Jan Simons simons@astron.nl

JS/2006/oct/09

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(C) ASTRON 2006

Jan Simons simons@astron.nl

10-oct-2006

Basic Detection Techniques Radio components

and system characterisation

Contents

Antenna - electromagnetics Transmission lines

S-parameters

Noise in RF components

Systems of RF components

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Radio frequency radiation

Characteristics

f = 30 MHz - 300 GHz λ = 10 m - 1 mm

300 GHz photon energy

h ν = k T = q

_e

V

=> 12 K , 1.25 mV

Sources cold matter .

Power detection limit

Minimum detectable noise power

T

sys

.

∆ T

_rms

= sqrt (∆ ν T

_int

) Minimum flux density

∆ P .

∆ S = η B A

_ant

Sensitivity of a radio telescope

T

_sys

/ A

_eff

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What is an antenna

Conversion of guided waves in a transmission line to waves in free space

Concentrator of waves from specific directions

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Wire antennas

Dipole

Loop antenna Helix antenna

X Y

Z

Horn antennas

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Reciprocity in antennas

Identical behaviour for transmit and receive situations

I

+

V

-

energy

antenna 1 antenna 2 transmit

V

+ -

energy

I

antenna 1 antenna 2 receive

Antenna circuit substitution

+ -

V₁

I₁

Receive and transmit antenna together form a two-port

+ -

V₂ I₂

For a two-port we have:

V

₁

= Z

₁₁

I

₁

+ Z

₁₂

I

₂

V

₂

= Z

₂₁

I

₁

+ Z

₂₂

I

₂

Z

₂₁

= Z

₁₂

(reciprocity)

Z

₂₂

= Z

₁₁

for identical antennas

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Antenna circuit substitution

Receive and transmit antenna together form a two-port

For a two-port we have:

V

₁

= Z

₁₁

I

₁

+ Z

₁₂

I

₂

V

₂

= Z

₂₁

I

₁

+ Z

₂₂

I

₂

Z

₂₁

= Z

₁₂

(reciprocity)

Z

₂₂

= Z

₁₁

for identical antennas

In this case: I

₂

<< I

₁

thus

V

₁

= Z

₁₁

I

₁

+ -

V₁

I₁

+ -

V₂ I₂

Antenna circuit substitution 2

I

₁

+

- V

₁

Z

₂₂

Z

₂₁

I

₁

I

₂

+

V

₂

+

- Z

₁₁

V

₁

= Z

₁₁

I

₁

V

₂

= Z

₂₁

I

₁

+ Z

₂₂

I

₂

transmit receive

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Antenna circuit substitution 3

Z

_ant

Z

_load

=Z

_ant^*

V

_open

Maximum power is received in Z

_load

when Z

_load

complex conjugated to Z

_ant

(network theory)

Why does an antenna radiate ?

Maxwell equations

• Every alternating current creates Electric and Magnetic fields

But why no radiation from “ordinary” electronics ?

• All currents run in closed loops

• The total current distribution can be split in many small line current segment

• For every current line segment there is an opposite

one of equal amplitude (red arrows)

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Why does an antenna radiate ? 2

Compare the contribution of the two opposite line current segments far away from the antenna

• At low frequencies there is no phase difference

• The two opposite contributions cancel out

⇒

no radiation

At high frequencies there are two effects:

• Phase difference between the line current segments is significant

• Amplitude of the line current segments may differ

⇒

radiation

Current distribution on a dipole

Current distribution from a two wire (open)

transmission line

⇒ varying magnetic fields

⇒ EM fields

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Dipole radiation principle

t=0

t=T/8 t=T/4=2/8 T

t=3/8 T t=T/2=4/8 T

t=T/4=2/8 T

Antenna field zones

• Reactive (near) zone

• E and H 90^o out of phase

•

Average power radiated during one period close to zero

• Energy storage in electric and magnetic fields

• Contributes to imaginary part of antenna impedance

• Radiating near zone (Fresnel zone)

• E and H field in phase

• Energy is radiated

• Radiating pattern strongly depends on distance

• Radiating far field (Fraunhofer zone)

φ

₂

φ

₁

∆φ=φ

₂

-φ

₁

Contributions from many line current segments. Phase differences between any two contributions depend strongly on distance.

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Antenna field zones 2

• Reactive (near) zone

• Radiating near zone (Fresnel zone)

• Radiating far field (Fraunhofer zone)

• E and H field in phase

• Energy is radiated

• Radiating pattern does not strongly depend on distance

• E and H components perpendicular to direction of propagation

∆φ

→

E

_→

H

→

S

E and H-vectors perpendicular

Poynting (E x H) vector indicates energy flow Ratio of |E| / |H| vectors is always 377 Ω (this is the free space wave impedance)

Antenna field zones 3

Near field

Far field

D

R=2D²/λ

R=0.62 sqrt(D³/λ)

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Radiated power density

Unit: W / m ²

Isotropic radiator:

r

Spherical surface:

A=4πr

²

Directivity

[W/sterad]

rad

0

P

) , ( 4 P

) , ( P

) , ( ) P

, (

D π θ φ

φ θ

φ φ θ

θ = =

P

₀

: reference antenna isotropic radiator:

.

1 ) ,

( θ φ = D

)

2

, ( )

,

( W r

P θ φ = θ φ ⋅

π 4 P ₀ = P ^rad

D

₀

= 1.5

= 1.8 dB

θ

Electric dipole radiation diagram

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Efficiency, Gain

Losses in an antenna (Ohmic, dielectric) Definition of antenna gain:

[dBi]

Efficiency

0 dBd = 2.15 dBi

) , ( D )

, (

G θ ϕ = η θ ϕ

Radiation diagram

angle (deg)

θ

ϕ

Main lobe

Side lobes

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Transmission line examples

Telegraph lines

Two-wire line

Coaxial cable

Microstrip

Transmission line examples

ground-layer below

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Stripline

Transmission line examples

ground-plane on top and bottom

Wave guides

Transmission line examples

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Electromagnetic fields in transmission lines

Transverse E and H fields

(perpendicular to direction of transport)

Poynting vector

^→

S = E

^→

× H

^→

Network approach to transmission lines

Substituting a short section

Equivalent schematic

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x y x y f

x t f

z u c t z u g t

z i

t z i l t z i r t

z u

x t

t

∂

= ∂

∂ ∂

⋅

−

⋅

−

=

∂

∂

⋅

−

⋅

−

=

∂ ( , )

) , ( where )

, ( )

, ( )

, (

) , ( )

, ( )

, (

z z

Telegraphers’ equations:

Combining ...

2 2 2

2 2

2

2

( , )

) , ( where 0

) , ( )

, (

0 ) , ( )

, (

x y x y f

x t f

z i lc t z i

t z u lc t z u

x t

z

t z

∂

= ∂

= ∂

∂

−

∂

=

∂

−

∂

Solutions: travelling wave equations

• Functions f(z,t) = g(x-vt) and g(x+vt)

• Travelling waves in +z and -z direction

• Propagating with speed v

c v l

= 1 ⋅

Network approach to transmission lines

RF signals

Voltage and current equations

} )

( Re{

) , ( and } )

( Re{

) ,

( z t U z e

^{j t}

i z t I z e

^{j t}

u = ⋅

^ω

= ⋅

^ω

Back substitution ...

) ( ) ) (

(

) ( ) ) (

(

z U c j dz g

z dI

z I l j dz r

z dU

ω ω +

−

=

+

−

=

⇒(Lossy) wave equations:

) ( } }{

) { (

) ( } }{

) { (

2 2 2

z I c j g l j z r

I d

z U c j g l j dz r

z U d

ω ω

ω ω

+ +

=

+ +

=

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RF signals 2 - lossy lines

( ) }

Re{

) ,

( z t U

₀

e

^z

e

^{j t z}

u

⁺

=

⁺

⋅

^−α

⋅

^{ω −β}

( ) }

Re{

) ,

( z t U

₀

e

^z

e

^{j t z}

u

⁻

=

⁻

⋅

^α

⋅

^{ω +β}

Complex coefficient

β α

ω ω

γ = ( r + j l )( g + j c ) = + j

Solutions

Wave in +z and -z direction

) , ( )

, ( )

,

( z t u z t u z t

u =

⁺

+

⁻

RF signals 2 - lossy lines

( ) }

Re{

) ,

( z t U

₀

e

^z

e

^{j t z}

u

⁺

=

⁺

⋅

^−α

⋅

^{ω −β}

( ) }

Re{

) ,

( z t U

₀

e

^z

e

^{j t z}

u

⁻

=

⁻

⋅

^α

⋅

^{ω +β}

)

(compare c = λ ⋅ f Complex coefficient

β α

ω ω

γ = ( r + j l )( g + j c ) = + j

Solutions

Wave in +z and -z direction

2 ) and

2 (

ω π β π λ

ω β _{= ⋅ =}

= f

v

z

z

e

e

^−α

and

^+α

Propagation speed Extinction

) , ( )

, ( )

,

( z t u z t u z t

u =

⁺

+

⁻

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Voltage / current relationship

Solution for i(z,t) similar to u(z,t)

Characteristic impedance

large

0

ω

ω ω

c l c

j g

l j

Z r ≈

+

= +

and

0 0 0

0 0

0

and

Z I U

Z I U

− −

+

=

+

= −

( )

( ) } Re{

) , (

} Re{

) , (

0 0

z t j z

z t j z

e e

I t

z i

e e

I t

z i

β ω α

β ω α

+

−

−

−

−

− + +

⋅

⋅

=

⋅

⋅

=

Voltage / current / power relationship

) (

) (

0 0

n n n

n n n

b a Z i

b a Z u

−

=

+

=

0 0

0 0

Z Z i

b u

Z Z i

a u

n n

n

n n

n

⋅

−

=

=

⋅

=

=

− − + +

} 2 {

1

} 2 Re{

1

2 2

*

n n

n

n n n

b a

P

i u P

−

=

⋅

= Power waves

(n=1, 2)

port_1 port_2

a₁ a₂

b₁ b₂

<=>

⎟⎟ ⎞

⎜⎜ ⎛

⎥ ⎤

⎢ ⎡

⎟⎟ =

⎜⎜ ⎞

⎛ b

₁

S

₁₁

S

₁₂

a

₁

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Reflections

Terminating with an impedance Z

_L

At z = 0

On the transmission line

)

0

0 (

) 0

( Z

I

U

₊⁺

=

At z=0

Z

L

I U =

No reflection if Z_L = Z₀

Reflections 2

If Z

_L

≠ Z

₀

An U^-(0) arises, fulfilling Ohm’s law at z=0

0

)

0

0 ( )

0

( Z Z

Z U Z

U

L L

+

⋅ −

=

⁺

−

Reflection coefficient

0

)

0

0

( Z Z

Z Z

L L

+

= − Γ

Along the transmission line we have

e

z

z ) ( 0 )

^2γ

( = Γ ⋅

Γ

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Reflections 3

Lossless line

Complex Γ-plane 4 )

)

(

0 ( )

( z e

^{j λ z}

⋅

π

Γ

= Γ

Standing waves

• The voltage U(z) follows (with no loss)

| )

0 ( 1

|

| ) 0 (

|

| ) ( )

(

|

| ) (

| U z = U

⁺

z + U

⁻

z = U

⁺

⋅ + Γ ⋅ e

^2γz

• The Voltage Standing Wave Ratio (VSWR) is

| ) 0 (

|

max

= 1 + Γ

= U VSWR

-L 0

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Line impedance

Input impedance at location z

) ( 1

) ( 1

) (

) ) (

(

₀

z Z z

z I

z z V

Z − Γ

Γ

⋅ +

=

=

If z= λ/4 we have special behaviour

R Z

_i

Z

2

=

0

Circuit characterisation

Two-port

+ V_s -

+ V₁

-

+ V₂ - Z_S

Z_L

I₁ I₂

Characterisation

• Signal amplification

• Frequency behaviour

• Noise behaviour

• Input impedance

• Output impedance

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Gain

Signal power

Periodic (sinusoidal) RF signals

) cos(

|

| } Re{

) (

) cos(

|

| } Re{

) (

2 0

1 0

0 0

ϕ ω

ϕ ω

ω ω

+

=

⋅

=

+

=

⋅

=

t I

e I t

i

t V

e V t

v

t j

t j

Power

∫ ^{⋅ = ⋅ ⋅}

^∗

=

^T

v t i t dt V I P T

0

} 2 Re{

) 1 ( ) 1 (

Impedance matching

2 2

,

2 | |

Re

|

|

L S

L S

L

S

Z Z

Z P V

+

⋅

= ⋅

=

_S∗

L

Z

Z

Z_S

Z_L +

V_s - Source output power into Z_L

Maximum when (available source power)

S S av

S

Z

P V

Re

|

| 8

1

²

,

= ⋅

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Power gain

in out

p

P

g = P

+ V_s -

+ V₁

-

+ V_O

- Z_S

Z_L

I₁ I₂

2 2

|

| 2

Re

|

|

in S

in S

in

Z Z

Z P V

+

⋅

= ⋅

Two-port with input and output impedances Z

_in

and Z

_out

2 2

|

| 2

Re

|

|

L out

L O

out

Z Z

Z P V

+

⋅

= ⋅

Input power

Output power

Gain

Power gain 2

Available power gain

Matching at in- and outputs

out S S

O av

in av out

av

Z

Z V

V P

g P

Re Re

|

|

|

|

2 2

,

,

= ⋅

=

+ V_s -

+ V₁

-

+ V_O

- Z_S

Z_L

I₁ I₂

out in

S

S in

O

Z Z

Z

Z Z

V V

Re

|

|

Re

|

|

|

|

|

|

2 2 2

1 2

⋅ +

⋅ ⋅

=

Z

_s

= Z

_in

Z

_L

= Z

_out

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Logarithmic units

dB 110 dB

60 dB 50 10

10

⁵

⋅

⁶

⇒ + =

(dB) decibel log

10 ¹⁰

in out

dB P

g = ⋅ P

• Decibel

(ref. Alexander Graham Bell) Large ranges of gain

Multiplication becomes addition

( ) dBm mW

log 1

10

¹⁰

⎟

⎠

⎜ ⎞

⎝

⋅ ⎛

= P

P

_dBm

• Absolute powers in decibel dBm: relative to 1 milli Watt

1 mW => 0 dBm 2 mW => 3 dBm 100 mW => 20 dBm

1 W => 30 dBm

dBi en dBc/Hz

• dBi

Antenna gain relative to isotropic transmitter

• dBc/Hz

Noise power per Hz of bandwidth

relative to the carrier

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Noise

Many different origins

• Thermal noise, resistors and lossy materials

• Shot noise, from electron particle behaviour

• 1/f noise, in semi conductors

• Generation-recombination noise (semi conductors)

• Phase noise (oscillators)

Thermal noise

R_N

Z_L +

V_N -

• Resistor noise, voltage V

_N

• With matching (Z_L = R_N) the noise dissipated in Z_L

T

R

k f

G ( ) = ⋅

k Boltzman’s constant T_R Resistor temperature

• Bandwidth limited noise power

Filter B = f

_h

-f

_l

G(f) N

Watt )

( f B k T B

G

N = ⋅ = ⋅

_R

⋅

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Noise figure

x i

o

i o

N N g N

S g S

+

⋅

=

⋅

=

• Signal to Noise Ratio (SNR) deterioration in a two port

(S/N)

_i

(S/N)

_o

N

_x

g

i x i

x i i

o o

i

N g

N N

g

N N g N g

N N

S N F S

+ ⋅

⋅ = +

= ⋅

= ⋅

= 1

) / (

) / (

• Noise figure F

N_i at T = 290 K

Noise temperature

At a fixed bandwidth noise is represented as an equivalent temperature

k f G B

k

T

_noise

N = ( )

= ⋅

Noise temperature with gain

N

_i

N

_o

=g.N

_i

+N

_x

T

_x

T

_i

g T

_o

=g.T

_i

+T

_x

N

_x

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Noise temperature of an amplifier

T

_i

T

_o

=g.T

_i

+T

_x

T

_x

g

Relate noise power back to the amplifier input

T

_i

+T

_x

/g T

_o

=g.(T

_i

+T

_x

/g)

thus

g T

_NF

= T

^x

Noise figure and noise temperature

0 0

1 1

1 T

T T

g T N

g

F N

^{x NF}

i

x

= +

+ ⋅

⋅ = +

=

)

0

1

( F T

T

_NF

= − ⋅

K 67 290 ) 1 23 , 1 ( 23

,

1 ⇒ = − ⋅ =

= T

_NF

F

Reference to the noise N

_i

at room temperature (T

₀

= 290 K)

For T

_NF

this gives

• Example

• Example T

_NF

= 25 K

dB 36 , 0 086 , 290 1

1 + 25 = =

F =

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Cascading

⋅

⋅

⋅

⋅

⋅ + +

+

=

2 1

3 1

2

1

g g

T g

T T T

_C

A series of amplifiers

T

₁

g

₁

T

₂

g

₂

T

₃

g

₃

Refer back to the input of the chain (Friis’ equation)

⋅

⋅

⋅

⋅

⋅ + + ⋅

+ ⋅ +

= +

=

0 2 1

3 0

1 2 0

1 0

1

1 g g T

T T

g T T

T T

F

_C

T

^C

+8

Smith chart

Line impedance Reflection coefficient

Combine the two ?

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Deduction

Normalise

0

0

Z

j X Z

x R j r

z = + = +

Position in Γ-plane

z z

z

and z

z z

Γ

− Γ

= + +

= −

Γ 1

1 1

1

• Biliniar transformation

• one-to-one

• can be inverted

• circles become circles

• direction and left/ right maintained

Example: impedance transformation

Six steps

1. Normalise Z

_L

by Z

₀

2. Determine position in Smith chart 3. Line from origin to z

_L

represents Γ

₀

4. Rotate Γ

₀

over -2βl

5. Read back Γ(-L) and z(-L)

6. Denormalise by Z

₀ (=> Z(-L))

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Example 6.1: impedance transformation

• Z_L = 100 + j 50 Ω

• Z₀ = 50 Ω

• f = 2,5 GHz

• L = 2 cm

• ε_r = 3,5 (thus v < light speed, v= c/√ε_r = 1,6 10⁸ m/s )

• Step 1

j j

z

_L

= + = 2 + 50

50 100

Example 6.1 (continued)

• Step 2: see figure

• Step 3

|Γ₀| = 0,46 Arg(Γ₀) = 26.6^o

• Step 4

-2βL = -224.5^o

• Step 5

z(-L) = 0,38 + 0,14 j

5 2

4

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Example 6.1: impedance transformation

• Z_L = 100 + j 50 Ω

• Z₀ = 50 Ω

• f = 2,5 GHz

• L = 2 cm

• ε_r = 3,5 (thus v < light speed, v= c/√ε_r = 1,6 10⁸ m/s )

• Step 6

Z(-L) = 18,9 + 6.8 j Ω

Useful tools

Smith.exe

http://www.hta-be.bfh.ch/~wwwel/projekte/cae/

Pasan (by Marien van Westen)

http://members.home.nl/mvanwesten/

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Two-port characterisation

Relate V₁, V₂, I₁ and I₂

Admittance matrix

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎥ ⎛

⎦

⎢ ⎤

⎣

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⇒ ⎛

⋅

=

2 1 22

21

12 11

2

1

V Y

I V

V Y

Y

Y Y

I I

Two-port

Two-port characterisation 2

Relate V₁, V₂, I₁ and I₂

Admittance matrix

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎥ ⎛

⎦

⎢ ⎤

⎣

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⇒ ⎛

⋅

=

2 1 22

21

12 11

2

1

V Y

I V

V Y

Y

Y Y

I I

Two-port

Impedance matrix

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎥ ⎛

⎦

⎢ ⎤

⎣

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⇒ ⎛

⋅

=

2 1 22

21

12 11

2

1

I Z

V I

I Z

Z

Z Z

V

V

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Two-port characterisation 3

Transmission ABCD-matrix

for cascading

⎟⎟⎠

⎜⎜ ⎞

⎝

⎥⎛

⎦

⎢ ⎤

⎣

=⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎥⎛

⎦

⎢ ⎤

⎣

=⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

3 3 2 2

2 2 2

2 2

2 1 1

1 1 1

1 and

I V D C

B A I

V I

V D C

B A I

V

⎟ ⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

⎥⎦ ⎡

⎢⎣ ⎤

= ⎡

⎟ ⎠

⎜ ⎞

⎝

⎛

3 3 2

2

2 2

1 1

1 1

1 1

I V D

C

B A

D C

B A I

V

Two-port characterisation 4

Relating ABCD and impedance matrices

( ) ⎥

⎦

⎢ ⎤

⎣

= ⎡

⎥ ⎦

⎢ ⎤

⎣

⎡

21 22

21

21 21

11

1

det

Z Z

Z

Z Z

Z D

C

B

A Z

Where det(Z) is the determinant

( )

_{11 22 12 21}

det Z = Z Z − Z Z

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Two-port characterisation 5

S-parameter matrix

• At high frequencies V and I can not be measured near two-port

• We can measure V

⁺

(z) and V

^-

(z)

• Determine amplitude and phase at some reference plane

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

+ +

−

−

2 1 22

21

12 11

2 1

V V S

S

S S

V V

ref.

plane_1 ref.

plane_2

port_1 port_2

Two-port characterisation 5

S-parameter matrix

• At high frequencies V and I can not be measured near two-port

• We can measure V

⁺

(z) and V

^-

(z)

• Determine amplitude and phase at some reference plane

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

+ +

−

−

12 1

1 11

V

S S

S V S

ref.

plane_1 ref.

plane_2

port_1 port_2

S11 Reflection at the input S21 Forward transmission S12 Backward transmission Ref. sheet 68-32, Voltage/

current/ Power relationship

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Example S-parameter matrix

Through connection

1 2

⎥ ⎦

⎢ ⎤

⎣

= ⎡

1 0

0 S 1

Amplifier

1 A 2

⎥ ⎦

⎢ ⎤

⎣

= ⎡

0 0

0 S A

Isolator (open)

1 2

⎥ ⎦

⎢ ⎤

⎣

= ⎡

0 0

0 S 1

Passive filter (R, L and C)

1 H(ω) 2

⎥ ⎦

⎢ ⎤

⎣

⎡

=

) ( 0

0 )

(

ω ω

H H

S

S-parameter matrix measurement

ref.

plane_1

port_1 requested ref.plane_1

ref.

plane_1 port_2

requested ref.plane_2

j k

Vk

j i

ij

V

S V

+= ∀ ≠

−

=

+

, 0

Network analyser

S

₁₁

• V₁⁺ input on port 1

• measure V₁^- (reflected wave)

• Keep V₂⁺ null (Z₀ terminated)

From S we can determine

Z

and

Y

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Reference plane and calibration

• Requested planes of reference

E.g. in- and output of the LNA (near the circuit)

• Practical reference planes

Connectors of Network Analyser

Interconnected with stable transmission lines

• Calibrate NWA at practical planes of reference

Using calibration kit references screwed directly on the cables Choose correct calibration procedure

• Check if calibration was successful Using calibration kit standards

Possibly measuring a (known) test impedance

• Repeat calibration often (to counter variability)

S-parameter matrix measurement

• Connect two port to practical planes of reference and measure

• Transform to requested planes of reference (de-embedding)

⎥ ⎦

⎢ ⎤

⎣

= ⎡

_{m m}

m m

S S

S S

22 21

12 11

S

m

⎥ ⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

= ⎡

⎥⎦ ⎤

⎢⎣ ⎡

β β

β β

2 1

2 1

0

0 0

0

22 21

12 11

22 21

12 11

jl jl

m m

m m

jl jl

e e

S S

S S

e e

S S

S S

• Network Analyser does this automatically after calibration

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Practical assignment

Measuring using a Spectrum Analyser and a noise source

Measure with a Network Analyser

Determining S

₁₁

of an unkown impedance

www.astron.nl

The end

Acknowledgement:

P. de Vrijer, M. Arts, W. v. Cappellen, R. Halfwerk, J.G. bij de Vaate, M. Bentum, D. Kant, J. Hamakers, J. Bregman, B. Woestenburg

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Literature

David M. Pozar; (Wiley 2005) Microwave engineering

C.R. Kitchin; (Taylor and Francis Group 2003) Astrophysical Techniques A.R. Thompson, J.M. Moran, G.W. Swenson Jr.; (Wiley 2004) Interferometry and Synthesis in Radio Astronomy

Web-books:

S. J. Orfanidis; Electromagnetic Waves and Antennas(802 pgs.) ECE Department, Rutgers University

http://www.ece.rutgers.edu/~orfanidi/ewa

D. Fisher; Basics of Radio Astronomy, for the Goldstone-Apple Valley Radio Telescope(91 pgs.)

Jet Propulsion Laboratory, JPL D-13835 (apr 1998) California Institute of Technology http://www.jpl.nasa.gov/radioastronomy

S. Mauch; Introduction to Methods of Applied Mathematics, or, Advanced Mathematical Methods for Scientists and Engineers(2321 pgs.)

CalTech 2004)

http://www.its.caltech.edu/˜sean