1.1.4 4 Diff Differe erence bet nce betwee ween ideal an n ideal and pract d practica ical passi l passive dev ve device ices s

1.1.4 4 Diff Differe erence bet nce betwee ween ideal an n ideal and pract d practica ical passi l passive dev ve device ices s

Practical devices have so-called parasitic elements at very high frequencies.

Practical devices have so-called parasitic elements at very high frequencies.

Resistor 

Resistor  Has an inductive parasitic action and acts Has an inductive parasitic action and acts like a low pass flike a low pass filtering function.iltering function.

Inductor 

Inductor  Has a capacitive and resistive parasitic, causing it to act like a damped parallelHas a capacitive and resistive parasitic, causing it to act like a damped parallel resonant tank circuit with a

resonant tank circuit with a certain self resonance.certain self resonance.

Capacitor 

Capacitor  Has an inductive and resistive parasitic, causing it to act like a damped tank circuitHas an inductive and resistive parasitic, causing it to act like a damped tank circuit with

with S S erieseries R R esonanceesonance F F requency (requency (SRF SRF ).).

The inductor’s and the

The inductor’s and the capacitor’s parasitic reactance causes self-resonances.capacitor’s parasitic reactance causes self-resonances.

 Figure 7:

 Figure 7: Equivalent models of Equivalent models of passive lumped passive lumped elementselements

The use of a

The use of a passive componenpassive component above its SRF is t above its SRF is possible, but must be critically evaluated. Apossible, but must be critically evaluated. A capacitor above its SRF appears as

capacitor above its SRF appears as an inductor with DC blocking capabilities.an inductor with DC blocking capabilities.

1.

1.1. 1.5 5 Th The e Sm Smit ith h Ch Char artt

As indicated in an example in

As indicated in an example in the former chapter, the the former chapter, the impedances of semiconductors are aimpedances of semiconductors are a combination of resistive and reactive parts caused by

combination of resistive and reactive parts caused by phase delays and parasitic. RF is phase delays and parasitic. RF is best analyzedbest analyzed in the frequency domain under the use

in the frequency domain under the use of vector algebraic expressions:of vector algebraic expressions:

Object 

Object  èè into into  èè Frequency domain Frequency domain  Resistor

Some useful basic vector algebra in

Some useful basic vector algebra in RF analysis:RF analysis:

Complex impedance 

Use of angle èè Polar Polar notationnotation Use of sum

Use of sum èè Cartesian (Rectangular) Cartesian (Rectangular) notationnotation

The same rules are used for other issues, The same rules are used for other issues,

e.g., the

e.g., the complex reflection coefficient complex reflection coefficient :: ^{(( bb f f ))}

 f 

§§ Resistive mismatch:Resistive mismatch: ϕϕ(( R R))

== ⁰⁰ °°

reflection coefficient:reflection coefficient: ϕϕ((r r ))

== ⁰⁰ °°

§§ Inductive mismatch:Inductive mismatch: ϕϕ(( L L))

== ++ ^{90 90} °°

reflection coefficient:reflection coefficient: ϕϕ((r r ))

== ++ ^{90 90} °°

§§ Capacity mismatch:Capacity mismatch: ϕϕ_((C C ))

== −− 90 90 °°

reflection coefficient:reflection coefficient: ϕϕ_((r r ))

== −− 90 90 °°

Im Im

Re{Z}

Re{Z}

Im{Z}

Im{Z}

ZZ ZZ

Re Re

Resistive-Axis Resistive-Axis

Reactive-Axis Reactive-Axis In applications RF designers try to remain close to a 50

In applications RF designers try to remain close to a 50

Ω Ω

resistive impedance. The polar diagram’sresistive impedance. The polar diagram’s origin is 0

origin is 0

Ω Ω

. In RF . In RF circuit’s, relative large impedcircuit’s, relative large impedances can occur but we try to ances can occur but we try to remain close to 50remain close to 50

Ω Ω

byby special network design for maximum power transfer. Practically, very low

special network design for maximum power transfer. Practically, very low and very high impedancesand very high impedances don’t need to be

don’t need to be known accurately. The Polar diagram can’t show simultaneous large impedances andknown accurately. The Polar diagram can’t show simultaneous large impedances and the 50

the 50

Ω Ω

region with acceptable accuracy, because of limited paper size.region with acceptable accuracy, because of limited paper size.

Dots on the Re-Line are 100% resistive Dots on the Re-Line are 100% resistive Dots on the Im-Line are 100% reactive Dots on the Im-Line are 100% reactive Dots some their above the

Dots some their above the Re-Line are inductive + resistiveRe-Line are inductive + resistive Dots som

Dots some their e their below the below the Re-Line Re-Line are capaciare capacity ty + + resistiveresistive 0°

180° 0°

180°

Using this fact Mr. Phillip Smith, an engineer at Bell Using this fact Mr. Phillip Smith, an engineer at Bell Laboratories develop

Laboratories developed in the ed in the 1930s the so-called1930s the so-called Smith Chart 

Smith Chart . The chart’s origin is at 50. The chart’s origin is at 50

Ω Ω

. Left and right. Left and right resistive values along the real axis end in 0

resistive values along the real axis end in 0

Ω Ω

and atand at

∞Ω ∞Ω

..

The imaginary reactive (imaginary axis, or Im-Axis) end i The imaginary reactive (imaginary axis, or Im-Axis) end i 100% reactive (L or C). Close to the 50

100% reactive (L or C). Close to the 50

Ω Ω

origin highorigin high resolution is offered. Far away of the chart’s centre does resolution is offered. Far away of the chart’s centre does the resolution dope down. From the centre of the chart, the resolution dope down. From the centre of the chart, the resolution / error increases. The standard Smith Char the resolution / error increases. The standard Smith Char only displays

only displays positive resistances positive resistances and has a unit radiusand has a unit radius (r=1).

(r=1). Negative resistances Negative resistances generated bygenerated by instability instability  (eg.

(eg. oscillation oscillation ) lay outside the unit circle. In ) lay outside the unit circle. In this non-this non-linear scaled diagram, the infinite dot of the Re-Axis is linear scaled diagram, the infinite dot of the Re-Axis is

“theoretically” bend to the zero point of the Smith Chart.

“theoretically” bend to the zero point of the Smith Chart.

Mathematically it can be shown that this will form the Mathematically it can be shown that this will form the Smith Chart’s unit circle (r=1). All dot’s laying on this Smith Chart’s unit circle (r=1). All dot’s laying on this circle represent a reflection coefficient magnitude of 1 circle represent a reflection coefficient magnitude of 1 (100% mismatch). Any positive L/C combination with a (100% mismatch). Any positive L/C combination with a resistor will be mathematically represented by it’s polar resistor will be mathematically represented by it’s polar notation reflection coefficient inside the Smith Chart’s notation reflection coefficient inside the Smith Chart’s unity circle. Because the Smith Chart is a transformed unity circle. Because the Smith Chart is a transformed linear scaled polar diagram we can use 100% of the pola linear scaled polar diagram we can use 100% of the pola diagram rules. The

diagram rules. The Cartesian-diCartesian-diagram rules are agram rules are changed,changed, because of the non-linear scaling.

because of the non-linear scaling.

∞Ω

∞Ω

0 0ΩΩ

Special cases:

Special cases:

§§ DoDots ats abobove tve the hhe hororizizonontatal axl axis ris repepreresesentnts ims impepedadancnce wie with ith indnducuctitive pve parartt ( 0° ( 0° <<

ϕϕ

< 180° )< 180° )

§§ Dots Dots below below the the horizontal horizontal axis axis represents represents impedance impedance with with capacitive capacitive part part ( ( 180° 180° <<

ϕϕ

< 360° )< 360° )

§§ DDootts s llaayyiinng g oon n tthhe e hhoorriizzoonnttaal l aaxxiis s ((oorrddiinnaattee) ) aarre e 110000% % rreessiissttiivvee ((

ϕϕ

= 0° )= 0° )

§§ Dots Dots laying laying on on the the vertical vertical axis axis (abscissa) (abscissa) are are 100% 100% reactive reactive ((

ϕϕ

= 90° )= 90° )

Figure 8: BGA2003 output Smith Chart (S  Figure 8: BGA2003 output Smith Chart (S 22 22  )  )  Illustrated are the special cases for

Illustrated are the special cases for ZERO and infinitely large impedance. The upper half circle ZERO and infinitely large impedance. The upper half circle is theis the inductive region. The lower half of the circle is the capacitive region. The origin is the 50

inductive region. The lower half of the circle is the capacitive region. The origin is the 50

Ω Ω

systemsystem reference (Z

reference (ZOO). To be more flexible, numbers printed in the chart are normalized to Z). To be more flexible, numbers printed in the chart are normalized to ZOO..

Normalizing impedance 

Normalizing impedance procedure:procedure:

o o  x  x norm norm  Z  Z 

 Z   Z   Z 

 Z 

==

ZZOO= System reference impedance (e.g., 50= System reference impedance (e.g., 50

Ω Ω

, 75, 75

Ω Ω

)) Example:

Example: Plot Plot a a 100100

Ω Ω

& 50& 50

Ω Ω

resistor into the upperresistor into the upper BGA2003 BGA2003 ’s output Smith chart.’s output Smith chart.

C

Caallccuullaattiioonn:: ZZnorm1norm1=100=100

Ω Ω

 /50 /50

Ω Ω

=2; Z=2; Znorm2norm2=25=25

Ω Ω

 /50 /50

Ω Ω

=0.5=0.5 R

Reessuulltt:: TThhe e 110000

Ω Ω

resistor appears as a dot on the horizontal axis at the location 2.resistor appears as a dot on the horizontal axis at the location 2.

The 25

The 25

Ω Ω

resistor appears as a dot on the horizontal axis at the location 0.5resistor appears as a dot on the horizontal axis at the location 0.5

Scaling rule for determine Scaling rule for determine the Magnitude (vector the Magnitude (vector distance) of the reflection distance) of the reflection coefficient

coefficient Z=0

Z=0ΩΩ ^Z=Z=∞Ω∞Ω

L-Area L-Area

C-Area C-Area

100 100ΩΩ

25 25ΩΩ

reactance @ 100MHz design frequency. Determine the value of

reactance @ 100MHz design frequency. Determine the value of the parts. Plot theirthe parts. Plot their impedance in to the

impedance in to the BFG425W BFG425W ’s output (S22) Smith Chart.’s output (S22) Smith Chart.

C

Ciirrccuuiitt:: RReessuulltt::

è è

Calculation:

Calculation: Case ACase A (constant resistance)(constant resistance)

In document Philips Rf Manual 4th Edition Appendix (Page 54-58)