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Dr. J.E. Rayas Sánchez1
Frequency Response
(Part 2)
Dr. J.E. Rayas Sánchez2
The FET High-Frequency Model
GD
S
gmvgs rοvgs
Cgd
Cgs
)(2 tGSm VVKg −=
DS
Ao I
Vr =
λ/1=AV
(input capacitance)gdgsiss CCC +=
(output capacitance)gddsoss CCC +=
(reverse capacitance)gdrss CC =
Dr. J.E. Rayas Sánchez3
The FET Unity-Gain Frequency (fT)
It is the frequency at which the magnitude of the short-circuit current gain of the Common Source configurationbecomes unity
)(2 gdgs
mT CC
gf+
=π
20 MHz ≤ fT ≤ 100 MHz for JFETs
100 MHz ≤ fT ≤ 2 GHz for MOSFETs
5 GHz ≤ fT ≤ 15 GHz for GaAs MESFETs
10 GHz ≤ fT ≤ 200 GHz for SiGe MOSFETs
Dr. J.E. Rayas Sánchez4
The BJT High-Frequency Hybrid π Model
T
Cm V
Ig =C
Ao I
Vr =mg
r βπ =
rπgmvπ rο
B C
vπ
rx
Cπ
Cµ
Eπrhr iex −= rehrr /πµ =
(Collector-Base capacitance)cbob CCC ==µ
(Base-Emitter capacitance)ebCC =π
Dr. J.E. Rayas Sánchez5
Typical Behaviour of Cπ
Dr. J.E. Rayas Sánchez6
Capacitancia de Difusión (repaso)
DT
d IV
C τ=
1IID = 12 IIID >=
)( jd CC >>ebCC =π
Dr. J.E. Rayas Sánchez7
Capacitancia de la Región de Desértica (repaso)
( ) 00
12 ψ
ε
AD
DAj NN
NqNAC+
=
0
0
1ψ
D
jj V
CC
−=
cbobj CCCC === 00µ
Dr. J.E. Rayas Sánchez8
Typical Behaviour of β
dB
20
40
60
20 log β o
fβfT
β
0
Frequency (Hz - Log Scale)109108107106105
f
- 3 dB
- 20 dB/decade
fT Unity-Gain Bandwidth
Dr. J.E. Rayas Sánchez9
The BJT Unity-Gain Bandwidth (fT)
It is the frequency at which the magnitude of theshort-circuit current gain of the common emitterconfiguration becomes unity
)(2 µππ CCgf m
T +=
2 MHz ≤ fT ≤ 100 MHz for general purpose BJTs
1 GHz ≤ fT ≤ 10 GHz for high speed BJTs
1 GHz ≤ fT ≤ 50 GHz for HBTs and HEMTs
Dr. J.E. Rayas Sánchez10
Typical Behaviour of fT
fT
log I C
fTmax
ICM
Dr. J.E. Rayas Sánchez11
High-Frequency Response of FET Amplifiers
Common Source
gmvgs rο||rLvgs
Cgd
Cgs vo
rS
virL vovi
rS
Sgs rR = )1(/ gs
oS
Sgs
ogsgd v
vrrvvv
R −=−
=
)||)(( LogsmS
gso rrvg
rv
v +−= )]/1)(||(1[ SmLoSgd rgrrrR ++=
)/(1 gdgdgsgsH CRCR +≈ω
Dr. J.E. Rayas Sánchez12
High-Frequency Response of FET Amplifiers
gmvgs
rLvgs Cgd
Cgs vo
rS
vi
roCommon Gate
rL vovi
rS
1|| ZrR Sgs =Lo
gs
rvv
Z/1
−= L
o
ogsgsmo r
rvv
vgv
−−+−= )(
oLLm
oLL
rrrgrrrZ/
)/1(1 +
+=m
Lo gZrr 1thenif 1 ≈>>
To calculate Rgd, assume ro >> rS , then vgs = 0 : Logd rrR ||=
)/(1 gdgdgsgsH CRCR +≈ω
Dr. J.E. Rayas Sánchez13
High-Frequency Response of FET Amplifiers
Common Drain
gmvgs rL
Cgs
Cgdvo
rS
vi+ vgsrL vo
vi
rS
Logsm
gsgs rvvg
vR
/−=
L
ogsm
S
ogs
rvvg
rvv
−=+
Sgd rR =
LS
gs
LS
Smgso rr
vrr
rgvv
/1/1)1(
+−
≈+
−=
)(1)/(11
SLm
SL
SLmgs rrg
rrrrg
R++
+=++
=
)/(1 gdgdgsgsH CRCR +≈ω
Dr. J.E. Rayas Sánchez14
High-Frequency Response of BJT Amplifiers
Common Emitter
rL vovi
rS rπgmvπ rovπ
rx
Cπ
CµrS
vorLvi
)1(/ π
πππ
πµ v
vRRvvvR oo −=−=ππ rrrR xS ||)( +=
)||)(( Lomo rrvgRvv π
π
π +−= )]/1)(||(1[ ππµ RgrrRR mLo ++=
)/(1 µµππω CRCRH +≈
Dr. J.E. Rayas Sánchez15
High-Frequency Response of BJT Amplifiers
Common Base
rL vo
rS
vi
It can be shown that
Sm
Sx
rgrrrR
++=
1||ππ LSxoL rrrrrR ≈+= )||(||µ
)/(1 µµππω CRCRH +≈
Dr. J.E. Rayas Sánchez16
High-Frequency Response of BJT Amplifiers
Common Collector
rL vovi
rS
It can be shown that
)||(1)||(||
oLm
oLSx
rrgrrrrrR
+++= ππ )(1
||xSLm
xSL
rrrgrrrrR+++
++= πµ
)/(1 µµππω CRCRH +≈
Dr. J.E. Rayas Sánchez17
Comparison of Low-Frequency FET Responses
CD
CG
CS
GCRSCR
DCR
mS g
R 1||21 || GGSS RRR + DL RR +
)1||(m
SSS gRR +21 || GG RR
LD RR +
Lm
S Rg
R +)1||( DR21 || GGSS RRR +
DCSCGCL CRCRCR
DSG
111 ++≈ω
Dr. J.E. Rayas Sánchez18
Comparison of Low-Frequency BJT Responses
CC
CB
CE
CCRBCR
ECR
1)||||(|| 21
++
βπ BBS
ERRRrR)||||( 21 πrRRR BBS + )||( oCL rRR +
)]||)(1([|||| 21 SEBB RRrRR βπ ++ mES grRR /1|||| π+ )||( oCL rRR +
)]||([|| LEoC RRrR +1
)||||(|| 21
+++
βπ BBS
ELRRRrRR)]||)(1([|||| 21 LEBBS RRrRRR βπ +++
CCECBCL CRCRCR
CEB
111 ++≈ω
Dr. J.E. Rayas Sánchez19
Comparison of High-Frequency FET Responses
gsR
)]/1)(||(1[ SmLoS rgrrr ++
CD
CG
CS
gdR
Sr
mS g
r 1|| Lo rr ||
)(1 SLm
SL
rrgrr++
+Sr
)/(1 gdgdgsgsH CRCR +≈ω
Dr. J.E. Rayas Sánchez20
Comparison of High-Frequency BJT Responses
πR
)]/1)(||(1[ ππ RgrrR mLo ++πrrr xS ||)( +
Sm
Sx
rgrrr
++
1||π )||(|| SxoL rrrr +
)||(1)||(||
oLm
oLSx
rrgrrrrr
+++
π
)/(1 µµππω CRCRH +≈
CC
CB
CE
µR
)(1||
xSLm
xSL
rrrgrrrr+++
++π
Dr. J.E. Rayas Sánchez21
Miller´s Theorem
V2
Z1
V1
Z2
ZV2
V1
1
2
VVAV =
VAZZ
−=
11 12 −=
V
V
AZAZ
V1
C1
V2
C2
CV2
V1)1(1 VACC −=
−=V
V
AACC 1
2
Dr. J.E. Rayas Sánchez22
Miller Effect
Common Source
gmvgs rο||rLvgs
Cgd
Cgs vo
rS
virL vovi
rS
)]/1)(||(1[ SmLoSgd rgrrrR ++=
)]/1)(||(1[ ππµ RgrrRR mLo ++=
rL vovi
rS rπgmvπ rovπ
rx
Cπ
CµrS
vorLvi
Common Emitter
Dr. J.E. Rayas Sánchez23
The Cascode Configuration
RC
R3
R2
vs
RS
CS
RE CE
VCC
RLvo
CCR1
CBQ2
Q1
Dr. J.E. Rayas Sánchez24
The Cascode Configuration (cont)
vs
RS RC || RLvo
Q2
Q1
R3 || R2
Dr. J.E. Rayas Sánchez25
The Cascode Amplifier at High Frequencies
23 |||| RRRr SS =23
23
||)||(
RRRRRvv
S
si +=
rS
vi ro1
gm1vπ 1
Cµ 1
Cπ 1
rx1
vπ 1
rπ 1 rπ 2
gm2vπ 2
ro2
vπ 2Cπ 2 Cµ 2rLvo
11
1'π
π
rrrrvvxS
ii ++
= 11 ||)(' πrrrr xSS +=
vi'gm1vπ 1
Cµ 1
Cπ 1 vπ 1
rπ 2
gm2vπ 2
ro2
vπ 2Cπ 2 Cµ 2rLvo
rS'
Dr. J.E. Rayas Sánchez26
The Cascode Amplifier at High Frequencies
22
21|| em
rg
r =π
vi'gm1vπ 1
Cµ 1
Cπ 1
rS'
rπ 2 vπ 2Cπ 2 1/gm2vπ 1
vi'gm1vπ 1
Cµ 1
Cπ 1
rS'
re2Cπ 2vπ 1
vx
'1 SrR =π 22 erR =π )1(''/ 11
11
ππ
πµ v
vrrvvvR x
SS
x −=−=
2111 )'
( emS
x rvgrvv π
π +−=2
221 '2)
'1(' eS
S
eemS rr
rrrgrR +≈++=µ
Dr. J.E. Rayas Sánchez27
The Cascode Configuration
RC
R3
R2
vs
RS
CS
RE CE
VCC
RLvo
CCR1
CBQ2
Q1
'1 SrR =π 22 erR =π
21 '2 eS rrR +=µ
)/(1 112211 µµππππω CRCRCRH ++≈
11 ||)(' πrrrr xSS +=
23 |||| RRRr SS =
Dr. J.E. Rayas Sánchez28
Frequency Response of a Differential Amplifier
ππ rrRR xS ||)( +=
)]/1)(||(1[ ππµ RgRrRR mCo ++=
)/(1 µµππω CRCRH +≈
vC1
RC
RS
+vd /2vC1 vC2
I
VCC
VEE
RCRC
vdRS
RS
Dr. J.E. Rayas Sánchez29
Cascode Differential Amplifier
RS RCvC1+vd /2
vC1 vC2
I
VCC
VEE
RCRC
vdRS
RS
R2
R1
VCC
VEE
])/[(1 21 µµπππω CRCRRH ++≈
erR =2π
erRR += 12 πµ
ππ rrRR xS ||)(1 +=
Dr. J.E. Rayas Sánchez30
Hybrid Microwave Integrated Circuits
M. Pozar (1998),Microwave Engineering. Amherst, MA: John Wiley and Sons.
Dr. J.E. Rayas Sánchez31
Hybrid Microwave Integrated Circuits (cont)
M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.
Dr. J.E. Rayas Sánchez32
Monolithic Microwave Integrated Circuits
M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.
Dr. J.E. Rayas Sánchez33
Active Device Models at Microwave Frequencies
rπgmvπ rοvπ
rx
Cπ
Cµ
B
E
C
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