JS/2006/oct/09

### R&D Division

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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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(C) ASTRON 2006

### 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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(C) ASTRON 2006

### 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

Iantenna 1 antenna 2 receive

### Antenna circuit substitution

### + -

V_{1}

I_{1}

### Receive and transmit antenna together form a two-port

### + -

V_{2}I

_{2}

### 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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(C) ASTRON 2006

### 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}

### + -

V_{1}

I_{1}

### + -

V_{2}I

_{2}

### 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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(C) ASTRON 2006

### 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=2D^{2}/λ

R=0.62 sqrt(D^{3}/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

*a*_{1} *a*_{2}

*b*_{1} *b*_{2}

**<=>**

### ⎟⎟ ⎞

### ⎜⎜ ⎛

### ⎥ ⎤

### ⎢ ⎡

### ⎟⎟ =

### ⎜⎜ ⎞

### ⎛ *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} = Z_{0}

### 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 )

^{2}

^{γ}

### ( = 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*

^{2}

^{γ}

^{z}

### • 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}
*-*

+
*V*_{1}

*-*

+
*V*_{2}
*-*
*Z*_{S}

*Z*_{L}

*I*_{1} *I*_{2}

### 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}
*-*

+
*V*_{1}

*-*

+
*V*_{O}

*-*
*Z*_{S}

*Z*_{L}

*I*_{1} *I*_{2}

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}
*-*

+
*V*_{1}

*-*

+
*V*_{O}

*-*
*Z*_{S}

*Z*_{L}

*I*_{1} *I*_{2}

*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 gainMultiplication 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

• Z_{0} = 50 Ω

• f = 2,5 GHz

• L = 2 cm

• ε_{r } = 3,5 (thus v < light speed, v= c/√ε_{r} = 1,6 10^{8} m/s )

• Step 1

*j* *j*

*z*

_{L}

### = + = 2 + 50

### 50 100

### Example 6.1 (continued)

• Step 2: see figure

• Step 3

|Γ_{0}| = 0,46
Arg(Γ_{0}) = 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

• Z_{0} = 50 Ω

• f = 2,5 GHz

• L = 2 cm

• ε_{r } = 3,5 (thus v < light speed, v= c/√ε_{r} = 1,6 10^{8} 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 V_{1}, V_{2}, I_{1} and I_{2}

### 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 V_{1}, V_{2}, I_{1} and I_{2}

### 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*

*V**k*

*j*
*i*

*ij*

*V*

*S* *V*

+= ∀ ≠

−

### =

+, 0

### Network analyser

### S

_{11}

• V_{1}^{+} input on port 1

• measure V_{1}^{-}
(reflected wave)

• Keep V_{2}^{+} null
(Z_{0} 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