### Noise Model of a High-Speed Operational Amplifier

```Noise Model of a High-Speed
Operational Amplifier Implementation in MATLAB SimRF
Application Note
Sheila P. Werth, Stephen J. Bitar, & Sergey N. Makarov
ECE Dept. WPI, Worcester, MA
July 5th 2011
1
Outline
1.
2.
3.
4.
5.
Concept of a noisy operational amplifier
Two basic op-amp circuits
Equivalent input noise
Extra contribution of noisy resistors R1, R2
Generic model of an op-amp circuit with noise –
noise figure
6. Noise figure of an op-amp
7. MATLAB script for finding the noise figure
8. How is the noise model of an operational amplifier
implemented in SimRF?
9. Test of op-amp model
10. Amplifier model example in SimRF
2
Concept of a noisy operational
amplifier
Op-amp datasheet reports:
en [nV/ Hz], in [pA/ Hz]
Resistor noise model:
eR  4kTRB [V]
Amplifier noise model:
eN  B  en [V], i N  B  in [A]
B – circuit bandwidth in Hz (bandwidth over which white noise is collected)
3
Two basic op-amp circuits (inv. and
non-inv. configurations)
Inverting configuration :
S (t )  0
Non- inverting configuration:
S (t )  0



R2  R1  RS  R2

out (t ) 
eN (t )  ( R1  RS )iN (t )

R1  RS 
R
2



equivalentinput noise e Ni



 R2  
R1 R2 

out (t )  1   eN (t ) 
iN (t )  RS iN (t )
R1  
R  R2

1 




equivalentinput noise e Ni
4
Equivalent input noise
Inverting configuration:
e Ni2
 R  R S  R2
  1
R2

Non-inverting configuration:
2
 2
2
 e N  ( R1  RS ) 2 i N

e Ni2
 RR
 e N2   1 2
 R1  R2
2
 2
2
 i N  RS2 i N

Example 1: Find noise voltage added by two operational amplifier circuits (inv. and
non-inv. op-amps) given that
1. en  14 nV/ Hz, iN  1.8 pA/ Hz (LM7171 of National Semiconductor);
2. RS  1k, R1  1k, R2  100k ;
3. The circuit bandwidth B is 20 kHz in every case;
4. The voltage noise of resistors is neglected.
Solution: Equivalent input noise becomes
eNi  2.08μV, eNi  2.01μV
for the inverting and non-inverting amplifier, respectively. At the output to the
amplifier, these values are transformed (through gain multiplication) to
eNout  0.104mV, eNout  0.203mV.
5
Extra contribution of noisy resistors R1, R2
Inverting configuration:
e Ni2
 R  RS
 e R21   1
 R2
Non-inverting configuration:
2
 2
 e R 2

e Ni2
 R2
 
 R1  R2
2
 2  R1
 e R1  

 R1  R2
2
 2
 e R 2

6
Extra contribution of noisy resistors R1, R2
(cont.)
Example 2: How do the results of the previous example change if the thermal noise of
the two resistors R1  1k, R2  100k is additionally taken into account? Assume
room temperature, T= 298K.
Solution: For the inverting amplifier, the contribution of noisy resistors into the
equivalent input noise is obtained using the superposition principle; it has the form
eNi  3.29 10 13  1.32 10 14  0.59 μV
at the input to the inverting amplifier. Similarly, one has
eNi  3.23 10 13  3.23 10 15  0.57μV
for the non-inverting amplifier.
7
Generic model of an op-amp circuit with
noise – noise figure
Noise factor: NF  1 

N 
Noise figure: NFdB  10log10 1  a 
 N in 
Na
N in
2
Added noise: N a  eNi
[V2 ]
Inf. input res.: Nin  e
 4kTRS B [V ]
2
RS
Input (reference) noise:
2
2
e RS
Matched input res.: N in 
 kTRS B [V 2 ]
4
(MATLAB SimRF)
8
Noise figure of an op-amp may be
surprisingly high…
Example 3: Determine the noise figure of an inverting amplifier at room
temperature taking into account extra noise contributions due to noisy resistors
R1 , R2 . It is given that
1. en  14 nV/ Hz, in  1.8 pA/ Hz (LM7171 high-speed op-amp);
2. RS  50, R1  50, R2  5k ;
3. The circuit bandwidth B is 20 kHz.
Solution: Total equivalent input noise to the amplifier includes semiconductor
noise and resistor noise; the result becomes
e Ni  4.10  10 12  1.71  10 14  2.02μV
The added noise power referenced to amplifier’s input is given by
N a  eNi2  3.941012 [V2 ]
Input or reference noise ( B  20 kHz ):
Ni  kTRS B  4.111015 [V2 ], RS  50
Therefore,
 N 
NFdB  10log10 1  a   10log10 1  9.96102  30dB !
 N in 


9
… but it decreases with a higher source resistance (causing a
higher input noise)
Example 4: Solve example 3 when RS  1k, R1  1k, R2  100k .
Solution:
NFdB  18dB
…or when a better IC chip is used
Example 5: Solve example 3 for LMH6624 op-amp from National Semiconductor
with en  0.9 nV/ Hz .
Solution:
NFdB  10dB
Question: Why use an op-amp then?
flexibility
10
MATLAB script for finding the noise figure
using the previous analysis:
clear all;
k
= 1.38066e-23;
T
= 298;
VT
= 4*k*T;
B
= 2e4;
%
%
%
%
Boltzmann constant [J/K]
temperature [K]
temperature constant [W/Hz]
system (noise) bandwidth, Hz (cancels out)
%
Amplifier parameters
en = 14e-9;
%
required (datasheet)
in = 1.8e-12;
%
required (datasheet)
RS = 1e3;
%
use an estimate when the exact value is not available
R1 = 1e3;
%
required
R2 = 100e3;
%
required
Nin= k*T*RS*B; %
reference input noise power (Pozar)
%
Inverting amplifier
inv.eNi = sqrt(B)*sqrt( (R1+RS+R2)^2/R2^2*en^2 + (R1+RS)^2*in^2);
inv.eR1 = sqrt(4*k*T*R1*B);
%
rms voltage noise
inv.eR2 = sqrt(4*k*T*R2*B);
%
rms voltage noise
inv.eR = sqrt(inv.eR1^2 + ((R1+RS)/R2)^2*inv.eR2^2); % inv
inv.eNi = sqrt(inv.eNi^2 + inv.eR^2);
inv.Na = inv.eNi^2;
inv.NF = 10*log10(1 + inv.Na/Nin); inv
%
Non-inverting amplifier
noninv.eNi = sqrt(B)*sqrt( en^2 + (R1*R2/(R1+R2))^2*in^2 +RS^2*in^2);
noninv.eR1 = sqrt(4*k*T*R1*B);
%
rms voltage noise
noninv.eR2 = sqrt(4*k*T*R2*B);
%
rms voltage noise
noninv.eR = sqrt((R2/(R1+R2))^2*noninv.eR1^2 + (R1/(R1+R2))^2*noninv.eR2^2); % non-inv
noninv.eNi = sqrt(noninv.eNi^2 + noninv.eR^2);
noninv.Na = noninv.eNi^2;
noninv.NF = 10*log10(1 + noninv.Na/Nin); noninv
11
How is the noise model of an operational
amplifier implemented in SimRF?
1. Run MATLAB script given above and calculate the op-amp noise figure in dB. If
you do not know source impedance RS exactly, use an estimated value.
2. Insert the noise figure value into
the amplifier block
3. Explore block “SimRF parameters”
4. Set noise reference impedance to be exactly equal to the value of your source
impedance RS identified previously.
5. Set noise bandwidth greater than
or equal to the expected system
bandwidth
12
Test of op-amp model
1. Construct the op-amp model in SimRF as described above
2. Short out its input
3. Measure rms added noise at the output
4. Compare this value with the corresponding theoretical prediction
Theoretical prediction (based on Example 3): For an inverting amplifier at room
temperature with
1. en  14 nV/ Hz, iN  1.8 pA/ Hz (LM7171 high-speed op-amp);
2. RS  50, R1  50, R2  5k ;
3. Circuit bandwidth of 20 kHz.
find total added output noise taking into account semiconductor/resistor noise.
Solution: Total equivalent input noise to the amplifier was found in example 3:
eNi  2.02μV
Total output noise to the amplifier is the input noise times the voltage gain:
eNout  G  eNi  101μV
13
SimRF set up
Parameters: You already calculated a noise figure of 30dB for this particular amplifier with a
voltage gain of 50. Now, enter these parameters:
You calculated the noise figure based upon a 50 ohm reference impedance and B= 20 kHz so:
14
Theory vs. simulations
The calculated rms output noise voltage is:
eNout  101V
The output from the experimental setup is a close match:
eNout  98.7V
The experimental setup calculates a running rms over a finite time window - this could be a
source of error:
Error 
101 98.7
101
100%  2.3%
15
Amplifier model example in SimRF: