This document discusses the design of low-noise amplifiers. It begins with an overview of the basic structure of transmitters and receivers in wireless communication systems. It then reviews the relationships between power and gain and introduces the concept of the available power gain circle. The document discusses a design method for amplifiers that does not require simultaneous conjugate matching of both ports. It also covers noise theory for two-port networks and the fixed noise figure circle. The key points are utilizing available power gain circles and fixed noise figure circles to design amplifiers through tradeoffs between gain and noise on the Smith chart.

1. 高頻電子電路
第六章 低雜訊放大器設計
李健榮 助理教授
Department of Electronic Engineering
National Taipei University of Technology

2. 大綱
• 無線收發機的基本架構
• 回顧：功率-增益關係式
• 可資用功率增益圓
• 非雙埠同時共軛匹配的放大器設計法：
可可可可資用功率增益設計資用功率增益設計資用功率增益設計資用功率增益設計法法法法
• 雙埠網路雜訊理論
• 固定雜訊指數圓
Department of Electronic Engineering, NTUT2/15

3. 無線收發機的基本架構
• 發射機(Transmitter, TX)
• 接收機(Receiver, RX)
高功率
低雜訊
Department of Electronic Engineering, NTUT3/15

4. 回顧：功率-增益關係式
2 2
2
212 2
22
1 1
1 1
s LL
T
AVS in s L
P
G S
P S
− Γ − Γ
= =
− Γ Γ − Γ
2 2
2
212 2
11
1 1
1 1
s LL
T
AVS s out L
P
G S
P S
− Γ − Γ
= =
− Γ − Γ Γ
2
2
212 2
22
11
1 1
LL
p
in in L
P
G S
P S
− Γ
= =
− Γ − Γ
2
2
212 2
11
1 1
1 1
sAVN
A
AVS s out
P
G S
P S
− Γ
= =
− Γ − Γ
• 功率轉換增益GT (Transducer Power Gain)
• 操作功率增益Gp (Operating Power Gain)
• 可資用功率增益GA (Available Power Gain)
Transistor
[S]
+
−
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
輸入總是匹配，考慮不同輸出匹配
輸出總是匹配，考慮不同輸入匹配
同時考慮不同輸入、輸出匹配
Department of Electronic Engineering, NTUT4/15

5. 功率轉換增益GT (Transducer Power Gain)
• 雙埠同時共軛匹配：最大轉換增益匹配
2 2 2 2
2 2
21 212 2 2 2
22 11
1 1 1 1
1 1 1 1
s L s L
T
s in L s out L
G S S
S S
− Γ − Γ − Γ − Γ
= =
− Γ Γ − Γ − Γ − Γ Γ
Transistor
[S]+
−
sE
sZ
LZ
見第五章投影片slide 32
Department of Electronic Engineering, NTUT
inΓ
1E
oZ
oZ
Transistor
oG
Output
matching
LG
Input
matching
sG
s in
∗
Γ = Γ L out
∗
Γ = ΓoutΓ
• 功率轉換增益 GT
inΓsΓ LΓoutΓ
輸出端的匹配目標輸入端的匹配目標
5/15

6. 可資用功率增益圓(I)
( )2 2
21 2
212
222
11
11
1
1 1
1
s
A a
s
s
s
S
G S g
S
S
S
− Γ
= = ⋅
 − ∆Γ
 − − Γ
 − Γ
 
• 無條件穩定雙向(bilateral)情況:
( ) ( )
2
2 2 2 2 2
21 22 11 1
1
1 2Re
sA
a
s s
G
g
S S S C
− Γ
= =
− + Γ − ∆ − Γ
1 11 22C S S∗
= − ∆
s a aC rΓ − =
( )
1
2 2
111
a
a
a
g C
C
g S
∗
=
+ − ∆ ( )
2 2
12 21 12 21
2 2
11
1 2
1
a a
a
a
K S S g S S g
r
g S
− +
=
+ − ∆
圓心 半徑
• 可資用功率增益圓(Available Power-Gain Circle):
其中
把GA改寫成只跟電晶體S參數與Γs有關：
Ga與ga為電晶體S參數與Γs的函數。可造成固定
ga的Γs值，其軌跡為一個圓形，也稱為可資用
功率增益圓(available power-gain circle)。
2
2
212 2
11
1 1
1 1
sAVN
A
AVS s out
P
G S
P S
− Γ
= =
− Γ − Γ
12 21 22 11 22 12 21 22
22
11 11 111 1 1
s s s s
out
s s s
S S S S S S S S
S
S S S
Γ − Γ + Γ − ∆Γ
Γ = + = =
− Γ − Γ − Γ
11 22 12 21S S S S∆ = −
Department of Electronic Engineering, NTUT6/15

7. 可資用功率增益圓(II)
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,max ,max_@A s GAG Γ
1 1@A sG Γ
2 2@A sG Γ
3 3@A sG Γ
Γs平面 Γs平面
18 dB
17 dB
16 dB
15 dB
14 dB
GaCircle
GaCircle1
GaCircle1=ga_circle(S,{18, 17, 16, 15 ,14} ,51)
GaCircle
MeasEqn
Meas1
GAmax=max_gain(S)
Eqn
Meas
GaCircle
GaCircle1
GaCircle1=ga_circle(S,GAmax ,51, 5, 1)
GaCircle ga_circle() 函 數 之 用 法
請參考ADS的Help說明
7/15

8. 設計程序
1E
oZ
oZ
Transistor
oG
Output
matching
LG
Input
matching
sG
sΓ LΓoutΓ
• 可資用功率增益設計法
18 dB
17 dB
16 dB
15 dB
14 dB
Γs平面
先選要配到的Γs (不一定在GA,max，待會
就會講到為什麼了)
選完 Γs 後可以得到 Γout
知道 Γout後，再讓Γout與其共軛匹配即可：
L out
∗
Γ = Γ
Department of Electronic Engineering, NTUT8/15

9. 雙埠網路雜訊理論
• 雜訊因子(noise factor)可由等效雜訊電阻與雜訊電導表示：
Noisy
Two-portsYsi
2
4
n
n
e
R
kTB
≡
2
4
u
u
i
G
kTB
≡
2
4
s
s
i
G
kTB
≡
( ) ( )
2 2
2
1 1
u c s c s nu c s n
s s
G G G B B RG Y Y R
F
G G
 + + + ++ +  = + = +
, ,and
Department of Electronic Engineering, NTUT
s c optB B B= − = 2u
s c opt
n
G
G G G
R
= + =and
2
min 1 2 1 2 u
n opt c n c c
n
G
F R G G R G G
R
 
 = + + = + + +  
 
( ) ( )
2 2
min
n
s opt s opt
s
R
F F G G B B
G
 = + − + −  
0
11
1
s
s
s
Y
Z
− Γ
=
+ Γ
0
11
1
opt
opt
opt
Y
Z
− Γ
=
+ Γ
( )
( )
2
min 22
0
4
1 1
s optn
s
s opt
R
F F
Z
Γ − Γ
Γ = +
− Γ + Γ
• 固定雜訊指數圓
9/15

10. 固定雜訊指數圓
Department of Electronic Engineering, NTUT
min ,@ s optNF Γ Γs平面 Γs平面
0.8 dB min 0.3 dBNF =
1.3 dB
1.8 dB
2.3 dB
1 1@ sNF Γ
2 2@ sNF Γ
3 3@ sNF Γ
ns_circle() 函 數 之 用 法
請參考ADS的Help說明
NsCircle
NsCircle1
NsCircle1=ns_circle(nf2,NFmin,Sopt,Rn/50,51)
NsCircle
VAR
VAR4
Num_NF_Circles=5
NF_Stepsize=0.5
Eqn
Var
NsCircle
NsCircle1
NsCircle1=ns_circle(NFmin+NF_Stepsize*[1::Num_NF_Circles],NFmin,Sopt,Rn/50,51)
NsCircle
min ,@ s optNF Γ
10/15

11. 低雜訊放大器設計(增益與雜訊的取捨)
GA circles
NF circles
Input
matching
Output
matching
Amplifier
sΓ LΓ
0Z
0Z
inΓ outΓ
outZinZ
Department of Electronic Engineering, NTUT
Min. noise figure, min ,, s optNF Γ
Max. available power gain, s in
∗
Γ = Γ
11/15

12. 利用ADS在史密斯圖上進行取捨設計
Department of Electronic Engineering, NTUT
GammaS
indep(GammaS)=
rhos=-0.11872 + j0.12612
impedance = 38.26607 + j9.95049
60
indep(rhos) (0.000 to 2000.000)
rhos
GammaSgain=18.937
gain=17.937
gain=16.937
gain=15.937
cir_pts (0.000 to 51.000)
GAcircles
indep(GammaLopt) (60.000 to 60.000)
GammaLopt
ns figure=1.404ns figure=1.604ns figure=1.804
Noise_circles
(0.000 to 0.000)
Sopt[fm1]
GammaLopt_NFmin
GammaS
indep(GammaS)=
rhos=-0.11872 + j0.12612
impedance = 38.26607 + j9.95049
60
Gamma_S (NFmin)
Gamma_L when NFmin
GA = 17.937 dB
GA = 16.937 dB
GA = 15.937 dB
GA = 18.937 dB
NF= 1.404 dB
NF= 1.604 dB
NF= 1.804 dB
NFmin= 1.204 dB
GammaS
indep(GammaS)=
rhos=-0.45577 + j0.18782
impedance = 17.56757 + j8.71721
486
indep(rhos) (0.000 to 2000.000)
rhos
GammaS
gain=18.937
gain=17.937
gain=16.937
gain=15.937
cir_pts (0.000 to 51.000)
GAcircles
indep(GammaLopt) (486.000 to 486.000)
GammaLopt
ns figure=1.404ns figure=1.604ns figure=1.804
Noise_circles
(0.000 to 0.000)
Sopt[fm1]
GammaLopt_NFmin
GammaS
indep(GammaS)=
rhos=-0.45577 + j0.18782
impedance = 17.56757 + j8.71721
486
Gamma_S (NFmin)
Gamma_L when NFmin
NF at GammaS (dB)
NF_at_GammaS
1.2042
Zsource2
38.2661 + j9.9505
Source Impedance at GammaS
zin(GammaLopt,Z0[fm1])
58.7305 + j15.5482
Optiomal Load Impedance at GammaS Transducer Power Gain (dB)
Gtrans_power
17.9575
(C) Matching Condition for Arbitray GammaS
NF at GammaS (dB)
NF_at_GammaS
1.4718
Zsource2
17.5676 + j8.7172
Source Impedance at GammaS
zin(GammaLopt,Z0[fm1])
57.1651 + j46.3908
Optiomal Load Impedance at GammaS Transducer Power Gain (dB)
Gtrans_power
18.7382
(C) Matching Condition for Arbitray GammaS
12/15

13. 在ADS建置完整的LNA設計環境(I)
Department of Electronic Engineering, NTUT
Move marker mBiasPtto desired bias point.
Smith Chartand data below will be updated.
2.400 GHz 50.000
System Impedance, Z0RFFrequency Move markers GammaSand GammaL to selectarbitrary source and load reflection coefficients The impedances,power gains,and noise figures
below will be updated.The transducer power gains are invalid ifthe markers are moved into the unstable regions.
Eqn num _NFc irc les =3
Eqn NFs tep_s iz e=0.2Eqn GAs tep_s iz e=1
Eqn num _GAc irc les =3
Zs ourc e,
Gam m a_S
Zload,
Gam m a_L
DUT*
Eqn num _GPc irc les =3
Eqn GPs tep_s iz e=1
indep( r hos) ( 0. 000 t o 2000. 000)
rhos
GammaS
indep( Sour ce_st abcir ) ( 0. 000 t o 51. 000)
Source_stabcir
gain=20. 728
gain=19. 728
gain=18. 728
gain=17. 728
cir _pt s ( 0. 000 t o 51. 000)
GAcircles
indep( G am m aLopt ) ( 161. 000 t o 161. 000)
GammaLopt
ns f igur e=0. 851ns f igur e=1. 051ns f igur e=1. 251
Noise_circles
Noise_circleMin
G am m aS
indep( G am m aS) =r hos=0. 15388 + j0. 23837
im pedance = 59. 49677 + j30. 84754
161
indep( r hos) ( 0. 000 t o 2000. 000)
rhos
GammaL
indep( Load_st abcir ) ( 0. 000 t o 51. 000)
Load_stabcir
gain=20. 728
gain=19. 728
gain=18. 728
gain=17. 728
cir _pt s ( 0. 000 t o 51. 000)
GPcircles
indep( G am m aSopt ) ( 246. 000 t o 246. 000)
GammaSopt
G am m aL
indep( G am m aL) =
r hos=0. 35071 / - 54. 37157
im pedance = Z0 * ( 1. 22760 - j0. 79805)
246
Available Gain Circle:
Noise Circles:
Source Stability Circle:
Source Gamma Corresponding Load Gamma (Black Dot)
Power Gain Circles:
Load Stability Circle:
Load Gamma Corresponding Source Gamma (Black Dot)
Load Stable Region
Outside
Eqn t index=[ 0: : 2000]
Eqn r hos=sqr t ( t index/ 2000) *exp( j*2*sqr t ( pi*t index) )
Eqn I Cindex2=f ind_index( I C[ VCEindex2] , m BiasPt )
Eqn VCEindex2=f ind_index( DC. VCE[ 0, : : ] , indep( m BiasPt ) )
Eqn Sour ce_st abcir =s_st ab_cir cle( S_bpm , 51)
Eqn Load_st abcir =l_st ab_cir cle( S_bpm , 51)
Eqn G am m aLopt =conj( S_22m +S_12m *S_21m *G am m aS/ ( 1- S_11m *G am m aS) )
Eqn G t _num =m ag( S_21m ) **2 *( 1- m ag( G am m aS) **2) *( 1- m ag( G am m aLopt ) **2)
Eqn G t _den=m ag( ( 1- S_11m *G am m aS) *( 1- S_22m *G am m aLopt ) - S_21m *S_12m *G am m aS*G am m aLopt ) **2
Eqn G am m aLopt _NFm in=conj( S_22m +S_12m *S_21m *Sopt _at _m BiasPt / ( 1- S_11m *Sopt _at _m BiasPt ) )
Eqn G t _num _NFm in=m ag( S_21m ) **2 *( 1- m ag( Sopt _at _m BiasPt ) **2) *( 1- m ag( G am m aLopt _NFm in) **2)
Eqn G t _den_NFm in=m ag( ( 1- S_11m *Sopt _at _m BiasPt ) *( 1- S_22m *G am m aLopt _NFm in) - S_21m *S_12m *Sopt _at _m BiasPt *G am m aLopt _NFm in) **2
Eqn G t r ans_power _NFm in=10*log( G t _num _NFm in/ G t _den_NFm in)
Eqn NF_lin_at _G am m aS=NFm in_lin+4*( Rn_at _m BiasPt / Z0_r ef ) *m ag( G am m aS- Sopt _at _m BiasPt ) **2/ ( ( 1- m ag( G am m aS) **2) *m ag( 1+Sopt _at _m BiasPt ) **2)
Eqn NFm in_lin=10**( NFm in_at _m BiasPt / 10)
Eqn NF_at _G am m aS=10*log( NF_lin_at _G am m aS)
Eqn NF_at _G am m aS_ConjM at ch=if ( st ab_f act ( S_bpm ) >1) t hen 10*log( NF_lin_at _G am m aS_ConjM at ch) else 1000
Eqn NF_lin_at _G am m aS_ConjM at ch=NFm in_lin+4*( Rn_at _m BiasPt / Z0_r ef ) *m ag( G am m aS_ConjM at ch- Sopt _at _m BiasPt ) **2/ ( ( 1- m ag( G am m aS_ConjM at ch) **2) *m ag( 1+Sopt _at _m BiasPt ) **2 +1e- 20)
( C) O pt im al G am m a_L when t he G am m a_S is at "m aker G am m aS"
( A) O pt im al G am m a_L when t he G am m a_S is at Sopt ( opt im al f or m inim um noise f igur e. )
( C) Noise f igur e f or an ar bit r ay G am m a_S ( m ar ker G am m aS)
( B) Noise f igur e f or sim ult aneously conjugat e m at ching. ( O nly def ined if K is >1. O t her wise t he noise f igur e is set t o 1000. )
( C) G t r ans_power : t r ansducer power gain wit h t he sour ce r ef lect ion coef f icient at m ar ker G am m aS, and t he load t hen conjugat ely m at ched.
( A) G t r ans_power _NFm in: t r ansducer power gain wit h t he sour ce r ef lect ion coef f icient Sopt f or m inim um noise f igur e, and t he load t hen conjugat ely m at ched.
Eqn G am m aSopt =conj( S_11m +S_12m *S_21m *G am m aL/ ( 1- S_22m *G am m aL) )
( D) O pt im al G am m a_S when t he G am m a_L at "m aker G am m aL"
Eqn G t load_num =m ag( S_21m ) **2 *( 1- m ag( G am m aSopt ) **2) *( 1- m ag( G am m aL) **2)
Eqn G t load_den=m ag( ( 1- S_11m *G am m aSopt ) *( 1- S_22m *G am m aL) - S_21m *S_12m *G am m aSopt *G am m aL) **2
Eqn G t r ans_power _load=if ( G t load_num >0) t hen 10*log( G t load_num / G t load_den) else 1e6
( D) G t r ans_load : t r ansducer power gain wit h t he load r ef lect ion coef f icient at m ar ker G am m aL, and t he sour ce t hen opt im um ly noise m at ched.( D) Noise f igur e f or an ar bit r ay G am m a_L ( t he sour ce r ef lect ion coef f icient is at G am m aSopt )
Eqn NF_lin_at _G am m aSopt =NFm in_lin+4*( Rn_at _m BiasPt / Z0_r ef ) *m ag( G am m aSopt - Sopt _at _m BiasPt ) **2/ ( ( 1- m ag( G am m aSopt ) **2) *m ag( 1+Sopt _at _m BiasPt ) **2)
Eqn NF_at _G am m aSopt =10*log( NF_lin_at _G am m aSopt )
Sour ce r ef lect ion coef f icientEqn G am m aS_ConjM at ch=sm _gam m a1( S_bpm )
Zsour ce is t he im pedance at m ar ker G am m aS.Eqn Zsour ce2=Z0[ 0, 0, 0] *( 1+G am m aS) / ( 1- G am m aS)
Eqn G t r ans_power =if ( G t _num >0) t hen 10*log( G t _num / G t _den) else 1e6
Eqn Noise_cir cleM in=ns_cir cle( NFm in_at _m BiasPt , NFm in_at _m BiasPt , Sopt _at _m BiasPt , Rn_at _m BiasPt / Z0_r ef , 51)
Eqn Noise_cir cles=ns_cir cle( NFm in_at _m BiasPt +NFst ep_size*[ 1: : num _NFcir cles] , NFm in_at _m BiasPt , Sopt _at _m BiasPt , Rn_at _m BiasPt / Z0_r ef , 51)
Eqn G Acir cleM ax=ga_cir cle( S_bpm , m ax_gain( S_bpm ) )
Eqn G Acir cles=ga_cir cle( S_bpm , m ax_gain( S_bpm ) - G Ast ep_size*[ 0: : num _G Acir cles] )
Eqn G Pcir cles=gp_cir cle( S_bpm , m ax_gain( S_bpm ) - G Pst ep_size*[ 0: : num _G Pcir cles] )
Set st ep size and num ber of cir cles t o plot
st ab_f act ( S[ I Cindex2, VCEindex2, 0] )
0. 6776
St abilit y K
t index is a vect or of num ber s 0, 1, 2, 3, . . . , 2000.
r hos ar e 2001 com plex r ef lect ion coef f icient s.
( B) G am m a_S f or sim ult aneous conjugat e m at ching at bias point m BiasPt .
NF at G am m aS ( dB)
NF_at _G am m aS
0. 6512
Zsour ce2
59. 4968 + j30. 8475
Sour ce I m pedance at G am m aS
. . . am m aLopt , Z0[ 0, 0, 0] )
31. 9360 + j31. 5019
O pt iom al Load I m pedance at G am m aS Tr ansducer Power G ain ( dB)
G t r ans_power
18. 6454
NFm in[ I Cindex2, VCEindex2, 0]
0. 6512
NFm in ( dB)
. . . ex2, VCEindex2, 0] , Z0[ 0, 0, 0] )
59. 0670 + j30. 3691
Sour ce I m pedance Zopt at NFm in
. . . m m aLopt _NFm in, Z0[ 0, 0, 0] )
31. 8982 + j31. 7136
O pt iom al Load I m pedance
f or sour ce Zopt at NFm in Tr ansducer Power G ain ( dB)
G t r ans_power _NFm in
18. 6761
NF_at _G am m aS_ConjM at ch
1000
. . . ex2, VCEindex2, 0] , Z0[ 0, 0, 0] )
50. 0000
. . . ex2, VCEindex2, 0] , Z0[ 0, 0, 0] )
50. 0000
. . . gain( S[ I Cindex2, VCEindex2, 0] )
20. 7283
NF wit h Zsour ce ( valid f or K>1)
Sim ult aneous Conjugat e M at ched ( valid f or K>1)
Zsour ce Zload M AG ( or M SG f or K<1) NF_at _G am m aSopt
0. 8436
. . . aSopt , Z0[ 0, 0, 0] )
29. 2563 + j12. 1537
zin( G am m aL, Z0[ 0, 0, 0] )
61. 3802 - j39. 9026
G t r ans_power _load
16. 9127
NF wit h opt im al Zsour ce O pt im al Zsour ce
when Zload is at G am m aL Zload at G am m aL Tr ansducer Power gain ( dB)
GAcircles
Noise_circles
Source_stabcir
GPcircles
Load_stabcir
Outside
Sourc e Stable Region
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
0.0
8.0
I BB=0. 000
I BB=2. 000E- 6
I BB=4. 000E- 6
I BB=6. 000E- 6
I BB=8. 000E- 6
I BB=1. 000E- 5
I BB=1. 200E- 5
I BB=1. 400E- 5
I BB=1. 600E- 5
I BB=1. 800E- 5
I BB=2. 000E- 5
I BB=2. 200E- 5
I BB=2. 400E- 5
I BB=2. 600E- 5
I BB=2. 800E- 5
I BB=3. 000E- 5
VCE
IC.i,mA
mBiasPt
m Bias Pt
VCE=
IC.i=5.417352m
IBB=0.000020
3.000000
(A) Matching Condition for Minimum Noise Figure
(B) Matching Condition for Simultaneously Conjugate Matched (C) Matching Condition for Arbitray GammaS (D) Matching Condition for Arbitray GammaL
Find t he index of VCE and I C of t he biased point m BiasPt
Show 2000 point s on Sm it h Char t
Equations to PlotNoise,Gain,and Stability Circles
Noise Circle
Available Power Gain Circle
Operating Power Gain Circle
Source and Load Stability Circles
Transducer Power Gain CalculationNoise Figure Calculation
Reflection Coefficients Calculation
4 DifferentMatching Condition:
(A) M atc h for m inim um NF
(D) M atc h for optim um NF with arbitray Gam m a_L (Output Power)
(B) Sim ulataneous ly Conjugate M atc h
I nput : m at ched m in. noise, out put : conjugat e m at ched
I nput : m at ched opt im um noise, O ut put : G am m aL
( A) NFm in_lin ( M im inum noise f act or )
( B) M ax. t r ansducer power gain is equal t o M AG ( or M SG ) when sim ulyaneously m at ched.
I nput : conjugat e m at ched, out put : conjugat e m at ched
(C) M atc h with arbitray Gam m a_S (Gain c ons ideration)
I nput : G am m aS, O ut put : conjugat e m at ched
Bias Point Selector
Eqn S_11m =S_bpm ( 1, 1)
Eqn S_12m =S_bpm ( 1, 2)
Eqn S_21m =S_bpm ( 2, 1)
Eqn S_22m =S_bpm ( 2, 2)
Eqn S_bpm =S[ I Cindex2, VCEindex2, 0]
Eqn NFm in_at _m BiasPt =NFm in[ I Cindex2, VCEindex2, 0]
Eqn Sopt _at _m BiasPt =Sopt [ I Cindex2, VCEindex2, 0]
Eqn Z0_r ef =Z0[ 0, 0, 0]
Eqn Rn_at _m BiasPt =Rn[ I Cindex2, VCEindex2, 0]
Transistor S-parameter atmBiasPt
O pt im um r ef lect ion coef f . ( NFm in)
Ref er ence im pedance
Rn at bias point
NFm in @ m BiasPt
13/15

14. 在ADS建置完整的LNA設計環境(II)
Department of Electronic Engineering, NTUT
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
1.0
1.5
2.0
2.5
0.5
3.0
I BB=0. 000
I BB=2. 00u
I BB=4. 00uI BB=6. 00uI BB=8. 00uI BB=10. 0uI BB=12. 0uI BB=14. 0uI BB=16. 0uI BB=18. 0u
I BB=20. 0uI BB=22. 0uI BB=24. 0uI BB=26. 0uI BB=28. 0u
I BB=30. 0u
VCE
NFmin[0]
m2
m 2
VCE=
NFm in[0]=727.6303m
IBB=0.000002
3.000000
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
-15
-10
-5
0
5
10
15
-20
20
I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00u
I BB=8. 00uI BB=10. 0u
I BB=12. 0u
I BB=14. 0uI BB=16. 0uI BB=18. 0u
I BB=20. 0uI BB=22. 0uI BB=24. 0uI BB=26. 0uI BB=28. 0uI BB=30. 0u
VCE
dB(S21[0])
m1
m 1
VCE=
dB(S21[0])=6.954
IBB=0.000002
3.000
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
-16
-14
-12
-10
-8
-6
-4
-2
-18
0 I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00u
I BB=8. 00u
I BB=10. 0u
I BB=12. 0u
I BB=14. 0uI BB=16. 0u
I BB=18. 0uI BB=20. 0u
I BB=22. 0uI BB=24. 0u
I BB=26. 0uI BB=28. 0uI BB=30. 0u
VCE
dB(S11[0])
I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00u
I BB=8. 00u
I BB=10. 0u
I BB=12. 0u
I BB=14. 0u
I BB=16. 0uI BB=18. 0u
I BB=20. 0uI BB=22. 0u
I BB=24. 0u
I BB=26. 0uI BB=28. 0u
I BB=30. 0u
dB(S22[0])
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
-20
-15
-10
-5
-25
0
I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00u
I BB=8. 00u
I BB=10. 0u
I BB=12. 0uI BB=14. 0uI BB=16. 0u
I BB=18. 0uI BB=20. 0u
I BB=22. 0uI BB=24. 0uI BB=26. 0u
I BB=28. 0uI BB=30. 0u
VCE
dB(S12)
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
0
5
10
15
20
-5
25
I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00uI BB=8. 00u
I BB=10. 0u
I BB=12. 0u
I BB=14. 0uI BB=16. 0u
I BB=18. 0uI BB=20. 0uI BB=22. 0u
I BB=24. 0uI BB=26. 0uI BB=28. 0uI BB=30. 0u
VCE
MAG,dB
M inim um Noise Figure versus IBB and VCETrans istor dB(S21) v ers us IBB and VCE
M axim um Av ailable Gain v ersus IBB and VCE
dB(S12) v ers us IBB and VCE
dB(S11) and dB(S22) v ers us IBB and VCE
0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 50. 0 4. 0
0
5
10
15
- 5
20
I B B = 0 . 0 0 0
I B B = 2 . 0 0 u
I B B = 4 . 0 0 u
I B B = 6 . 0 0 u
I B B = 8 . 0 0 u
I B B = 1 0 . 0 u
I B B = 1 2 . 0 uI B B = 1 4 . 0 uI B B = 1 6 . 0 u
I B B = 1 8 . 0 uI B B = 2 0 . 0 uI B B = 2 2 . 0 uI B B = 2 4 . 0 u
I B B = 2 6 . 0 uI B B = 2 8 . 0 uI B B = 3 0 . 0 u
VCE
Pgain_assoc
m 4
m 4
VCE=
Pgain_as soc=19.273
IBB=0.000030
3.000
As soc iated Power Gain (input m atc hed for NFm in,
output then c onjugately m atc hed) v ers us IBB and VCE
Eqn
M AG =m ax_gain( S) M axim um av ailable gain at all frequenc ies
Eqn f r equency=SP. f req[ 0, 0, 0]
Eqn
I Cindex=f ind_index( I C[ VCEindex] , m 3)
Eqn VCEindex=f ind_index( DC. VCE[ 0, : : ] , indep( m 3) )
Eqn I C=-SRC1. i
Eqn
DC_power =m3*indep( m 3)
Eqn G am maS_at _bias_pt =sm_gam ma1( S_bp)
Eqn G am maL_at _bias_pt =sm_gam ma2( S_bp)
Eqn Zopt =zopt ( Sopt _at _bias_pt , Z0[ 0, 0, 0] )
Eqn S_11=S_bp( 1, 1)
Eqn S_12=S_bp( 1, 2)
Eqn
S_21=S_bp( 2, 1)
Eqn S_22=S_bp( 2, 2)
Eqn
S_22p_at _bias=S_22p[ I Cindex, VCEindex]
Eqn
Pgain_assoc_at _bias=Pgain_assoc[ I Cindex, VCEindex]
Eqn Zload_wSopt =zopt ( conj( S_22p_at _bias) , Z0[ 0, 0, 0] )
Eqn K=st ab_f act ( S_bp)
Eqn Pgain_assoc=pwr _gain( S[ 0] , zopt (Sopt [ 0] , Z0[ 0, 0, 0] ) , zopt ( conj(S_22p), Z0[ 0, 0, 0] ), Z0[ 0, 0, 0] )
Eqn
S_22p=S22[ 0] +( S12[ 0] *S21[ 0] *Sopt [ 0] ) / ( 1-S11[ 0] *Sopt [ 0] )
Eqn G am maL_wSopt =conj( S_22p_at _bias)
Eqn S_bp=S[ I Cindex, VCEindex, 0]
Eqn NFmin_at _bias_pt =NFm in[ I Cindex, VCEindex, 0]
S-param eters at the bias point s pec ified by m arker m 3.
Source im pedanc e for m inim um nois e figure at the bias
point s pec ified by m arker m 3.
Stability fac tor at the bias point m 3.
Zsourc e and Zload are the s ourc e and load im pedanc es to pres ent to
the dev ice for s im ultaneous conjugate m atching, at the bias point m 3.
Thes e are not defined and return 0 if K<1.
S_22p : reflection look ing into the output of the dev ice,
when the sourc e is optim al for m inim um nois e figure.
Gam m aL_wSopt is the c om plex c onjugate of S22_p, and
is the optim al load reflection c oeffic ient when Sopt is the s ource
reflec tion coeffic ient. Zload_wSopt is the c orres ponding im pedance.
Sim ultaneous c onjugate m atch s ource and load reflec tion c oefficients
at bias point m 3. These are not defined and return 0 if K<1.
Trans duc er power gain with the s ourc e reflec tion c oeffic ient Sopt for m inim um nois e figure, and the load
then c onjugately m atc hed. zopt() is jus t us ed to c onv ert a reflec tion coeffic ient to an im pedance.
Collec tor DC current
Find index for the swept v ariable VCE and ICE
acc ording to m ark er "m 3" x-axis .
M inim um nois e figure at the m 3 bias point.
DC power c om s um ption when biased at m arker "m 3" (bas e current is ignored)
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0
1.00m
2.00m
3.00m
4.00m
5.00m
6.00m
7.00m
0.000
8.00m
I BB=0. 000
I BB=2. 00u
I BB=4. 00u
I BB=6. 00u
I BB=8. 00u
I BB=10. 0u
I BB=12. 0u
I BB=14. 0u
I BB=16. 0u
I BB=18. 0u
I BB=20. 0u
I BB=22. 0u
I BB=24. 0u
I BB=26. 0u
I BB=28. 0u
I BB=30. 0u
VCE
IC.i,A
m3
m 3
VCE=
IC.i=5.417352m
IBB=0.000020
3.000000
I/V Curv e (Selec t Biasing Point via m aker m 3)
Eqn
Sopt _at _bias_pt =Sopt [ I Cindex, VCEindex, 0]
Eqn Zsour ce=sm _z1( S_bp, Z0[ 0, 0, 0] )
Eqn Zload=sm _z2(S_bp, Z0[ 0, 0, 0] )
Source reflection c oeffic ient for m inim um nois e figure
at frequenc y s pec ified by m ark er m 3. Sopt is the s-param eter
for optim um noise perform ance.
(1) (2)
Bas ic inform ation at the bias point m 3.
Optim um reflec tion c oeffic ient(im pedanc e) for m inim um noise at the bias point m 3.
Output Conjugately M atching Im pdeance Calculation (when input is nois e m atc hed)
Input/Output Sim ultaneous ly Conjugate M atc hed (input is NOT nois e m atc hed)
Move marker m3 to selectbias point.
All listings and impedances on Smith Chartwill be updated.
Matching for Gain Zsourc e Zload
DUT*
(0.000 to 0.000)
Sopt_at_bias_pt
GammaS_at_bias_pt
GammaL_at_bias_pt
GammaL_wSopt
Optim al Sourc e Reflection Coeffic ients for M ininum NF, Sim ultaneous Conjugate M atching,
and Load Reflec tion Coeffic ient for Sim ultaneous Conjugate M atc hing, and with s ource
m atc hed for NFm in
Note: if the dev ic e (or circ uit) is uns table at the bias point, the s im ultaneous c onjugate m atc hing im pedances
are undefined and Gam m aL_at_bias _pt and Gam m aS_at_bias_pt default to 0. Als o, M AG is set equal to the
m ax im um stable gain, |S21|/|S12|.
1.00m
2.00m
3.00m
4.00m
5.00m
6.00m
7.00m
0.000
8.00m
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.4
2.0
IC
NFmin,dB
m 5
m 5
indep(m 5)=
v s (NFm in[VCEindex ,0],IC.i[VCEindex])=0.651189
0.005417
NFmin versus IC, at VCE (set by m3)
1.00m
2.00m
3.00m
4.00m
5.00m
6.00m
7.00m
0.000
8.00m
-15
-10
-5
0
5
10
15
-20
20
IC
dB(S21)
dB(S21) v ersus IC, at VCE (s et by m 3)
indep( m3)
3. 0000
m 3[ 0]
5. 4174 m
DC_power [ 0]
16. 252 m
f r equency
2. 400 G
VCE IC DC power (W)
dB( S_11)
-6. 7279
dB( S_12)
- 23. 460
dB( S_21)
17. 996
dB( S_22)
-7. 0302
Trans is tor S-param eter at bias point m 3
K
0. 6776
Stability Fac tor
Z0[ 0, 0, 0]
50. 0000
Charac teris tic s Im pedanc e
m ax_gain(S_bp)
20. 7283
M ax Av aliable Gain (dB) Zsource
50. 0000
Zload
50. 0000
Sim ultaneous M atc h
Matching for Noise Figure
NFm in_at _bias_pt
0. 6512
M inim um Nois e Figure (dB)
Sopt _at _bias_pt
0. 2799 / 57. 8169
Soure Reflec tion Coeff. for NFm in
Zopt
59. 0670 + j30. 3691
Zopt for NFm in
Zload_wSopt
31. 8982 + j31. 7136
Conjugate M atc hed Load
(for input m atched to NFm in)
Zopt Zload_wSopt
DUT*
Pgain_assoc_at _bias
18. 6761
Power Gain (dB)
at this nois e m atc hed condition
Gam m a_S (NFm in)
Gam m a_L when NFm in
Bias Point Selector
Updated Information according to the Bias Point m3
14/15

15. 在ADS建置完整的LNA設計環境(III)
Department of Electronic Engineering, NTUT
Move marker mBiasPt to desiredfrequency point.
Smith Chart and data below will be updated.
Move markers GammaSand GammaL to selectarbitrary source and load reflection coefficients The impedances,power gains,and noise figures
below will be updated.The transducer power gains are invalid ifthe markers are moved into the unstable regions.
Eqn num _NFc ircles=3
Eqn NFs tep_s iz e=0.2Eqn GAs tep_s iz e=1
Eqn num _GAc ircles=3
Zs ourc e,
Gam m a_S
Zload,
Gam m a_L
DUT*
Eqn num _GPcirc les =3
Eqn GPs tep_siz e=1
G am m aS
indep(G am m aS) =
r hos=-0. 25766 - j0. 01061
im pedance = 29. 50724 - j0. 67091
133
indep(rhos) (0. 000 t o 2000. 000)
rhos
GammaS
indep(Sour ce_st abcir) (0. 000 t o 51. 000)
Source_stabcir
g a in = 2 1 . 0 0 4
g a in = 2 0 . 0 0 4
g a in = 1 9 . 0 0 4
g a in = 1 8 . 0 0 4
cir_pt s (0. 000 t o 51. 000)
GAcircles
indep(G am m aLopt ) (133. 000 t o 133. 000)
GammaLopt
n s f ig u r e = 0 . 8 6 7
n s f ig u r e = 1 . 0 6 7
n s f ig u r e = 1 . 2 6 7
Noise_circles
Noise_circleMin
G am m aS
indep(G am m aS) =
r hos=-0. 25766 - j0. 01061
im pedance = 29. 50724 - j0. 67091
133 G am m aL
indep(G am m aL) =
r hos=0. 35071 / -54. 37157
im pedance = Z0 * (1. 22760 - j0. 79805)
246
indep(rhos) ( 0. 000 t o 2000. 000)
rhos
GammaL
indep( Load_st abcir ) (0. 000 t o 51. 000)
Load_stabcir
g a in = 2 1 . 0 0 4
g a in = 2 0 . 0 0 4
g a in = 1 9 . 0 0 4
g a in = 1 8 . 0 0 4
cir_pt s (0. 000 t o 51. 000)
GPcircles
indep(G amm aSopt ) (246. 000 t o 246. 000)
GammaSopt
G am m aL
indep(G am m aL) =
r hos=0. 35071 / -54. 37157
im pedance = Z0 * (1. 22760 - j0. 79805)
246
Available Gain Circle:
Noise Circles:
Source Stability Circle:
Source Gamma Corresponding Load Gamma (Black Dot)
Power Gain Circles:
Load Stability Circle:
Load Gamma Corresponding Source Gamma (Black Dot)
Load Stable Region
Eqn t index=[ 0: : 2000]
Eqn r hos=sqrt (t index/ 2000)*exp(j*2*sqrt (pi*t index))
Eqn Source_st abcir=s_st ab_circle(S[ f m 1] , 51)
Eqn Load_st abcir =l_st ab_cir cle(S[ f m 1] , 51)
Eqn
G amm aLopt =conj(S22[ f m1] +S12[ f m 1] *S21[ f m 1] *G am m aS/ (1-S11[ f m 1] *G am maS))
Eqn G t _num=m ag( S21[ f m 1] )**2 *( 1-m ag(G am m aS) **2) *(1-m ag(G am m aLopt ) **2)
Eqn G t _den=m ag(( 1-S11[ f m 1] *G am m aS) *(1-S22[ f m 1] *G am maLopt ) -S21[ f m 1] *S12[ f m 1] *G am maS*G amm aLopt )**2
Eqn
G amm aLopt _NFm in=conj(S22[ f m 1] +S12[ f m 1] *S21[ f m1] *Sopt [ f m1] / (1- S11[ f m1] *Sopt [ f m1] ))
Eqn G t _num_NFmin=m ag( S21[ f m1] ) **2 *( 1-m ag(Sopt [ f m1] )**2) *(1- mag( G amm aLopt _NFm in)**2)
Eqn G t _den_NFm in=mag(( 1- S11[ f m 1] *Sopt [ f m 1] )*(1- S22[ f m1] *G amm aLopt _NFm in) -S21[ f m 1] *S12[ f m1] *Sopt [ f m1] *G amm aLopt _NFm in) **2
Eqn G t rans_power_NFm in=10*log(G t _num _NFm in/ G t _den_NFm in)
Eqn
NF_lin_at _G am m aS=NFmin_lin+4*(Rn[ f m 1] / Z0[ f m1] ) *m ag( G am maS-Sopt [ f m1] )**2/ (( 1-m ag(G am m aS)**2)*m ag( 1+Sopt [ f m1] )**2)
Eqn NFm in_lin=10**(NFmin[ f m1] / 10)
Eqn
NF_at _G amm aS=10*log(NF_lin_at _G am m aS)
Eqn
NF_at _G amm aS_ConjM at ch=if (st ab_f act ( S[ f m1] ) >1) t hen 10*log(NF_lin_at _G am m aS_ConjM at ch) else 1000
Eqn NF_lin_at _G am m aS_ConjM at ch=NFm in_lin+4*( Rn[ f m 1] / Z0[ f m 1] )*mag(G amm aS_ConjM at ch- Sopt [ f m 1] ) **2/ ( (1- m ag( G amm aS_ConjM at ch)**2)*m ag( 1+Sopt [ f m1] )**2 +1e-20)
( C) O pt im al G am ma_L when t he G am m a_S is at " maker G am m aS"
( A) O pt im al G am ma_L when t he G am m a_S is at Sopt (opt im al f or m inimum noise f igure. )
(C) Noise f igur e f or an ar bit ray G am m a_S ( m arker G am maS)
(B) Noise f igur e f or sim ult aneously conjugat e m at ching. (O nly def ined if K is >1. O t her wise t he noise f igure is set t o 1000. )
( C) G t rans_power : t ransducer power gain wit h t he source ref lect ion coef f icient at marker G amm aS, and t he load t hen conjugat ely mat ched.
( A) G t rans_power _NFmin: t ransducer power gain wit h t he sour ce ref lect ion coef f icient Sopt f or m inim um noise f igur e, and t he load t hen conjugat ely mat ched.
Eqn G amm aSopt =conj(S11[ f m1] +S12[ f m 1] *S21[ f m 1] *G am m aL/ (1- S22[ f m1] *G amm aL))
( D) O pt im al G am ma_S when t he G am m a_L at " m aker G am m aL"
Eqn G t load_num =m ag( S21[ f m1] )**2 *( 1-m ag(G am m aSopt )**2) *( 1-m ag(G am m aL) **2)
Eqn G t load_den=mag(( 1-S11[ f m 1] *G am m aSopt )*( 1-S22[ f m 1] *G am m aL) - S21[ f m 1] *S12[ f m 1] *G am maSopt *G amm aL)**2
Eqn
G t rans_power_load=if (G t load_num>0) t hen 10*log( G t load_num / G t load_den) else 1e6
( D) G t rans_load : t r ansducer power gain wit h t he load ref lect ion coef f icient at m ar ker G am m aL, and t he sour ce t hen opt imumly noise m at ched.
(D) Noise f igur e f or an ar bit ray G am m a_L (t he source ref lect ion coef f icient is at G am m aSopt )
Eqn
NF_lin_at _G am m aSopt =NFmin_lin+4*(Rn[ f m 1] / Z0[ f m 1] ) *m ag( G am maSopt - Sopt [ f m 1] ) **2/ ( (1- mag(G amm aSopt )**2) *m ag(1+Sopt [ f m 1] ) **2)
Eqn NF_at _G amm aSopt =10*log(NF_lin_at _G am m aSopt )
Sour ce ref lect ion coef f icient
Eqn G amm aS_ConjM at ch=sm _gam m a1( S[ f m 1] )
Zsource is t he im pedance at m ar ker G am m aS.
Eqn Zsource2=Z0*(1+G am m aS) / ( 1-G am m aS)
Eqn
G t rans_power=if ( G t _num >0) t hen 10*log(G t _num / G t _den) else 1e6
Eqn
Noise_circleM in=ns_circle(NFm in[ f m 1] , NFm in[ f m 1] , Sopt [ f m 1] , Rn[ f m1] / Z0[ f m 1] , 51)
Eqn Noise_circles=ns_circle(NFm in[ f m 1] +NFst ep_size*[ 1: : num _NFcir cles] , NFm in[ f m 1] , Sopt [ f m 1] , Rn[ f m 1] / Z0[ f m 1] , 51)
Eqn G AcircleM ax=ga_cir cle( S[ f m 1] , m ax_gain(S[ f m 1] ))
Eqn G Acircles=ga_cir cle( S[ f m 1] , max_gain(S[ f m 1] )-G Ast ep_size*[ 0: : num _G Acircles] )
Eqn G Pcircles=gp_cir cle( S[ f m 1] , max_gain(S[ f m 1] )-G Pst ep_size*[ 0: : num _G Pcircles] )
Set st ep size and num ber of circles t o plot
st ab_f act (S[ f m 1] )
0. 7083
St abilit y K
t index is a vect or of numbers 0, 1, 2, 3, . . . , 2000.
r hos are 2001 com plex ref lect ion coef f icient s.
( B) G am m a_S f or sim ult aneous conjugat e m at ching at bias point m BiasPt .
NF at G amm aS (dB)
NF_at _G amm aS
0. 9252
Zsource2
29. 5072 - j0. 6709
Sour ce I m pedance at G am maS
. . . am m aLopt , Z0[ f m 1] )
34. 8292 + j54. 1030
O pt iom al Load I m pedance at G amm aS Tr ansducer Power G ain (dB)
G t r ans_power
20. 3030
NFmin[ f m 1]
0. 6669
NFmin (dB)
zopt (Sopt [ f m 1] , Z0[ f m1] )
58. 8848 + j26. 9719
Source I mpedance Zopt at NFm in
. . . maLopt _NFm in, Z0[ f m 1] )
32. 4007 + j30. 7066
O pt iom al Load I m pedance
f or source Zopt at NFm in
Tr ansducer Power G ain (dB)
G t rans_power_NFm in
18. 8942
NF_at _G amm aS_ConjM at ch
1000
sm _z1(S[ f m 1] , Z0[ f m1] )
50. 0000
sm _z2(S[ f m 1] , Z0[ f m 1] )
50. 0000
m ax_gain(S[ f m 1] )
21. 0038
NF wit h Zsour ce (valid f or K>1)
Sim ult aneous Conjugat e M at ched (valid f or K>1)
Zsour ce Zload M AG ( or MSG f or K<1) NF_at _G am maSopt
0. 8562
. . . aSopt , Z0[ f m1] )
29. 1731 + j10. 0394
zin(G am m aL, Z0[ f m 1] )
61. 3802 - j39. 9026
G t rans_power_load
17. 1906
NF wit h opt imal Zsource
O pt imal Zsource
when Zload is at G am m aL Zload at G am maL Transducer Power gain ( dB)
GAcircles
Noise_circles
Source_stabcir
GPcircles
Load_stabcir
Source Stable Region
(A) Matching Condition for Minimum Noise Figure
(B) Matching Condition for Simultaneously Conjugate Matched (C) Matching Condition for Arbitray GammaS (D) Matching Condition for Arbitray GammaL
Find t he index of VCE and I C of t he biased point mBiasPt
Show 2000 point s on Smit h Char t
Equations to PlotNoise,Gain,and Stability Circles
Noise Circle
Available PowerGain Circle
Operating PowerGain Circle
Source and Load Stability Circles
Transducer Power Gain CalculationNoise Figure Calculation
Reflection Coefficients Calculation
4 DifferentMatching Condition:
(A) M atc h for m inim um NF
(D) M atc h for optim um NF with arbitray Gam m a_L (Output Power)
(B) Sim ulataneous ly Conjugate M atch
I nput : m at ched min. noise, out put : conjugat e m at ched
I nput : m at ched opt im um noise, O ut put : G amm aL
(A) NFm in_lin ( M iminum noise f act or)
( B) Max. t r ansducer power gain is equal t o M AG ( or MSG ) when sim ulyaneously m at ched.
I nput : conjugat e mat ched, out put : conjugat e mat ched
(C) M atc h with arbitray Gam m a_S (Gain c ons ideration)
I nput : G amm aS, O ut put : conjugat e m at ched
Frequency Point Selector
fm1
indep(fm1)=
plot_vs([0::sweep_size(frequency)-1],frequency)=6.000000
2.360000G
2.32E9
2.34E9
2.36E9
2.38E9
2.40E9
2.42E9
2.44E9
2.46E9
2.48E9
2.30E9
2.50E9
0. 0
1. 0E6
f requency
fm1
fm1
indep(fm1)=
plot_vs([0::sweep_size(frequency)-1],frequency)=6.000000
2.360000G
15/15