freq response
DESCRIPTION
Nice explanationTRANSCRIPT
ALP_Rotondaro EE5321/EE7321 1
EE5321/EE7321Semiconductor Devices and Circuits
Frequency Response Part1
ALP_Rotondaro EE5321/EE7321 2
Impedance network transfer function • Impedance network transfer function:
where H(ω), Vout(ω) and Vin(ω) are phasors
( ) ( )( )ωωω
in
out
VV H =
( ) ( )( ) C R j 1
1 C j
1 R
C j1
VV H
in
out
ωω
ωωωω
+=
+
==
ALP_Rotondaro EE5321/EE7321 3
H(ωωωω) in polar coordinates• H(ω) is represented by its amplitude and phase• Amplitude |H(ω)|
• Phase ∠θ
• If H(ω) = N(ω) / D(ω) then:
Re[H(ω)] = Re[N(ω)·D*(ω)]
Im[H(ω)] = Im[N(ω)·D*(ω)]
( ) ( ) )(HH H * ωωω •=
( ) ( )[ ]( )[ ]
=
ωωωθ
HReHImarctan
ALP_Rotondaro EE5321/EE7321 4
H(ωωωω) for the RC circuit• Amplitude
• Amplitude in Decibels
• For a -3dB reduction on the magnitude
( ) ( )
−
•
+
=•C R j1
1C R j1
1 HH *
ωωωω
( ) ( ) ( )
+=• 2
*
C R 11 HH
ωωω
( ) ( )[ ]ωω Hlog20 H dB •=
( )[ ]dB3Hlog20 3 ω•=− ( ) 0.7079H 3 =dBω
ALP_Rotondaro EE5321/EE7321 5
Bode Plot RC circuit – Amplitude
( ) ( ) 0.7079 C R 1
1 H 23
3dB =
+=
dBωω CR
1 p3dB == ωω
( )
+
= 2
3
1
1 H
dBωω
ω For ω >> ω3dB ( )
≈ 2
3
1 H
dBωω
ω
( )ω
ωω dB3 H ≈
Amp drops by 2 when f doubles
Amp drops by 10 every decade
ALP_Rotondaro EE5321/EE7321 6
Bode Plot RC circuit - Phase
• H(ω) phase ( ) ( )[ ]( )[ ]
=
ωωωθ
HReHImarctan
( ) 2
3dB
3dB
3dB
3dB
3dB3dB 1
j 1
j 1
j 1
j 1
1 j 1
1 H
+
−=
−
−•
+=
+=
ωω
ωω
ωω
ωω
ωω
ωωω
( ){ } 2
3dB
1
1 H Re
+
=
ωω
ω ( ){ } 2
3dB
3dB
1
H I
+
−
=
ωω
ωω
ωm
ALP_Rotondaro EE5321/EE7321 7
Bode Plot RC circuit - Phase• And the Phase is given by:
( ) ( )[ ]( )[ ]
=
−=
=
3dB3dB
-arctanarctan HReHImarctan
ωω
ωω
ωωωθ
ALP_Rotondaro EE5321/EE7321 8
RC circuit – sine wave• The output wave has amplitude and phase altered by
the circuit
In Out
( )C R j 1
1 Hω
ω+
=
ALP_Rotondaro EE5321/EE7321 9
Bode Plots – 1 pole• RC circuit
( )ω
ωω dB3 H ≈
( )
=3dB
-arctan ωωωθ
CR1 p3dB == ωω
ALP_Rotondaro EE5321/EE7321 10
SPICE SIM – RC circuit• Run AC Sweep with 1V amplitude and freq: 10Hz to
100MHz• Output DB[V2(C1)/V1(V1)] and P[V2(C1)]
ALP_Rotondaro EE5321/EE7321 11
SPICE SIM – RC circuit• ωp = 1/RC = 10k → fp = 1.6kHz
ALP_Rotondaro EE5321/EE7321 12
RC circuits in series – 2 poles• The combination of two RC circuits in series is going
to result in 2 poles
( )
p2p1
j 1
1
j 1
1 H
ωω
ωωω
+•
+= where: ωp1 = 1/R1C1 and
ωp2 = 1/R2C2
ALP_Rotondaro EE5321/EE7321 13
RC circuits in series – 2 poles• Overall transfer function ( ) ( ) ( )ωωωωω p2p1 HHH •=
( ) ( )[ ] ( )[ ]p2p2p1p1 jexpH jexpH H ωωωω θθω •••=
( ) [ ]( )p2p1p2p1 jexpH H H ωωωω θθω +••=
( ) ( ) ( )( )ωθωω jexpH H •=
( ) ( ) p2p1p2p1 and H H H ωωωω θθωθω +=•=∴
ALP_Rotondaro EE5321/EE7321 14
Amplitude Bode Plot – 2 poles• Second pole “accelerates” the amplitude reduction
20dB/Dec
40dB/Dec
( ){ } ( ) ( ){ } HH log20 H log20 21 pp ωωω ••=•
( ){ } ( ){ } ( ){ } H log20 H log20 H log20 21 pp ωωω •+•=•
ALP_Rotondaro EE5321/EE7321 15
Phase Bode Plot – 2 poles• Second pole adds to the phase shift
( ) p2p1 ωω θθωθ +=
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2 poles circuit – 180° phase shift• A phase shift of 180° can be a problem
• If in a feedback loop, a 180° phase shift will turn a negative feedback into a positive feedback
• This results in an unstable system if the loop gain is > 1
ALP_Rotondaro EE5321/EE7321 17
Bode Plots – 3 Superimposed Poles• The phase shift
is quite “fast” and “strong”
• When used in a feedback loop will probably result in an unstable circuit
ALP_Rotondaro EE5321/EE7321 18
C R circuit – H(ωωωω)• Circuit has:
1 Zero at ω = 01 Pole at ω = 1/R C
( )
1/RC j 1
CRj C R j 1
C Rj C j
1 R
R H ωω
ωω
ω
ω+
=+
=
+
=
( ) ( )( )ωωω
in
out
VV H =
( ) ( )2C R 11C R H
ωωω
+•=
ALP_Rotondaro EE5321/EE7321 19
C R circuit – Bode plot amplitude• At ω = 0 → |H(ω)| = 0 and since
-3dB
( ) ( )[ ]ωω Hlog20H dB •=
( ) ( ) ∞→
+••= -
C R 11C R log20 H 2dB ω
ωω
( ) ( ) 3dB- 1 1
11 log20 H 2dB=
+••=pω
( )( )
01log20
HdB
=•=
>> pωω
ALP_Rotondaro EE5321/EE7321 20
SPICE SIM – C R circuit• ωp = 1/RC = 10k → fp = 1.6kHz
ALP_Rotondaro EE5321/EE7321 21
Zero’s phase response• The phase response of a Zero depends on which
half plane the Zero is located
( )zss-1 sH = ( )
zss1 sH +=zz j-s ω=
ALP_Rotondaro EE5321/EE7321 22
Zero’s gain response• For Zero in either half plane the amplitude
response is the same
0
( )2
z
1 H
+=
ωωω ( )
+•=
2
zdB 1 log10 H
ωωω
( )
dB/dec20
log20
H
z
dB
•≈
>>
ωω
ωω z
ALP_Rotondaro EE5321/EE7321 23
Transfer function – Other circuits• 1 Pole
• 1 Pole, 1 Zero
( ) ( )CR||Rj11
RRRH
2121
2
ωω
+•
+=
( ) ( )CR||Rj1CRj1
RRRH
21
1
21
2
ωωω
++
•+
=
ALP_Rotondaro EE5321/EE7321 24
1 Pole, 1 Zero response• The response depends
on the relative location of the Pole and the Zero ( )
p
z
j1
j1 H
ωωωω
ω+
+=
ALP_Rotondaro EE5321/EE7321 25
MOSFET capacitances - circuit• Specs: tox (Cox), CGSO, CGDO, CGBO, CJ, PB (φB)
• Typical Values
Cox = 10-4 F/m2
CGSO = 5x10-10 F/m
CGDO = 5x10-10 F/m
CGBO = 4x10-10 F/m
CJ = 10-4 F/m2
PB = 0.8 V
ALP_Rotondaro EE5321/EE7321 26
MOSFET capacitances - equationsSaturation Linear
with: PS = Perimeter of Source, AS = Area of Source
MJ = ½ (default), MJSW = 3 (default)
CGB = CGBO · L
WCGSOWLC32 C oxGS •+•••= WCGSO
2WLC C ox
GS •+••=
WCGDO CGD •= WCGDO2
WLC C oxGD •+••=
MJSWBS
MJBS
SB
PBV1
PCJSW
PBV1
ACJ C
+
•+
+
•= SS a similar equation is used to calculate CDB
ALP_Rotondaro EE5321/EE7321 27
MOSFET – classic layout• Area of Source = AS = 4λ·W
• Area of Drain = AD = AS = 4λ·W
• Perimeter of Source = PS = 8λ+W
• Perimeter of Drain = PD = 8λ+W
ALP_Rotondaro EE5321/EE7321 28
MOSFET – SPICE attributesM1 1 2 3 4 NMOS L=2U W=2U+ AS=4p AD=4p PS=6U PD=6U
• Overlap capacitances are calculated using W
• Capacitance to body have area and perimeter terms
ALP_Rotondaro EE5321/EE7321 29
Miller approximation• Capacitance between
input and output appears multiplied by the gain at the input
inout vA-v •=
( )
( )
( )dtdvA1C i
vAvdtdC i
v-vdtdCi
inc
ininc
outinc
•+•=
•+•=
•=
ALP_Rotondaro EE5321/EE7321 30
Miller approximation – Common source
( )[ ]CRRg1CRj1Rg
vv
outoutmin
outm
in
out
+•++•
=ω
( )outminp Rg1 CR
1•+••
=ω
Miller Capacitor
ALP_Rotondaro EE5321/EE7321 31
Common Source• CD can be ignored
sometimes
• Rout = RL || ro
• CG = CGB + CGS
Rout
ALP_Rotondaro EE5321/EE7321 32
Common source – small signal• Using impedances
Rout
0Z
v-vZv
Rv-v
GD
out1
G
1
in
in1 =++
( )[ ]{ } GDGinout2
GDLinGoutmGD
m
GD
outmin
out
CCRRCRRCRg1C j1gC-1
R-gvv
ωω −+•+•++••=
p2p1
2
p2p1
m
GD
p2p1
m
GD
in
out
1111j1
gCj-1
j1j1
gCj-1
vv
ωωω
ωωω
ω
ωω
ωω
ω
••−
++
=
+•
+
=
ALP_Rotondaro EE5321/EE7321 33
Common source – Poles and Zeros• From the transfer function:
( )[ ] GDLGoutmGDinp1 CRCRg1CR
1-++•+•
=ω
( ) Ggm1
inoutGDoutp2 C ||R ||R
1CR1- −=ω
GD
mz C
g=ω
( )
++
+
+
=
p2p1
z
j1j1
j1H
ωω
ωω
ωω
ω
ALP_Rotondaro EE5321/EE7321 34
Common source – Poles and Zeros• Converting to s space:
sz = -jωz sp1 = -jωp1 sp2 = -jωp2
( )
++
+
+
=
p2p1
z
j1j1
j1H
ωω
ωω
ωω
ω ( )
+
=
p2p1
z
-1-1
-1H
ss
ss
ss
s
ALP_Rotondaro EE5321/EE7321 35
Diode connected and Pole Splitting
ALP_Rotondaro EE5321/EE7321 36
Common source – Capacitance Cases• Relative magnitude of
the capacitors result in different scenarios
• Case1: Miller Cap small
RinCπ, RoutCD >> RinCMiller
π
ωCR1
inp1 =
DCR1
outp2 =ω oLout r||R R =
ALP_Rotondaro EE5321/EE7321 37
Common source – Small Miller capacitance• Output Impedance, Zout
• Stage gain, Av
• Output pole
Dout
DoLout Cj
1 ||R Cj1 ||r ||R Z
ωω==
Dout
out
Dout
D
out
moutmv CRj1R
Cj1 R
CjR
g- Zg- Aω
ω
ω+
=+
•=•=
Doutp CR
1 =ω
ALP_Rotondaro EE5321/EE7321 38
Common source – Small Miller capacitance• Input transfer function
• Input pole
π
π
π
ωω
ωCRj1
1
Cj1 R
Cj1
vv
inin
in
'in
+=
+=
π
ωCR1
inp =
ALP_Rotondaro EE5321/EE7321 39
Common source – Other cases• Case 2: Large CD
RoutCD >> RinCMiller, RinCπ
• Case 3: Large Cµ
RinCMiller >> RoutCD, RinCπ
Doutp1 CR
1 =ω ( )µπ
ωCCR
1 in
p2 +=
( ) µ
ωCRg1R
1 outmin
p1 +=
CMiller
( ) ( )D
m
Dm
p2 CCg
CCg1
1 +
=+
=π
π
ω
ALP_Rotondaro EE5321/EE7321 40
Poles and Zeros• Usually the multiplying factor on the Miller
capacitor results in poles far apart from each other than in other cases.
• The pole splitting is used to compensate the circuit.
MILLERinp1 CR
1 ≈ω
ALP_Rotondaro EE5321/EE7321 41
Common drain (source follower)• Small circuit analysis
vOUT
( ) outoutinin
g vv-vCRj1
1v +•
+
=πω
ALP_Rotondaro EE5321/EE7321 42
Common drain – Small signal analysis
( ) outmgmoutg
s
out vg1-Vg
Cj1v-v
Rv ••+•+= χ
ω π
( ) ( ) gmms
out vgCjCjg1R1v •+=
+•++• ππ ωωχ
( )( ) sm
smin
m
sm
sm
s
out
Rg11Rg1CRj1
gCj1
Rg11Rg
Rv
•++•+•+
+
••++
=
χχω
ω
χπ
π
π
ωCg m
z = ( )A1CR1
inp1 −
=π
ω ( ) sm
sm
Rg11Rg Aχ++
=
ALP_Rotondaro EE5321/EE7321 43
Common drain (source follower)• Effect of CSB
• The Body is Grounded
ALP_Rotondaro EE5321/EE7321 44
Common drain – small signal
( ) GSoutgGgin
gin Cjv-vCjvR
v-vωω •+•=
( ) ( ) SBouts
outoutgmGSoutg Cjv
Rvv-vg-Cjv-v ωω •+=••
ALP_Rotondaro EE5321/EE7321 45
Common drain – Small signal analysis
( ) ( )
+
++−
+
++
+++
+•
+=
sm
GSGSBGGSins
2
sm
SBGSs
sm
GSinGin
m
GS
sm
sm
in
out
Rg1CCCCCRR
Rg1CCR
Rg1CRCRj1
gCj1
Rg1Rg
vv
ωω
ω
• Having the denominator to be in the format:
• The poles are:
p2p1
2
p2p1p2p1
11 j1 j1j1ωω
ωωω
ωωω
ωω −
++=
+•
+
( ) ( )SBGSOsm
GSinGin
sm
SBGSs
sm
GSinGin
p1CCR
Rg1CRCR
1
Rg1CCR
Rg1CRCR
1
+++
+=
+++
++
=ω
( )[ ]GSSBGSBGGSinO
SBGSOsm
GSinGin
p2 CCCCCCRR
CCRRg1
CRCR
++
+++
+=ω s
mO R ||
g1R =
ALP_Rotondaro EE5321/EE7321 46
Common drain - Cases
• Case 1:
• Case 2:
( )SBGSOsm
GSGin CCR
Rg1CCR +>>
+
+
+
+=
sm
GSGin
p1
Rg1CCR
1ωsm
sm
Rg1Rg A
+= ( )A-1CC GSMiller •=
( )
+
+>>+sm
GSGinSBGSO Rg1
CCR CCR
( )SBGSOp1 CCR
1+
=ω
ALP_Rotondaro EE5321/EE7321 47
Common Gate
• Assuming ro → ∞
ALP_Rotondaro EE5321/EE7321 48
Common gate – small signal• Using KCL @ vs and @ vout
• No Zeros
smsss
sin vgCjvR
v-v+•= ω
L
outDoutsm R
vCjv vg +•= ω
( )
+
+•+
+=
sm
ssDL
sm
Lm
in
out
Rg1CRj1CRj1
Rg1Rg
vv
ωω
DLp1 CR
1=ω
sm
sssm
sp2
Cg1 ||R
1
CRg1
R1
=
+
=ω