Download Small-signal noise simulation

Noise Figure Calculation Exercise Using Microwave CAD Software
1.0 Derivation of Noise Figure Expression for Computer Simulation
A simple noise analysis of the resistive network is carried out.
Linear 2-port network
Port 1
R1
Port 2
R2
RS
RL
Vout
Vin
Figure 1 – A two-port resistive network.
The output voltage at Port 2 is given by:
R L // R2
Vout =
Vin = A ⋅ Vin
R L // R2 + R1 + RS
(1)
In equation (1) A is the effective voltage gain. Deactivating the source voltage Vin and
considering the thermal noise from the resistors, the equivalent circuit of Figure 2 is
obtained.
Port 1
R1
Port 2
R2
RS
4kTR2
Vn1
RL
Vout
Vn2
4kT (RS + R1 )
Figure 2 – Equivalent circuit with thermal noise source, signal source deactivated.
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The noise sources Vn1 and Vn2 are white noise, and the Power Spectral Density (PSD) Sn1
and Sn2 which are assumed constant with respect to frequency.
S n1 ( f ) = 4kT (RS + R1 )
(2a)
S n 2 ( f ) = 4kTR2
(2b)
Assuming a bandwidth of ∆f.
Using the fact that Vn1 = ∫ S n1 ( f )df
2
and
Vn 2 = ∫ S n 2 ( f )df , the contribution of the noise sources individually at the output node is
2
given as follows:
R2 // RL
R2 // RL + R1 + RS
2
Vout ( n1) = Vout ( n1) =
2
Vout (n 2 ) = Vout (n 2 ) =
4kT (RS + R1 )∆f
(R1 + RS )// RL
(R1 + RS ) // RL + R2
4kTR2 ∆f
(3a)
(3b)
The random process due to Vn1 and Vn2 can be combined using Superposition Theorem
for linear circuit. Since noise sources Vn1 and Vn2 are uncorrelated, the total PSD at port
2 is (See [1], Section11.4):
S ( f ) = S n1 ( f ) + S n 2 ( f )
(4a)
Or
2
2
Vout (noise ) = Vout ( n1) + Vout ( n 2 )
This implies:
Vout (noise ) = Vout (noise )
2
= [V

R2 // RL
= 
 R2 // RL + R1 + RS
2
out ( n1)
(4b)
2
+ Vout ( n 2 )
2
]
1
2

 (R1 + RS ) // RL
4kT (RS + R1 )∆f  + 

 (R1 + RS ) // RL + R2
2
1
2
(5)
 
4kTR2 ∆f  
 
2
Not that in computing the total noise voltage, the contribution from the termination at
port 2, e.g. RL is not considered. The signal-to-noise ratio at the input, SNRin is given by:
Vin2
SNRin =
(6)
4kTRS
The signal-to-noise ratio at the output, SNRout is given by:
( A ⋅ Vin )2
SNRout =
2
Vout ( noise )
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(7)
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Thus the noise factor of the network, from (6) and (7) is:
2
Vout (noise )
SNRin
= 2
SNRout
A ⋅ 4kTRs
The noise figure (F) in dB is then given by
 SNRin 
F = 10 ⋅ log 10 

 SNRout 
NF =
(8)
(9)
Equations (8) and (9) are usually employed by computer simulation software to
determine the noise figure F of a linear circuit.
2.0 Numerical Example
Let RS = RL = 50, R1 = 50 and R2 = 100. T = 25o C or 298K, k = 1.380×10-23 JK-1.
Bandwidth ∆f = 1Hz (spot noise calculation).
Then:
Vout ( n1) = 0.25 ⋅ 1.28256 × 10 −9 = 320.64 pV
Vout ( n 2 ) = 0.25 ⋅ 1.28256 × 10 −9 = 320.64 pV
A = 0.25
2
2 × (320.64 pV )
NF =
≅ 4.00
2
0.25 2 ⋅ (906.91 pV )
F = 10 log10 (4.00 ) = 6.02
3.0 ADS Simulation of Small-signal Noise
In ADS the noise simulation is typically performed during AC or S-parameter simulation.
Therefore it will be convenient if the noise factor NF can be expressed in terms of the Sparameters.
Port 1
I1
R1
V2
Linear 2-port
network
 s11
s
 21
Vs1
s12 
s 22 
Port 2
I2
V2
R2
Vs2
Figure 3 – Two-port network.
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Consider a two-port network as shown in Figure 3, the normalized incident and reflected
voltage waves can be defined according to Chapter 1 of [3]:
ai =
bi =
Vi + Ri I i
2 Ri
Vi − Ri I i
2 Ri
Vsi
=
(10a)
2 Ri
2Vi − Vsi
=
(10b)
2 Ri
Where i = 1 or 2. When measuring parameter sij, the source Vsi would be deactivated
(e.g. shorted). For s21:
b
s 21 = 2
a1 a = 0
2
Generator Vs2 = 0. Hence using the definition (10a) and (10b):
2V2
 V  R1 
2 R2

s 21 = V
= 2 2 

s1
V
R
2 
 s1 
2 R1
s 21
2
V
=4 2
V1
2
RS
2 RS
=4A
RL
RL
(11)
Since R1 = RS, R2 = RL, V1=Vin and V2 = Vout. Now consider equation (8) again,
2
NF =
Vout (noise )
Vn ( 2− port ) + 4kTRS A
2
2
A ⋅ 4kTRs
=
2
RL
(4 A R )⋅ kT
2
S
RL
Here the total mean-square noise voltage at the output is divided into contribution from
the source resistance RS and the noise sources within the two-port network. Vn(2-port) is
the noise voltage at the output due to noise sources within the two-port network. Now
using (11) this can be written as:
2
NF =
Vn
+ kT s 21
RL
kT s 21
2
(12)
2
Equation (12) is the expression used by ADS software for noise figure computation when
noise calculation is enabled during S-parameter simulation. It can be easily extended to
multi-port network as illustrated in the online documentation in ADS.
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4.0 ADS Simulation Example
Here the numerical example of Section 3.0 is repeated in ADS (version 2003C is used).
The schematic and data display are shown in Figure 4 and Figure 5 respectively. The
S-parameter is performed at a single frequency of 1.0GHz.
Figure 4 – The schematic and “Noise” tab setting in the S-parameter control.
Figure 5 – The data display.
Here we are only interested in the noise figure with output being taken at port 2. So we
select nf(2) to be shown in the list. Note that the software also calculates the minimum
noise figure NFmin and the corresponding optimum source reflection coefficient Sopt. See
[4] on how to find Sopt and NFmin. In Figure 5, the contribution of the individual noise
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source at the output port is also displayed. We are only interested in the listing under
port2.NC.name and port2.NC.vnc for output taken at port 2. Compare the values with
those obtained in Section 3.0.
References
1. B. P. Lathi, “Modern digital and analog communication systems”, 3rd edition 1998,
Oxford University Press.
2. B. Razavi, “RF microelectronics”, 1998, Prentice-Hall Inc.
3. G. D. Vendelin, A. M. Pavio, U. L. Rohde, “Microwave circuit design using linear
and nonlinear techniques”, 1990, John Wiley & Sons.
4. R. Ludwig, P. Bretchko, “RF circuit design – theory and applications”, 2000,
Prentice-Hall Inc.
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