Lecture 11 Passive Components - McMaster University · Lecture 11 Passive Components (Capacitors,...

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)


parallel-plate type

gap type

interdigitated type

Lumped (On-chip) Planar Inductors


meander type

spiral type

Terminations


• 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


• 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


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


• 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


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


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


• 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


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


• 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


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.


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


input

output

output

3-dB Wilkinson Power Dividers for 50-Ω System Impedance


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


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


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


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

3-dB Wilkinson Power Divider: Frequency Responses


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


• 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



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


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


(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)


(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


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)


Directional Couplers: Performance Parameters

, dBI D C

the ideal coupler, I D


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


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


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


• 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


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


assume same phase velocity for even and odd modee o

Single-section Coupled-line Directional Coupler


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


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


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


• 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


• 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


• 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


design formulas

Single-section Coupled-line Coupler

microstrip single-section coupler


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°


“unfolded” equivalent


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


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


1

4

2

3

through

coupled

Coupled Planar Line Couplers (Multi-layer PCB designs)


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