IN5240 S Parameters, Impedance Matching and Smith Charts · •Signal flow and Mason’s rule to calculate input reflection, transducer gain of a two-port network IN5240: Design of

Institutt for Informatikk

IN5240 S Parameters, Impedance Matching and

Smith Charts

Sumit Bagga* and Dag T. Wisland**

*Staff IC Design Engineer, Novelda AS**CTO, Novelda AS


Outline

• Scattering parameters*• Impedance matching• Smith chart

*Covered in detail in amplifier design

IN5240: Design of CMOS RF-Integrated Circuits, Dag T. Wisland and Sumit Bagga


Scattering Parameters

• Difficult to measure voltages/currents at RF àS-parameters w/ ‘power’ flow– "# = "%&,( − "*, where under matched conditions "%&,( = +(,/8/0

• Signal flow and Mason’s rule to calculate input reflection, transducer gain of a two-port network


[Niknejad, EECS 242]


Return Loss and Mismatch Loss

• Absolute impedance: !" + $"

• Reflection coefficient, Γ: (()*+)-./-

((.*+)-./-, where 01 is

the source impedance

• Voltage standing wave ratio (VSWR): 234254

• Return loss (S11): −20log(Γ)• Mismatch loss: −10(1 − Γ")

IN5240: Design of CMOS RF-Integrated Circuits,

Dag T. Wisland and Sumit Bagga


Smith Chart

• Graphical tool invented in 1939 by Phillip H. Smith to evaluate input-output transfer functions, complex functions, such as:– Complex voltage and current transmission and reflections

coefficients, power reflection and transmission coefficients, reflection and return loss, standing wave loss factor, !"#$ and !"%&

– Reflection coefficient for a loss-less line is a circle of unitary radius in the complex plane à Smith chart domain

• Identify all impedances of the reflection coefficient



Frank Lynch, W4FAL

Page 12

24 April 2008

The officalversion!



Transmission Line &Impedance Admittance

• Characteristic impedance– "# = "%("'+ "% tanh , -)/("%+ "' tanh , -), where , =0 + 23

– 45/6 = 4748• Impedance is resistance + j(reactance)

– " = 9 + 2:• Admittance is conductance + j(susceptance)

– Y= ; + 2<



Smith Chart – Impedance Form


Constant resistance ‘circle’

Constant reactance ‘arc’


Real Part of Smith Chart w/ |"| ≤ $

• Normalized resistance %, center is &'(& , 0 & radius is '

'(&


[Amanogawa, 2006 - Digital Maestro Series]

Transmission Lines

© Amanogawa, 2006 - Digital Maestro Series 170

The result for the real part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized resistance r are found on a circle with

1, 01 1rr r+ +

Center = Radius =

As the normalized resistance r varies from 0 to ∞ , we obtain a family of circles completely contained inside the domain of the reflection coefficient | Γ | ≤ 1 .

Im(Γ )

Re(Γ )

r = 0

r →∞

r = 1

r = 0.5

r = 5


Imaginary Part of Smith Chart w/ |"| ≤ $

• Normalized reactance %, center is 1, () & radius is ()


[Amanogawa, 2006 - Digital Maestro Series]

Transmission Lines

© Amanogawa, 2006 - Digital Maestro Series 171

The result for the imaginary part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized reactance x are found on a circle with

1 11,x x

Center = Radius =

As the normalized reactance x varies from -∞ to ∞ , we obtain a family of arcs contained inside the domain of the reflection coefficient | Γ | ≤ 1 .

Im(Γ )

Re(Γ )x = 0

x →±∞

x = 1

x = 0.5

x = -1x = - 0.5


Smith Chart Admittance

• Impedance and admittance à opposite sides of the Smith chart à imaginary parts w/ opposite signs – Positive (inductive) reactance à negative (inductive)

susceptance– Negative (capacitive) reactance à positive (capacitive)

susceptance

• For !" = $ + &' and (" = ) + &* = +,-./, then

) = ,,0-/0 and b = 2/

,0-/0



Basics of Smith Chart

• Impedance, ! " à reflection coefficient, #(")• Reflection coefficient, #(")à impedance, ! "• Impedance, ! " à admittance, &(")• Admittance, Y " à impedance, !(")• VSWR (voltage standing wave ratio)

– Maximum and minimum locations ("()* and "(+,) for

the voltage standing wave pattern

IN5240: Design of CMOS RF-Integrated

Circuits, Dag T. Wisland and Sumit Bagga


! " à #(")

• Step 1: Normalize the impedance – '( " = ! " /!+ = ,/!+ + .//!+ = 0 + .1/!+

• Step 2: On the circle of constant normalized resistance, find 0

• Step 3: On the arc of constant normalized reactance, find 1

• Intersection of two curves à reflection coefficient à magnitude and the phase angle of Γ



! " à VSWR

• Step 1: Find the reflection coefficient and the normalized impedance on Smith chart

• Step 2: Draw circle of constant reflection coefficient amplitude

• Step 3: Find intersection of this circle with the real positive axis for the reflection coefficient à "'()

• Eq. *+ "'() = -./ 0123-4/ 0123

= -./5-4/5

= 6789

•IN5240: Design of CMOS RF-Integrated Circuits, Dag T. Wisland and Sumit Bagga

Institutt for InformatikkIN5240: Design of CMOS RF-Integrated Circuits, Dag T. Wisland and Sumit Bagga



!(#)à % #

• Step 1: Find complex point on the chart for !(#)• Step 2: Normalized impedance

– '( # = * + ,-• Step 3: Actual impedance

– % # = '( # %. = %. * + ,- = %.* + ,%.x




! " à #(")

• Step 1: Find load reflection coefficient and the

normalized load impedance on chart

• Step 2: Draw circle of constant reflection

coefficient amplitude

• Step 3: Normalized admittance is the point on the

circle of constant |Γ| diametrically opposite to the

normalized impedance

– ) = 180° or λ/4





Transmission Line



• ‘Short lines’ à lumped element distributed model

– Shorted line à magnetic flux, ! = #$– Open line à electric field, Q = &'

• “… signals travel instantly, ( → 0”


Transmission Line contd.

• Velocity of the sine wave is !"#

• If $ and % remain constant with &, the velocity of all sine waves will be the same!– Ideal transmission line: '(), +) = '(0, + − 0)with 0 =) 12

• Relationship between ' and 3 at the input of our transmission line à characteristic impedance, 4 =5(6,7)8 6,7 = "

#



Transmission Line contd.


• Finite transmission line with a termination resistance– Current on T-line ≠ current at load à discontinuity à

reflected wave– Γ$ = −1 (($ = 0), Γ$ = 1 (($ = ∞) & Γ$ = 0 (($ = +,)

• Steady-state, eq. circuit à -.. = /.( 1212314

)à T-line

– T-line is visible if we disconnect the source or load!

Transmission Line Termination

Rs

i+ =v+

Z0

+Vs

−

+v+

−

ℓ

Z0, td i =vL

RL

Consider a finite transmission line with a terminationresistanceAt the load we know that Ohm’s law is valid: IL = VL/RL

So at time t = ℓ/v, our pulse reaches the load. Sincethe current on the T-line is i+ = v+/Z0 = Vs/(Z0 + Rs)and the current at the load is VL/RL, a discontinuity isproduced at the load.

University of California, Berkeley EECS 117 Lecture 2 – p. 5/22


Transmission Line Summary

• For !"= !$ à Γ"= 0, and there is no reflection

– Forward wave carries energy à distributed '’) and *’)absorb energy temporarily à electrical energy to !$

– Ideal transmission line does not dissipate energy, only

transport energy!

• For !"= 0, à Γ"= −1– Voltage on the load is 0; current flowing into the load is

twice the current of the forward wave

• For !"= ∞, à Γ"= 1– Current into the load is 0; voltage on the load is double the

forward wave's voltage




Wavelength

• Wire length, ℓ > #/10 of RF signal à transmission line effects (variation of current/voltage along signal path)– @ 60 GHz à 1/16th rule of thumb

• Complex plane, a circle with '{0,0} and radius |Γ-|à possible reflection coefficients along T-line àvalues of the line impedance at any location

• Phase, . = 212 = 2(24λ)Δ(180/4)à find 29:;and 29<= of VSWR





Matching

• Absorb or resonate imaginary part of source (or load) impedance

• Transform real part for maximum power transfer



L-Matching

ECE145A/ECE218A Impedance Matching Notes set #5 Page 2

“L” Matching Networks 8 possibilities for single frequency (narrow-band) lumped element matching networks.

Figure is from: G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, Second Ed., Prentice Hall, 1997. These networks are used to cancel the reactive component of the load and transform the real part so that the full available power is delivered into the real part of the load impedance. 1. Absorb or resonate imaginary part of Zs and ZL . 2. Transform real part as needed to obtain maximum power transfer.

Rev. January 22, 2007 Prof. S. Long, ECE, UCSB


[G. Gonzalez, 1997]


LC Matching

Frank Lynch, W4FAL

Page 19

24 April 2008

Series L (increasing L)

Series Cdecreasing Series R (increasing)

Parallel L (decreasing)Parallel Rdecreasing

Parallel Cincreasing

What Components do on the Smith Chart….


[F. Lynch, W4FAL]


Stubs

Frank Lynch, W4FAL

Page 25

24 April 2008

OC Stub

SC Stub

Stubs can rotate all the way around the chart (unlike shunt L’s and C’s),but along circles of constant conductance (Like L’s and C’s).


[F. Lynch, W4FAL]




Series – Parallel Transformation

• Recall, !" = !$(1 + ()) & +" = +$(1 + 1/()), where ( is the unloaded quality-factor– !||/: +" = 1/ω/" and +$ = 1/ω/$

• Conjugate match à design matching network to match 23 to 24 and cancel reactances



Example L-Matching

• LPF with known !" and !#

• Quality factor, $ = &'&(− *à +# = ,/!# and

+" = !"/,– / = 0(/1 and 2 = */10'

• Through parallel to series transformation, reactances are equal and opposite, with !# = !"∗



Example L-Matching contd.

• Match at ! = 1590MHz with '( = 50 Ω and

'* = 500 Ω– , = 2 3.14 159016 = 1e10 rad/s

• 5 = 67767 − 1 = 3, so 9( = 3'( = 150 Ω and

9* = '*/3 = 500/3 = 167 Ω• @ ,, C = 0.6 pF and L = 15 nH




ECE145A/ECE218A Impedance Matching Notes set #5 Page 9

Figure is from: G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, Second Ed., Prentice Hall, 1997.

Rev. January 22, 2007 Prof. S. Long, ECE, UCSB

[G. Gonzalez, 1997]


Broadband Match

Frank Lynch, W4FAL

Page 29

24 April 2008

Using Many Lumped Elements

Although the graph below was done on a software program, this complex (5L’s, 5C’s) matching network could have easilybeen done on a paper smith chart. The same calculations to do this would have been very time consuming.


[F. Lynch, W4FAL]


Key References

1. Amanogawa, Digital Maestro Series, 20062. A. M. Niknejad, EECS 1173. F. Lynch, W4FAL4. G. Gonzalez, Microwave Transistor Amplifiers:

Analysis and Design, Second Ed., Prentice Hall, 1997


Documents

IN5240 S Parameters, Impedance Matching and Smith Charts · •Signal flow and Mason’s rule to calculate input reflection, transducer gain of a two-port network IN5240: Design of