distributed circuit design in rf ics
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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
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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?
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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.
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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
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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
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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
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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
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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
− = + =
− = + =
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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).
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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
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Vx
Envelope≈ Voe-αz
α β
+ - z
x oV = V e cos(ω t - z)
z
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b l l d b b i i l f (6) & ( ) i (3 &3 b)
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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.
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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
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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
−= +
= − + +=
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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:
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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
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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
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THE SMITH CHART
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Bandwidth estimation techniques in IC design
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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
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Gate Drain
Bulk Source
Some Necessary Formulations
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Estimate the bandwidth of operation
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Methodology
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Basic Concepts in RF Design
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….(1)
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Harmonics such as
Eq’n (1)
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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
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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
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Impedance Matching
Th S i P ll l T f
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The Series Parallel Transforms
The L Match
series parallel
BW/ωo =1/Q
E l
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Example
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The π Match
The T Match
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The T Match
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