Distributed Circuit Design in RF ICS

8/13/2019 Distributed Circuit Design in RF ICS

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DISTRIBUTED SYSTEMS

IN RF IC DESIGN

By

Pragnan Chakravorty

Director, CARETM.Tech (IIT Kharagpur), Member-IEEE(USA), ACM(USA)

Member IEEE :-

Communication. Soc,

Microwave Theory and Techniques Soc,

Antenna & Wave Propagation. Soc

Clique for Applied Research in Electronic Technology

Advaita Corporation



What are Distributed System?

At first place they are all passive systems made out of Resistors

Capacitors and Inductors.

Is this not a good news?



Why do we need passives in IC design which is perhaps

grossly active?

Match / transform and modify impedances for effective power

transfer.

Cancel out transistor parasitics to increase gain, stability etc.

Modify the bandwidth of operations and to make circuits act as filters

Couple or decouple AC with DC.

Stabilize or destabilize a system.

All actives are indeed passives with dependent sources.





When can a lumped become distributed ?

When the size of a passive become comparable to the wavelength of

the signal(which it takes) then it changes from lumped to distributed

Passive Passive

L< = λ/12 L> = λ/8

LUMPED DISTRIBUTED



Why distributed ?

Lot of lumped elements inevitably become distributed athigh RF frequencies and hence need to be treated differently.

High RF frequencies mean higher bandwidth of operation

which can be practically achieved with distributed elementsonly



When signals at high RF frequencies start behaving as waves

then simple wires and interconnects become distributed

systems and are particularly known as transmission lines.

Transmission Lines



Transmission Lines

L∆z

I(z,t)

G∆z

∆I

C∆z

I(z +∆z),t)

z + ∆z

V(z +∆z),t)V(z,t)

I(z,t)

Main

node

Note that ∆z→0 ;( R≡ Ω/m; L ≡H/m) in conductor ;( G ≡ S/m, C ≡ F/m) in Dielectric

R∆z

z

Applying Kirchhoff’s voltage law to the outer loop of the above circuit we get:

I(z, t) V(z z, t) V(z, t) I(z, t)V(z, t) R zI(z, t) L z V(z z, t) RI(z, t) L

t z t

- ......(1)

∂ + ∆ − ∂= ∆ + ∆ + + ∆ ⇒ − = +

∂ ∆ ∂

⇒ V(z, t) I(z, t)

= RI(z, t) + L

z t



Applying Kirchhoff’s current law to the main node of the above circuit we get:

Equations (1) & (2) are known as telegraphist’s equation. Assuming

time harmonic variations

Equations (1) & (2) are known as telegraphist’s equation. Assuming time harmonicvariations, Equations (1) & (2) can be modified as:

Differentiating with respect to z and combining the above differential equations only

in terms of Vs or Is we get

I(z z, t) I(z, t) V(z z, t)I(z, t) I(z z) I GV(z z, t) C

z t

)......(2)

+ ∆ − ∂ + ∆= + ∆ + ∆ ⇒ − = + ∆ +

∆ ∂

⇒ I(z, t V(z, t)

- = GV(z, t) + Cz t

j t j t

s sV(z, t) Re[V (z)e ] and I(z, t) Re[I (z)e ]......(3)ω ω = =

(4) (5) where j (R j L)(G j C)γ α β ω ω = + = + +

2 2

2 2s s

s s2 2

d V d I-γ V = 0 ...... or - γ I = 0 ......

dx dx

ss s

s

s s

dV(R j L)I ZI ......(3.a )

dz

dI

(G j C)V YV ......(3.b)dz

ω

ω

− = + =

− = + =



Equations (4) & (5) are known as the wave equations for transmission lines where:

γ→ is propagation constant( in per meter ); α→ is attenuation constant( in

nepers/meter or db/m) and β→ is phase constant(in radians/meter)

Solutions of the differential equations (4) & (5):

Here + and – superscripts indicate wave motion along positive and negative z

direction respectively. Keeping the above values of Vs (z) and Is (z) in the equation(3):We obtain:

This should be noted that the ‘-’ sign associated with z indicates wave motion along

positive z. This is because to maintain ωt -βz = constant , with increase in time, z in ‘-βz’

moves in positive direction. ωt -βz = constant indicates the motion of equiphase along

positive z direction with increase in time.

2 22 2s s

s s2 2

d V d I

V 0 and I 0 aredx dx

......(6)

......(7)

γ γ − = − =+ -γz - +γz

s 0 0

+ -γz - +γz

s 0 0

V (z) = V e + V e

I (z) = I e + I e

......(8)

.....(9)

+ -αz - αz

0 0

+ -αz - αz

0 0

V(z,t) = V e cos(ωt - βz) + V e cos(ωt + βz)

I(z, t) = I e cos(ωt - βz) + I e cos(ωt + βz).





z

V or

I

t1

t2

t3

Equiphases, ωt -βz = constant at time t 1 t 2 and t 3

t z constant

differentiating with respect to time

dz dz0 ;Sinceβis phaseconstant; 2

dt dt

2; f ; here is wavelenght,f is linear frequency, is wave velocity

ω β

ω ω β ω βν ν λβ π

β

π ω λ ν λ λ ν

β β

⇒ − =

∴

− = − = ∴ = = =

∴ = = =

......(10)+ -

0 0

o o o+ -

0 0

V V R + jωL γ R + jωL ZZ = = - = = = = = R + jX

I Iγ G + jωC G + jωC Y



Vx

Envelope≈ Voe-αz

α β

+ - z

x oV = V e cos(ω t - z)

z





b l l d b b i i l f (6) & ( ) i (3 &3 b)



Zo can be calculated by substituting values from eqns (6) & (7) into eqns (3.a &3.b)

In Lossless Transmission line(R = G = 0): the conductors are perfect(R = 0) and dielectric

separating them is lossless (G = 0).

In distortion-less Transmission line (R/L = G/C): α should be frequency independent and β

should linearly vary with frequency to avoid distortion.

Transmission Line Characteristics (Table-1)

Type Propagation Constant

γ = α + jβ

Characteristics

Impedance

Zo = Ro+jXo

General

Lossless(R = G= 0)

Distortionless(R/L = G/C)

(R j L)(G j C)ZY

ω ω + +=

(R j L) Z(G j C) Y

ω ω

+ =+

/ 1 / LC

0 j LC j

ν ω β

ω β

= =

+ =

∴

L j0C

+RG j LCω +

L j0

C+

Note that for the case of lossless line, in γ the real part becomes zero where as in Zo

the imaginary part becomes zero.



Input Impedance:

Let’s consider a transmission line, of length l with γ and Zo specified which is

connected to a load ZL . From equations (6), (7) & (10) the corresponding voltage and

current equations will be:

From equations (11) & (12), calculating Vs (z) and Is (z) for z = 0 (Sending-end) and z = l(Receiving-end) ,

......(11)

......(12)

+ -γz - +γz

s 0 0

+ -γz - +γz0 0

s

0 0

V (z) = V e + V e

V e V eI (z) = -

Z Z

Vg

ZgIo

+

VL

_

+

Vo

_

ZL

z = 0 z = l

Zin

......(13)

.....(14)

+ -

s o 0 0

in + -

s 0 0

+γl - -γl

0 L o L 0 L o L

V (z = 0) Z (V + V )Z = =

I (z = 0) (V - V )

1 1V = (V + Z I )e and V = (V - Z I )e .

2 2



Substituting the values Eqn.(14) in Eqn.(13) we get:

Standing Wave Ratio (SWR):

Any two waves with same polarization, traveling in opposite directions will always

form standing waves. If they have same magnitude then they form pure standing

waves. The two waves interact in phase in some points and 1800 out of phase atsome other points. The ratio of maximum amplitude (in phase interaction) to the

minimum amplitude (180 0 out of phase interaction) of the standing wave is known as

standing wave ratio (SWR).It is often called VSWR (Voltage standing wave ratio).

It should be noted that the equiphase is lost and hence the wave become non

propagating(standing).Time and space can vary independently

{ }

z j t j t

x 1 2

x 1 2

V (z, t) V e e V e .....(17)

Re V (z, t ) V cos( t z) V cos( t z); for Lossles medium

2 cos( t ) cos( z)

γ ω ω

ω β ω β

ω β

−= +

= − + +=



V1 + V2

V1 - V2

1 2

1 2

V Vs SWR

V V

+= =

−

s = ∞ for purely standing waves

s = 1 for purely traveling waves

1≤ |s| ≤ ∞

z

r

i

V e Reflected wave......(18)

V e Incident Wave

1 s 1

s ; ......(19)1 s 1

z

γ

γ

−

+ −Γ = =

+ Γ −

= Γ =− Γ +

L oL

L o

Z Z......(20)

Z Z

−Γ =

+

Reflection coefficient:

If the oppositely traveling wave is a result of reflection from theincident wave then the reflection coefficient is defined as:

Reflection coefficient at load in the transmission l ines considered above

wil l be:



Type of Load Reflection Coefficient Standing Wave Ratio

S

Shorted Line(ZL = 0) -1 ∞ (pure standing wave)

Open Circuited Line(ZL = ∞) 1 ∞ (pure standing wave)

Matched Line (ZL = Zo) 0 1 (pure traveling wave)

LΓ

Reflection characteristics at different load conditions (Table-2)

Shorted Line(ZL = 0) Open Circuited Line(ZL = ∞) Matched Line (ZL = Zo)

1= −L

Γ 1=L

Γ 0=L

Γ

I d M hi d Q W (λ/4) f



2

L o o

in oo L L

'oino in o

o L

Z Z tan / 2 ZZ Z [here / 4 or (2 / )( / 4) / 2]......(21)

Z Z tan / 2 Z

Therefore :

ZZ Now if Z is selected such that Z Z then

Z Z

......(22)

l lπ

λ β π λ λ λ π

′ ′+′= = = = =

′ +

′= =

′

'

o o LZ = Z Z

Impedance Matching and Quarter Wave (λ/4) transformer:

It is obvious from the above table that if the load impedance is not

matched with the characteristics impedance then reflections are

bound to happen and standing waves are bound to form. So to avoid

the formation of standing waves thereby losses in the transmissionline, ZL is matched (apparently made equal) to Zo. This can be done

with the help of Quarter Wave (λ/4) transformer.

λ/4

ZLZo

Zin = Zo



THE SMITH CHART













Bandwidth estimation techniques in IC design







Before we begin to estimate the bandwidth we must look into MOSFET device and its model

for the sake of comprehension. We apply the Oct to the device model after deducing some

formulation on the device model



Gate Drain

Bulk Source

Some Necessary Formulations



Estimate the bandwidth of operation







Methodology



Basic Concepts in RF Design



….(1)









Harmonics such as

Eq’n (1)





Signals can not be time limited or band limited at the together thus a

time limited signal gives rise to a n infinite bandwidth and finite

bandwidth gives a time domain non-limitation causing intersymbolinterference





Noise which is distributed arbitrarily in a circuit can be equivalently represented as

an input noise source usually known as input referred noise, it is modeled as an

equivalent voltage and current source





Impedance Matching

Th S i P ll l T f



The Series Parallel Transforms

The L Match

series parallel

BW/ωo =1/Q

E l



Example



The π Match

The T Match



The T Match

Documents

Distributed Circuit Design in RF ICS