IIT Bombay Course Code : EE 611 Department: Electrical Engineering - - PowerPoint PPT Presentation

IIT Bombay

Course Code : EE 611 Department: Electrical Engineering Instructor Name: Jayanta Mukherjee Email: jayanta@ee.iitb.ac.in

EE 611 Lecture 2 Jayanta Mukherjee Page 1

Lecture 2

Overview of the course (Recap)

• In this course we will study the basic passive devices used

in microwave systems

• Passive devices are those which do not produce any power

themselves i.e. there is never any gain involved

• These include impedance matching networks, couplers, filters,

attenuators, phase shifters etc

• Electromagnetic theory combined with Network Theory
• Basic parameters used in designs of microwave systems

e.g., S parameters, impedance issues etc.

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 2

Distributed circuit theory

At low frequencies circuits are designed using circuit theory. Circuit theory relies on basic lumped elements derived from Maxwell’s equations such as:

• inductors
• capacitors
• resistors

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 3

Distributed circuit theory

Similarly at microwave frequencies we will design microwave Circuits using distributed circuit theory. Distributed circuit theory relies on basic elements also derived from Maxwell’s Equations, such as:

• transmission lines
• shorted stubs (generalized inductors)
• open stubs (generalized capacitors)
• coupled lines
• Tapered lines

Besides using lumped elements of regular circuit theory

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Attributes of Distributed circuit theory

• Unlike Maxwell’s equation based analysis and like traditional

circuit theory, it is simpler to use

• The traditional circuit theory is a subset of distributed circuit

theory

• New design techniques and synthesis theorems exist in

distributed circuit theory without equivalent in traditional circuit theory

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Guided Wave System

• Transmission line system featuring 2 or more conductors

guiding the waves eg, co-axial cable, telegraph lines, parallel plate

• Closed metallic waveguides consisting of hollow conductive

pipes guiding the waves along the z axis, eg rectangular waveguide

• Dielectric waveguides typically consisting of a material with

a high dielectric constant (slab of rod) sandwiched by materials with low dielectric constants eq microstrip, stripline

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Guided waves as opposed to waves in free space can flow through the following types of systems

Guided Wave System

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 7 Microstrip Rectangular Waveguide Two wire Coaxial Cable

Electromagnetic Waves

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 8

Maxwell’s Equations

= ⋅ ∇ = ⋅ ∇ B D ρ J t D H t B E + ∂ ∂ = × ∇ ∂ ∂ − = × ∇

3 2 2 2

Coul/m in density, charge electric the is ρ A/m in density, current electric the is J in Wb/m density, flux magnetic the is B Coul/m in density, flux electric the is D A/m in intensity, field magnetic the is H V/m in intensity, field electric the is E

Electromagnetic Waves

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 9

E D B H ε µ = =

Assuming µ and ε are scalars then Maxwell’s equations in free space becomes (ρ=J=0)

= ⋅ ∇ = ⋅ ∇ B D t D H t B E ∂ ∂ = × ∇ ∂ ∂ − = × ∇

Derivation of wave equation in free space

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 10

( )

2 2 2

t E E E B t E ∂ ∂ − = ∇ − ⋅ ∇ × ∇ × ∇ ∂ ∂ − = × ∇ × ∇ µε

Wave equation in free space

2 2 2 2 2 2 2

t E E H t E E ∂ ∂ − = ∇ − ∇ = ∂ ∂ − ∇ µε µε

Sinusoidal Time Dependence

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So far time dependence has not been specified E is a function of space and time {E(x,y,z,t)} We will assume cosine time dependence

[ ] [ ]

t j t j

e ) z , y , x ( H Re ) t , z , y , x ( H e ) z , y , x ( E Re ) t , z , y , x ( E

ω ω

= =

Solution only for spatial components

Sinusoidal Time Dependence

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The wave equations are then modified to

H H E E

2 2 2 2

= µεω + ∇ = µεω + ∇

Helmhotz equations

Modes in Waveguide Systems

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Various modes of electromagnetic waves can propagate in Transmission lines and waveguide systems

• TEM waves (Transverse Electromagnetic) :
• TE waves (Transverse Electric) or H waves:
• TM waves (Transverse Magnetic) or E waves:

H E

z z

= = E , H

z z

= ≠ H , E

z z

= ≠

TEM Waves (Refer Pozar Ch 3)

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• TEM waves require at least two conductors
• Other propagation modes TE and TM are possible in systems

supporting TEM mode but are usually not desired

• Example of the coaxial line

The TEM mode will be the only mode propagating in the frequency range

( )

r r 11 c

b a c ) TE ( f f µ ε + π = < <

Where fc(TE11) is the cutoff frequency at which the TE11 mode (transverse electric mode) starts to propagate in the coaxial cable

a b

2 conductor waveguiding systems supporting the TEM mode

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 14

Microstrip

E H I I 2 wire I I Co-axial

An example – Parallel Plate Waveguide

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 15

TEM wave solution

jkz jkz

e d V y z y x E e d V x z y x H

− −

− = = ˆ ) , , ( ˆ ) , , ( η plates between Voltage ,

0 =

= V ε µ η

d W y z x

An example – Parallel Plate Waveguide

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 16

TM mode

2 2

, 2 , , cos cos sin ) , , (

c c y x z j n c y z j n c x z j n z

k k k d n k H E e d y n A k j E e d y n A k j H e d y n A z y x E − = = = = = − = = =

− − −

β λ π π π β π ωε π

β β β

• Various “modes” are present
• A cutoff frequency fc such that for f<fc , k<kc,and β is imaginary
• Treatment for TE mode is similar

TEM Mode in a Coaxial System

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BNC f < 1 or 4 GHz APC 7mm (sex less) f < 18 GHz APC 3.5 mm f < 34 GHz SMA f < 24 GHz SSMA f < 38 GHz

APC 7mm APC 7mm SMA BNC

a b

Modes in hollow waveguides

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No TEM mode can propagate in waveguides (only one conductor) Condiser the rectangular waveguide. The dominant mode is (lowest cutoff)TE10 . The mode only propagates for frequencies Verifying:

µε λ λ a f f a

c c

2 1 2

10 , 10 ,

= > = <

E H H E

λ/2

Rectangular Waveguide Fundamental Mode

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The propagation constant β for TE10 is given in terms of the frequency ω by εµ β β ω π ω π β ω ω µε β / , 2 ,

2 2 2 2

= = =       − = − = c f with a

c c c

Dispersion curves for a waveguide (full line) and a TEM mode (dashed line)

ω c vp vg ωc = β0c

β

ω

Phase and Group Velocity

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The phase velocity (cordal slope) and the group velocity (differential Slope) are respectively larger and smaller than the speed of light In the dielectric material considered:

β ω µε β ω ∂ ∂ = > = > =

g d p

v c v 1

• Faster than light ?
• No information is carried by the wave with the phase velocity vp

ω c vp vg ωc = β0c

β

ω

Group Velocity and EM modes

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• Group velocity vg gives the velocity of the information

transmission

• The group velocity can be frequency dependent (e.g.

waveguide) As a result signals at different frequencies will be transmitted at different speeds leading to the distortion of the wave packets transmitted Such a communication medium is said to be dispersive

• On the contrary TEM modes are not dispersive as they verify

β ∂ ω ∂ = = µε = = β ω =

g d p

v 1 c v

Distributed Circuit Theory

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 22

The development of a distributed circuit theory is based on the Introduction of a local voltage V(z,t) and local current I(z,t) from the Electric field E(x,y,z,t) and magnetic fields H(x,y,z,t) In a transmission line system a voltage between the conductors C1 and C2 can be obtained from the following path integration

∫

⋅ − =

2 1

) , , , ( ) , (

C C

dl t z y x E t z V

Distributed Circuit Theory

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 23

Similarly the current flowing in conductor C1 or C2 is obtained from The closed path integration:

dl H(x,y,z,t) t z I ⋅ = ∫

C2

• r

C1

) , (

For TEM modes these integrations are path independent and The voltage and current so defined assume a unique value For TE and TM modes the integrations are path dependent and Hence the voltages , currents and impedance are not uniquely define.

Wave Equations for Transmission Lines Using Maxwell Equations

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 24

The solution of the wave equation derived from Maxwell’s Equation leads to a voltage wave of the form:

( ) ( )

z t j z t j

e V e V t z V

β ω β ω + − − +

+ = ) , (

And a current wave of the form:

( ) ( )

z t j z t j

e Z V e Z V t z I

β ω β ω + − − +

− = ) , (

Where is the propagation constant And where Z0 is the characteristic impedance

εµ ω β =

Proof

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 25

Wave equation in free space

and

2 2 2 2 2 2

= ∂ ∂ − ∇ = ∂ ∂ − ∇ t H H t E E µε µε

TEM waves are of the following form ( )

) ( ) ( ) ( ) (

) , ( ) , ( ) , , , ( ) , ( ) , ( ) , , , (

z t j z t j z t j z t j

e y x H e y x H t z y x H e y x E e y x E t z y x E

β ω β ω β ω β ω + − − + + − − +

+ = + =

εµ ω β =

Proof

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 26

Voltage wave between conductors C1 and C2

) , ( ) , (

• )

, , , ( ) , (

) ) ( ) ( 2 1 ) C2 C1 2 1

e V e V e dl y x E e dl y x E dl t z y x E t z V

z j(ω( z t j z t j C C z t j( C C β β ω β ω β ω + − − + + − − +

+ = ⋅ − ⋅ = ⋅ − =

∫ ∫ ∫

Proof

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 27

Current wave in conductors C1 or C2

) , ( ) , ( ) , (

) ) ( C2

• r

C1 ) ( C2

• r

C1 ) ( C2

• r

C1

e I e I e dl y x H e dl y x H dl H(x,y,z,t) t z I

z j(ω( z t j z t j z t j β β ω β ω β ω + − − + + − − +

+ = ⋅ + ⋅ = ⋅ =

∫ ∫ ∫

Characteristic Equation from Maxwell Equation

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Using equation 3.18 in Pozar

( )

( ) ( )

( ) ( ) ( ) ( )

∫ ∫ ∫ ∫

± ± ± ± ± ± ± ± ± ± ± ±

⋅ ε µ = ⋅ ⋅ = = µ ε ± = ⋅ × µ ε ± = ⋅ × µ ε ± = ⋅ × µ ε ± =

2 C 2 C 1 C 2 C 2 C 1 C `

y , x E n dl y , x E dl y , x H dl y , x E I V Z : ratios qamplitude wave current to voltage the from defined Z impedence stic characteri The C2. to normal vector normalized a n with n E E z ˆ dl dl E z ˆ dl H and E z ˆ H

Quasi-TEM mode in Microstrip line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 29

t W h dielectric Strip conductor

εr

Ground plane

Quasi-TEM mode in Microstrip line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 29

• The TEM derivation holds for 2 conductor systems with a uniform

dielectric between the two conductors

• In the case of a microstrip system, two different dielectric

materials with dielectric constant ε0 (air) and ε1 are used

• In such a case an effective dielectric constant can be defined for

the quasi-TEM mode such that if it were the only dielectric material present in the system, the same capacitance per unit length would result

ε0 ε1 C C0 C ε 0 εeff

Effective Dielectric Constant

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 30

ε0 ε1 C C0 C ε 0 εeff

The effective dielectric constant can be obtained from the Capacitance per unit length C and C0 from the above systems From C’/εeff = C0/ ε0 we have: εeff = ε0(C/C0) For such a quasi-TEM mode the propagation constant is then Now given by

C C v 1 Z : by impedance stic characteri the and

eff p eff

µε = = µ ε ω = β

Approximate Formula for Microstrip

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 2 Jayanta Mukherjee IIT Bombay Page 31

t W h dielectric Strip conductor

εr

Ground plane

1 w/h for 444 . 1 h W ln 667 . 393 . 1 h W 1 120 Z 1 w/h for h 4 W W h 8 ln 60 Z W h 12 1 1 2 1 2 1

eff eff r r eff

≥       + + + × ε π = ≤       + ε = + − ε + + ε = ε

Characteristic Impedance for Waveguides

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The definition of a unique voltage is not possible in a waveguide depending of the choice selected the characteristic impedance for the TE mode in a rectangular waveguide is found to be

E H H E

λ/2

Characteristic Impedance for Waveguides

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( )

Ω < < Ω = ε µ = η − ε µ = 754 Z 377 with f / f 1 a / b Z Z

v r r 2 c r r v TE 0,

10

• The characteristic impedance is not uniquely defined in

waveguides

• However once a definition has been selected the voltage and

current waves can be productively used to design waveguide circuits.