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Johan Wernehag, EIT
Lecture 5
Radio Electronics
Johan WernehagElectrical and Information Technology
1
Lecture 5
• RF Amplifier Analysis and Design– The hybrid-p-model
• transistor configurations• amplifier classes• example of a tuned amplifier
– Two-Port Network Representation• y, z, and ABCD parameters, definition and
applications• measurement of parameters• power gain definitions• stability
– Diagnostic test
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 2
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 3
Transistor Models
Physical model
Simplified hybrid-p-model
Rcvin Cp
Cm
rp roic
b c
e
Ib
Vout
S-Parameter model
2-porta1
b1
b2a2
S11 S22
S21
S12
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 4
Transistors
• Example: c ¢b c = 0.5pF c ¢b e = 9pF cce = 1pFr ¢b b = 10W b = 300 VA = 50V IC = 1mA
What happens when the frequency escalates from 0 to 1 GHz?
r ¢b e = rpC ¢b e = Cp
rbc = rmCbc = Cm
rce = r0
The hybrid-p model for bipolar transistors
DC:
7.8kΩ gmvb’cvbe
+
-
e
c
v ce
+
-
b
50kΩ
15MΩ
b’10Ω
+
-
50Ω
V s 1kΩ
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 5
Transistors (cont.)
1 GHz:
7.8kΩj18Ω– gmvb’e
vbe
+
- e
c
vce
+
-
j159Ω–j318Ω–b
50kΩ
15MΩ
b’10Ω
+
-
50Ω
V s 1kΩ
1 MHz:
7.8kΩj18kΩ– gmvb’e
vbe
+
-
c
vce
+
-
j159kΩ–j318kΩ–b
50kΩ
15MΩ
b’10Ω
+
-
50Ω
V s 1kΩ
e
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 6
Voltage Gain
Av
f
f3dB
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 7
Tuned Amplifiers
• Conventional design of amplifiers results in low-pass behaviour due to capacitances and this causes problems at higher frequencies
• i.e. connect an inductor in parallel to the capacitance to form a resonant circuit.The rest of the amplifier is designed by conventional low-frequency methods
• By neutralizing these capacitances the high-frequency behaviour may be improved
– with preserved bandwidth!
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 8
The Result of Neutralization
Av
Vcc
Vcc
Both amplifiers have same bandwidth, don’t let the log-scale foul you. = / = / = / = ⇒ =1/2
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 9
Example of a Tuned Amplifier
Compare tolab 3.2!
Vcc
Vin
Vout
Rc
R1
R2 R3
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 10
Rc
b
e
cCQI
in
cc
v
V
vout
A Small Signal Model for Bipolar Transistors
ic = b ib = gm vin
gm =ICQ
kTq»
ICQ
0,026AVéëê
ùûú
vid T = 290KThe model is valid at small input signals:
vin << vT = kTq » 0,026V vid T = 290K
Signal diagram
Rcvin C
C
r roic
b c
e
b
v
i i c
pp
m
out
The simplified hybrid-p-model
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 11
Common Emitter (CE) Coupled Transistor
• High voltage gain
Av = -gm Rc
• High input resistance and capacitance
Rin =b
gm= b +1( ) re Cin = Cp + 1+ Av( )Cm
• High output resistance and low output capacitance
Rout = ro / /Rc Cout » Cm
Result of the Miller theorem
Rcb
e
c
inv
v
CE
out
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 12
Common Base (CB) Coupled Transistor
• High voltage gain
Av = gm Rc
• Low input resistance and capacitance
Rin = re » 1gm
Cin = Cp
• High output resistance and low output capacitance
Rout = ro / /Rc Cout = Cm
Rc
b
e
c
in
v
v
C B
out
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 13
• High input resistance and low input capacitance
Rin = b 1gm
+Re( ) » b re+Re( ) Cin » Cm
Common Collector (CC) Coupled Transistor
• Low voltage gain
Av » 1
• Low output resistance and capacitance
Rout = re / /Re Cout » 0
Re
b
e
c
invv
CC
out
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 14
Amplifier Classes
• Class A– linear– suitable for all signals– efficiency
vBE
iC
ICQ A
• Class B– non-linear– only signals at high and constant amplitude– efficiency
B
h £ 78%
h £ 50%
• Class C– non-linear– only signals at high and constant amplitude (e.g. FM)– efficiency
C
h £ 90%
VBEQ
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 15
When the Frequency Increases Further...
• Simple neutralization is not efficient– more reactive elements affects the properties of the transistor– more elements generally affects the properties and therefore the rough
hybrid-p-model doesn’t last
• It is more important to consider power to “preserve” available performance
• Design at high frequencies by lumped components therefore utilizes two-port models such as S-parameters that more often are measured quantities than calculated...
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 16
Two-Port Network Representation
• voltage and current– z, impedance parameters – y, admittance parameters– ABCD, chain parameters– ...
2-port
i1 i2
v1 v2
• waves– S, scattering parameters– T, transmission parameters– X, large signal scattering
parameters2-port
a1b1 b2
a2
ax = incident wave
bx = reflected wave
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 17
Two-Port
2-port
i1 i2
v1 v2
• If– the two-port is linear and – port voltages and port currents are complex quantities
the small signal properties may be characterized by four complex parameters
– these are only valid at the specified frequency and bias conditions
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 18
Admittance (y) Parameters
2-porti1
v1
i2
v2y12v2 y22 y21v1
Two-port schematic
y11
Definition = += +
or in matrix format:
=
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 19
Impedance (z) Parameters
Two-port schematic
2-porti1
v1
i2
v2z12i2
z22z21i1+ +
- -
z11
Definition
= += +
or in matrix format:
=
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 20
Chain (ABCD) Parameters
Two-port schematic
2-port i1
v1
i2
v2Bi2A-1( )v2
+ - +-
D -1( ) i2Cv2
Definition
= + −= + −
or in matrix format:
= −
= += +
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 21
Measuring Two-Port Parameters
ex. y parameters
y11 =i1v1 v2 =0
• at the same time as port 2 is short-circuited a signal is applied to port 1. The current voltage ratio at port 1 is then:
• A parameter is simply derived when a port voltage is turned to zero, i.e. a port is short-circuited:
2-porti1 i2
v2y11y12v2 y22 y21v1v1
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 22
Measuring Two-Port Parameters (cont.)
• Two-port parameters are generally measured by means ofopen (i = 0) or short-circuited (v = 0) ports.
• Open or short-circuited termination is however hard to achieve at high frequencies, especially at the desired location close to the component.
• Open or short-circuited termination may also cause instability and self-oscillation when measurements are performed on active components.
• At higher frequencies there is obviously a need for a model (e.g.S parameters) that doesn’t require open or short-circuited ports.
• y, z and ABCD parameters are still valid for calculations at higher frequencies as well.
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 23
Circuit Analysis by Two-Port Parameters
i1i2
é
ëêê
ù
ûúú=
i1a + i1
b
i2a + i2
b
é
ë
êê
ù
û
úú=
y11a + y11
b y12a + y12
b
y21a + y21
b y22a + y22
b
é
ë
êê
ù
û
úú
v1
v2
é
ëêê
ù
ûúú
• parallel connection:use y parameters
2-port ai1
v1
i2
v2y11y12v2 y22 y21v1
2-port b
y12v2 y22 y21v1
• series connection:use z parameters
2-port ai1
av1
i2av2
z1 1 z1 2i2z2 2z2 1i1
+ +
- -
2-port b
bv1
bv2
z1 1 z1 2i2z2 2z2 1i1
+ +
- -
v1
v2
é
ëêê
ù
ûúú=
v1a + v1
b
v2a + v2
b
é
ë
êê
ù
û
úú=
z11a + z11
b z12a + z12
b
z21a + z21
b z22a + z22
b
é
ë
êê
ù
û
úú
i1i2
é
ëêê
ù
ûúú
y11
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 24
Circuit Analysis by Two-Port Parameters (cont.)
• cascade connection: use ABCD parameters
v1
i1
é
ëêê
ù
ûúú=
v1a
i1a
é
ë
êê
ù
û
úú= Aa Ba
Ca Da
é
ëêê
ù
ûúú
v2a
-i2a
é
ë
êê
ù
û
úú= Aa Ba
Ca Da
é
ëêê
ù
ûúú
Ab Bb
Cb Db
é
ëêê
ù
ûúú
v2b
-i2b
é
ë
êê
ù
û
úú
2-port a
v1
i2
v2Bi2A-1( )v2
+ - + -
Cv2 D -1( ) i2
2-port b
v1 v2Bi2A-1( )v2
+ - + -
Cv2 D -1( ) i2
i1 i2i1
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 25
Parameters for Various Transistor Configurations
• The parameters may easily be converted between different configurations without any loss of information. (Table 8.1 in the textbook)
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 26
Power Gain Definitions
G
• How should we define the power gain?
ZS
ZLESPAVS
• PAVS = AVailable power from Source
PIN
• PIN = power delivered to the INput of the two-port
PAVN
• PAVN = AVailable power from Network (the output of the two-port)
PL
• PL = power delivered to the Load
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 27
Power Gain Definitions (cont.)
G
ZS
ZLESPAVS PAVN
available gain GA =PAVN
PAVS
PIN PL
operating gain GP =PL
PIN
transducer gain GT =PL
PAVS
Important definitions!
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 28
Power Gain from Several Stages
total transducer gain GT tot =PL
PAVS
G1
ZS
ZLES G2 G3PAVS PL
What is the total power gain?
GT tot =PL
PAVS
= POUT1
PAVS
× POUT 2
PIN2
× PL
PIN3
=GT1 ×GP2 ×GP3
Extract the expression by using the gain definitions:
Can GT be calculated in alternative ways? How?
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 29
The Gain Expressed by y parameters
GA =gS y21
2
yS + y112 Re y22 -
y12y21
yS + y11
é
ëê
ù
ûú
available gain
GP =gL y21
2
yL + y222 Re y11 -
y12y21
yL + y22
é
ëê
ù
ûú
operating gain
GT =4gSgL y21
2
yS + y11( ) yL + y22( )- y12y212
transducer gain
yiS yL
yS = gS + jbS yL = gL + jbL
yS
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 30
Port Admittances for a Two-Port
yin = y11 -y12 × y21
y22 + yL
input admittance
output admittance yout = y22 -y12 × y21
y11 + yS
ySiS yL
yin
®yout
¬
2-porti1
v1
i2
v2y11y12v2 y22 y21v1 y12 connect
output to input
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 31
Stability
• If the circuit isn’t able to self-oscillate it is consideredto be stable.
• This is not an obvious quality at RF design because– several feedback paths (parasitic elements)– several reactive circuit elements (parasitic elements)
• Stability analysis based on poles is possible only if the complete circuit is completely known.
• It is not possible to calculate poles and zeros if only two-port parameters are available.There is obviously a need for alternative methods to study stability…
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 32
Stability - an Example
Stable when RP > 0 - RP denotes the circuit losses
Unstable when RP < 0 - RP supplies power to the circuit
Poles?
RP LPCP
ZP
ZP =1
sC+1sL
+1R
=sL
s2LC+1+ sLR
=
sC
s2 + s 1RC
+sLR
R > 0 : poles in the left-half planeR < 0 : poles in the right-half plane
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 33
Unconditional Stability
yin = y11 -y12 × y21
y22 + yL
yout = y22 -y12 × y21
y11 + yS
Linvill’s stability factor CL =y12 × y21
2 ×g11 ×g22 -Re y12 × y21[ ]
The two-port is unconditionally stable if
Note! The result is only valid at the actual frequency where the stability test is performed
0 < CL < 1
A two-port is defined to be unconditionally stable if no yS or yL results in a negative port admittance
Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 5 34
Conditional Stability
• The two-port is stable if the loop resistance rloop > 0– Note! The result is only valid at the actual frequency where
the calculation is done.
rloop = Re zP[ ]+Re zL[ ]xloop = Im zP[ ]+ Im zL[ ]
• If the port impedance still is negative– examine the loop impedance:
• If it’s not unconditionally stable the two-portmay be conditionally stable:
– some yS or yL causes a negative port admittance– avoid these yS and yL if possible
yin = y11 -y12 × y21
y22 + yL
yout = y22 -y12 × y21
y11 + yS
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