-port devices designed to reduce the transferred signal power
• the power reduction is achieved by dissipation
• impedance match is required at both ports
• attenuators of fixed attenuation are called pads
L12: PASSIVE COMPONENTS
Lumped-resistor Pads: T-pads
ElecEng4FJ4 7L12: PASSIVE COMPONENTS
• a pad is characterized by its attenuation factorin
ou
PKP
01 01 021
( 1) 21
Z K KZ ZRK
unbalanced T-pad
balanced T-pad
02 01 022
( 1) 21
Z K KZ ZRK
01 023
21
KZ ZRK
dB dBdB 1010log in
in ouou
PK P PP
design formulas
Lumped-resistor Pads: Π-pads
ElecEng4FJ4 8L12: PASSIVE COMPONENTS
unbalanced Π-pad
balanced Π-pad
01 021
02 01
( 1)( 1) 2
Z K ZRK Z KZ
02 012
01 02
( 1)( 1) 2
Z K ZRK Z KZ
01 023
12
K Z ZRK
design formulas
Lumped-resistor Pads: Limitation on K
ElecEng4FJ4 9L12: PASSIVE COMPONENTS
• if Z01 ≠ Z02, there is a minimum attenuation factor that can be achieved
01 01 01min
02 02 02
2 1 21
Z Z ZKZ Z Z
• if Z01 = Z02, Kmin = 1 and any attenuation factor can be achieved
Distributed Attenuators
ElecEng4FJ4 10L12: PASSIVE COMPONENTS
coaxial pad
microstrip pad
lossy material
3-port Networks: T-junctions and Power Dividers
• power dividers are often realized as T-junctions of transmission lines
• characterized by a 3 by 3 S-matrix
• limitations of 3-port networks: can a 3-port network be simultaneously reciprocal, loss-free and matched at all ports? No
2 212 13 12 1312 13
2 212 23 12 23 23 12
2 213 23 13 23 13 23
| | | | 1 000 | | | | 1 and 0
0 | | | | 1 0
S S S SS SS S S S S SS S S S S S
S
12 13 23
at least 2 of the 3parameters , ,must be zero
S S S
• a 3-port device is either lossy, or non-reciprocal, or mismatched on at least 2 portsElecEng4FJ4 11L12: PASSIVE COMPONENTS
loss-free conditions cannot be satisfiedmatched and reciprocal
loss-free
12
• TYPE 1: 3-port devices can be matched on all 3 ports and can be reciprocal if they are lossy (e.g., Wilkinson power divider, resistive power dividers)
• TYPE 2: 3-port devices can be loss-free and reciprocal but only 1 port is matched (e.g., the input of the power divider)
• TYPE 3: 2 ports (say, 1 and 2) are matched and reciprocal while the other port (say, port 3) is mismatched – amounts to a piece of TL plus another reactively loaded completely decoupled piece of TL
3-port Networks: T-junctions and Power Dividers – 2
ElecEng4FJ4 L12: PASSIVE COMPONENTS
13
• TYPE 3, cont.: ports 1 and 2 matched, port 3 mismatched, cont.
3-port Networks: T-junctions and Power Dividers – 3
2 212 13 12 13 23 3312 13
2 212 23 12 23 23 12 33 13
2 2 213 23 33 13 23 33 13 23
| | | | 1 000 | | | | 1 and 0
| | | | | | 1 0
S S S S S SS SS S S S S S S SS S S S S S S S
S13 23| | | |S S
13
23
| | 0 or| | 0SS
33| | 1S 12| | 1S
0 00 0
0 0
j
j
j
ee
e
S
not a very useful device
ElecEng4FJ4 L12: PASSIVE COMPONENTS
matched on port 1&2 and reciprocal
13 23| | | | 0S S
T-junction Transmission-line Power Dividers (TYPE 2)
• E-plane waveguide T-junction • H-plane waveguide T-junction
ElecEng4FJ4 14L12: PASSIVE COMPONENTS
• microstrip T-junction
• loss-free reciprocal T-junctions cannot be matched simultaneously on all 3 ports; they are matched only at the input port: input cannot be used as output and vice versa
1 21 1 22 2 2 0
11E
S
1 21 1 22 2
112 0
H
S
1
2
3
21
3input
input
T-junction Transmission-line Power Dividers: Circuit Model
• loss-free divider
2 3 0
1 1 1inY jB
Z Z Z
matched to Z0
• if we ignore B (the susceptance of the junction discontinuity)
2 3 0
1 1 1Z Z Z
port 1
port 2
port 3
2Z
3Z
ElecEng4FJ4 15L12: PASSIVE COMPONENTS
11 0S
22 0S
33 0S
T-junction TL Power Dividers: Power Division Ratio
2 2 20 0 0
2 30 2 3
1 1 1, , 2 2 2in
V V VP P PZ Z Z
2 3
3 2
P ZKP Z
2 2 0
1 1 1Z KZ Z
2 0
3 0
11
( 1)
Z ZK
Z Z K
• example: 3-dB power divider for input matched to Z0 = 50 Ω0 2 31, 50 100K Z Z Z
ElecEng4FJ4 16L12: PASSIVE COMPONENTS
• let P2/P3 = K (power-division ratio or power-split ratio)
T-junction TL Power Dividers: Port Matching
• Z2 and Z3 can be transformed to Z0 via impedance-matching networks, e.g., quarter-wavelength impedance transformers
• if ports 2 and 3 have matched loads, then port 1 is matched (S11 = 0); however, there will be mismatch looking into the output ports (S22 ≠ 0, S33 ≠ 0)
• example: 3-dB power divider for input matched to Z0 = 50 Ω
ElecEng4FJ4 17L12: PASSIVE COMPONENTS
2 3 1 11100 50 0inZ Z Z S 2
22 33 22
22 33
100 100 50 100 where 100 100 50 3
0.5
inin
in
ZS S ZZ
S S
• major shortcoming: output ports are not isolated (S23 = S32 ≠ 0)Find all the elements of the generalized S-matrix of the loss-free 3-dB T-junction power divider for an input matched to Z0 = 50 Ω.
ElecEng4FJ4 18L12: PASSIVE COMPONENTS
3-dB resistive splitter
Z
T-junction Resistive Power Dividers (TYPE 1)• all ports are matched to Z0 (advantage)• network is lossy (disadvantage)• output ports are not isolated (disadvantage)
0 0 0000 0
21 22 3 3 33 3in
Z Z ZZ ZZ Z Z Z Z
1 10
2/ 3 3ZV V V
Z Z
2 313
4 2V V V V
1 11 1 12 1 1
00
0
S
2 3 4inPP P loss ?P
why?
0 portsinZ Z
Wilkinson Power Dividers• all ports are matched to Z0 (advantage)
• network is lossy for signals arriving from output ports or if unbalanced (not a disadvantage)
• output ports are isolated (advantage)
Wilkinson 3-dB power divider
ElecEng4FJ4 19L12: PASSIVE COMPONENTS
input
output
output
3-dB Wilkinson Power Dividers for 50-Ω System Impedance
ElecEng4FJ4 20L12: PASSIVE COMPONENTS
this device can work as a power divider and a power combiner• divider: splits the power at port 1 equally between ports 2 and 3• combiner: adds up the input signals at ports 2 and 3 to produce
the output signal at port 1
21L12: PASSIVE COMPONENTSElecEng4FJ4
3-dB Wilkinson Power Divider: Even/Odd Mode Analysis
midplane
1r
?Z
?Z
?r
?r
all impedances normalized to Z0
every pair of voltages (Vg2,Vg3) can be represented as the superposition of even-mode voltages (Ve,Ve) and odd-mode voltages (Vo,−Vo)
2 2 3
3 2 3
( ) / 2( ) / 2
e o eg g g
e o og g g
V V V V V VV V V V V V
3-dB Wilkinson Power Divider: Even Mode Analysiseven mode
2 3 02g gV V V
2
2 2ein
ZZ
• for a matched port 2 (or port 3) in an even-mode regime2
2 21 2e
in ZZZ
• to find S12 in the even mode, we need the voltage at port 1 1( )eV
0x / 4x
( ) ( )j x j xV x V e e
x
( /4) 02 (1 )exV V V j V
1 (0) (1 )eV V V
01
01
1 , 2 2
2
2 21
e
eV jV
V jV
ElecEng4FJ4 22L12: PASSIVE COMPONENTS
λ/4 impedance transformer
0 02 21
eine
ein
ZV V VZ
1 112
022
e ee
eV VS jV V
3-dB Wilkinson Power Divider: Odd Mode Analysisodd mode
2 3 02g gV V V
short
open2 2
oin
rZ
• for a matched port 2 (or port 3) in an odd-mode regime
2 21 2o
in rrZ
• to find S12 in the odd mode, we need the voltage at port 1 1( )oV
2 0 00.52
0.5 1o rV V V
r
1 (port 1 is shorted, all power delivered to resis )0 toroV
ElecEng4FJ4 23L12: PASSIVE COMPONENTS
why?
112
2 0
0 0o
oo
VSV V
3-dB Wilkinson Power Divider: S-parameters• input impedance at port 1 when ports 2 and 3
have matched loads2
,2 ,21 2in inZ Z Z Z
,2 / 2 1in inZ Z
o port 1 is matched if ports 2 and 3 have matched loads 11 0S
• since input impedances at ports 2 and 3 are 50 Ω in both even- and odd-mode regimes, for any regime
22 33 0S S
ElecEng4FJ4 24L12: PASSIVE COMPONENTS
circuit is symmetric when matched on ports 2 &3
,2inZ
,2inZ
,2inZ
,2inZ
no loss when balanced
3-dB Wilkinson Power Divider: S-parameters (2)
01 10 122 2
02 2
2 0 2 2
e oe o
e o
V V V j jV V V SV V V
• for both even and odd modes, the incident voltage at port 2 is V0
12 21
13 12
(reciprocity) (symmetry)
S SS S
0 1 11 1 0 02 1 0 0j
S
• due to short or open mid-plane, ports 2 and 3 are decoupled (isolated) 32 23 0S S
• Wilkinson’s power divider is loss-free if ports 2 and 3 are matched
• it has loss only if power is reflected from the output ports – it is dissipated in the shunt resistorElecEng4FJ4 25L12: PASSIVE COMPONENTS
Unequal 2-way Wilkinson Power Divider
• let the power-split ratio be 23 2/P P K
2 202 03 0
2
03 0 3
0
(1 )1
1
Z K Z Z K KKZ Z
K
R Z KK
ElecEng4FJ4 27L12: PASSIVE COMPONENTS
• using even/odd mode analysis the following expressions are obtained
N-way Equal-split Wilkinson Power Dividers
• corporate arrangement of 2-way splitters
• parallel equal-split N-way Wilkinson power divider
ElecEng4FJ4 28L12: PASSIVE COMPONENTS
ElecEng4FJ4 29L12: PASSIVE COMPONENTS
4-port Networks: Scattering Parameters
• consider reciprocal matched loss-free 4-port network
12 13 14
12 23 24
13 23 34
14 24 34
00
00
S S SS S SS S SS S S
S
13 23 14 24*
14 13 24 23
12 23 14 34
14 12 34 23
13
24
12
34
0 0 0 0
//
//
S S S SS S S SS S S SS S S
SSS
SS
2 214 13 24
2 223 12 34
| | | | 0| | | | 0
S S SS S S
unitary conditions
• one possible solution is 14 23 0S S
directional-coupler solution
12 13
12 24
13 34
24 34
0 00 00 0
0 0
S SS SS S
S S
S
ElecEng4FJ4 30L12: PASSIVE COMPONENTS
4-port Networks: Directional Couplers or Hybrids
• directional-coupler solution 14 23 0S S
2 212 13
2 212 24
2 213 34
2 224 34
| | | | 1 ( )| | | | 1 ( )| | | | 1 ( )| | | | 1 ( )
S SS SS SS S
13 24
12 34
| | | || | | |S SS S
common symbols
2nd set of unitary conditions
Directional Couplers: Scattering Matrix
• choose reference planes so that
1 212 34
13 24, jB jBS SS e S e
1 2
1 2
1
12 13 24 34
2
0 0
2 (set 0)jB jB
jB jB
e e
S S S S e e
n nB B
case 1: B1 = B2 = π/2 case 2: B1 = 0, B2 = π
0 00 00 0
0 0
jj
jj
S0 0
0 00 0
0 0
S
symmetrical coupler anti-symmetrical coupler
2 2 1
ElecEng4FJ4 31L12: PASSIVE COMPONENTS
(unitary conditions)
plane of symmetry or anti-symmetry
(through)(coupled)
Directional Couplers: Scattering Matrix (2)
• similar result is obtained if we choose the reference planes so that1 212 34
13 24
, jA jAS e S eS S
2 2 1
1 2
1 2
12 13 24 34 0 0jA jA
jA jA
e e
S S S S e e
case 1: A1 = A2 = π/2 case 2: A1 = 0, A2 = π
0 00 00 0
0 0
jj
jj
S0 0
0 00 0
0 0
S
always: if “through” parameters are in phase, phases of “couple” parameters add to π (previous slide); vice versa if “couple” parameters are in phase, phases of “through” parameters add to π (this slide)
32
symmetrical coupler anti-symmetrical coupler
1 2 2 (set 0)n nA A
(unitary conditions)
ElecEng4FJ4 L12: PASSIVE COMPONENTS
(through)(coupled)
2 214 13 24
2 223 12 34
| | | | 0| | | | 0
S S SS S S
• another solution to the above equations: 12 313 4 42 , | || | | | | |S S SS
• this is also satisfied by the loss-free directional-coupler solution but here we also assume that 14 23| | 0, | | 0S S
(no isolation)• from the unitary conditions
214
?223
14 2322
213
22
32
212
212
234
23
42
14
132
424
| | 1| | 1 | | | | 0| | 1| |
| || |
| || |
| || |
| || | 1
SS
S
SS S SSS
SS
SSS
ElecEng4FJ4 33L12: PASSIVE COMPONENTS
Directional Couplers: Scattering Matrix (3)
Directional Couplers: Scattering Matrix (4)• choose plane references such that “through” S-parameters are in phase
1 212 34
13 24, jB jBS SS e S e
12 13 24 34 1 20 2S S S S B B n (set n = 0)
1 1( )13 23 14 24 23 14 23 14
12 23 14 34 23 14
0 0 00 0
jB j BS S S S e S e S S SS S S S S S
14 23 0S S
• the same result would be obtained if we set reference planes so that1 212 34
13 24
, jA jAS e S eS S
the directional-coupler solution is after all the only solution
a reciprocal, loss-free and matched 4-port network is always a directional coupler with one pair of ports decoupled (input/isolated)
ElecEng4FJ4 34L12: PASSIVE COMPONENTS
Directional Couplers: Performance Parameters
, dBI D C
the ideal coupler, I D
ElecEng4FJ4 35L12: PASSIVE COMPONENTS
1
10 10 313
3 3110 10
4 41
110 10 41
4
Coupling: 10log 20log | | dB
| |Directivity: 10log 20log dB| |
Isolation: 10log 20log | | dB
PC SP
P SDP S
PI SP
More on the physical meaning of Directivity: The difference in dB of the power output at a coupled port, when power is transmitted in thedesired direction, to the power output at the same coupled port when the same amount of power is transmitted in the opposite direction.[from Mini-Circuits Application Note]
Hybrid Couplers• a particular case of a directional coupler with C = 3 dB (equal
power split between the through and coupled ports)1/ 2
• S-matrix of the quadrature hybrid (symmetrical coupler of C = 3 dB): 90° phase difference between the through and coupled ports
0 1 01 1 0 0
0 0 120 1 0
jj
jj
S
• anti-symmetrical hybrid (180° hybrid): 0° phase difference between the through and coupled ports if port 1 (or 3) is excited; 180° phase difference if port 2 (or 4) is excited
ElecEng4FJ4 36
0 1 1 01 1 0 0 1
1 0 0 120 1 1 0
S
Quadrature Hybrid
0 1 01 1 0 0
0 0 120 1 0
jj
jj
S
equivalent circuit(normalized to Z0)
even-odd mode analysis of the equivalent circuit helps understand how this hybrid works
ElecEng4FJ4 37L12: PASSIVE COMPONENTS
Quadrature Hybrid: Even-Odd Mode Analysis
even-mode excitation
odd-mode excitationElecEng4FJ4 38L12: PASSIVE COMPONENTS
Quadrature Hybrid: Even-Odd Mode Analysis (2)
shunt shunt/4 TLSC stub SC stub
/8 /8
11 0 1 00 / 2 111 1/ 2 0 2o
l l
A B j jjC D j jj
shunt shunt/4 TLOC stub OC stub
/8 /8
11 0 1 00 / 2 111 1/ 2 0 2e
l l
A B j jjC D j jj
y j
y j
ElecEng4FJ4 39L12: PASSIVE COMPONENTS
find the ABCD matrices in both modes 1 2 2
1 2 2
V AV BII CV DI
• even mode
• odd mode
Quadrature Hybrid: Even-Odd Mode Analysis (3)
1
2
3
4
0.5( )0.5( )0.5( )0.5( )
e o
e o
e o
e o
BB T TB T TB
, , , ,, 11 ,
, , , ,
, 21 ,, , , ,
2
e o e o e o e oe o e o
e o e o e o e o
e o e oe o e o e o e o
A B C DSA B C D
T SA B C D
(1 )0, 2
10, 2
e e
o o
jT
jT
1
2
3
4
0/ 2
1/ 20
BB jBB
ElecEng4FJ4 40L12: PASSIVE COMPONENTS
• scattered waves at all four ports
• obtain reflection and transmission coefficients from ABCD matrices
0 1 01 0 0
0 0 120 1 0
jj j
jj
S
Quadrature (90°) Hybrid Performance
• relatively narrow-band (10% to 20%)
microstrip 90° hybrid
ElecEng4FJ4 41L12: PASSIVE COMPONENTS
1 2
4 3
Coupled-line Directional Couplers: Coupled Lines
strip lines / edge-coupled strip lines / broadside-coupled
microstrip lines / edge-coupled
equivalent circuit for TEM propagation
42L12: PASSIVE COMPONENTSElecEng4FJ4
Coupled Lines: Even / Odd Mode Analysis (Symmetric Lines)
even mode
odd mode
11 22
0
1
e
ee
e e
p e
C C CL LCZ
C C
v C
11 12
0
21
o
op o
C C C
Zv C
strip
ElecEng4FJ4 43L12: PASSIVE COMPONENTS
assume same phase velocity for even and odd modee o
Single-section Coupled-line Coupler: Even / Odd Mode Analysis
1 3 2 4
1 3 2 4
,,
e e e e
e e e eI I I IV V V V
1 3 2 4
1 3 2 4
,,
e e e e
e e e eI I I IV V V V
einZ
oinZ
0 00
0 0
tan( )tan( )
eeein
e
Z jZ LZ ZZ jZ L
0 00
0 0
tan( )tan( )
oooin
o
Z jZ LZ ZZ jZ L
001 1
0 0,
eine e
e ein in
Z VV V IZ Z Z Z
001 1
0 0,
oino o
o oin in
Z VV V IZ Z Z Z
ElecEng4FJ4 45L12: PASSIVE COMPONENTS
even mode
odd mode
Coupled-line Coupler: Even / Odd Mode Analysis (2)
21 01 1
01 01 1
2( )2
e oe oin in
in ine o e oin in
Z Z ZV V VZ Z ZI I I Z Z Z
• to match port 1, Zin = Z0; then, the 1st condition for Z0e and Z0o is obtained
20
e oin inZ Z Z
0 0 0e oZ Z Z
ElecEng4FJ4 46L12: PASSIVE COMPONENTS
set to zero
0 0 0 020 00
0 0 0 0
tan( ) tan( )tan( ) tan( )
e oe o
e o
Z jZ L Z jZ LZ Z ZZ jZ L Z jZ L
Coupled-line Coupler: Even / Odd Mode Analysis (3)
• the coupled-port voltage is then
3 03 3 1 10 0
0 03 0
2 0 0
tan( ) , where 1 tan( )
e oin ine o e o
e oin in
e o
e o
Z ZV V V V V VZ Z Z Z
C L Z ZV V j CZ ZC j L
ElecEng4FJ4 47L12: PASSIVE COMPONENTS
• the through-port voltage is obtain in a similar way2
2 02
11 cos( ) sin( )
CV VC L j L
• the isolation-port voltage is obtained as 4 0V
Coupled-line Coupler: Even / Odd Mode Analysis (4)
• if L = λ/4, then the 2nd condition for Z0e and Z0o is obtained
3 2 24
0 0, 1 , 0V VC j C V
V V 0 0
0 0
e o
e o
Z ZCZ Z
desired coupling
• for given Z0 and C
0 00 0 0
0 0
0 00 0
11 11
ee o
e o
oe o
CZ ZZ Z ZCZ ZC CZ ZZ ZC
ElecEng4FJ4 48L12: PASSIVE COMPONENTS
design formulas
Single-section Coupled-line Coupler
microstrip single-section coupler
ElecEng4FJ4 49L12: PASSIVE COMPONENTS
Multi-section Coupled-line Couplers• broadband performance – bandwidths over decades (advantage)
• low coupling levels (limitation)
• significant length – same phase velocity for even and odd modes is important!
3-section binomial
ElecEng4FJ4 L12: PASSIVE COMPONENTS 50
51
Lange (Quadrature) Couplers
• interdigitated line geometry
• strong coupling possible: 6 dB, 3 dB
• wide bandwidths: octave, decade is possible
• the 3-dB Lange coupler is a quadrature hybrid
• the design is based on that of a coupled-line directional coupler
90°
ElecEng4FJ4 L12: PASSIVE COMPONENTS
“unfolded” equivalent
ElecEng4FJ4 52L12: PASSIVE COMPONENTS
Lange Couplers: Equivalent Circuits
for 4 lines of same widths and spacings
0 04 0
0 0
0 04 0
0 0
3
3
o ee e
o e
o eo o
o e
Z ZZ ZZ ZZ ZZ ZZ Z
0 0, - even and odd impedances of a pair of lines
e oZ Z
180° Hybrids
0 1 1 01 0 0 11 0 0 120 1 1 0
j
S
hybrid• input signal at port 1 is split equally into two in-phase signals at
ports 2 and 3 (port 4 is isolated)• input signal at port 4 is split equally into two out-of-phase signals at
ports 2 and 3 (port 1 is isolated)
power combiner• signals at ports 2 and 3 are added to produce the signal at port 1 • signals at ports 2 and 3 are subtracted to produce the signal at port 4
ElecEng4FJ4 53L12: PASSIVE COMPONENTS
modes of operation:
180° Ring (Rat-race) Hybrid• rigorous analysis through even/odd mode analysis – ring-line
impedance must be
• narrow-band performance02Z
54ElecEng4FJ4 L12: PASSIVE COMPONENTS
Tapered Coupled Line Hybrid
• wide bandwidth (decade or more)
• any power-division ratio can be achieved in principle
• the hybrid is analyzed using even-odd mode analysis
ElecEng4FJ4 55L12: PASSIVE COMPONENTS
1
4
2
3
through
coupled