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Noise Model of a High-Speed Operational Amplifier - Implementation in MATLAB

SimRF

Application Note

Sheila P. Werth, Stephen J. Bitar, & Sergey N. MakarovECE Dept. WPI, Worcester, MA

July 5th 2011

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Outline1. Concept of a noisy operational amplifier2. Two basic op-amp circuits3. Equivalent input noise4. Extra contribution of noisy resistors R1, R25. 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 model10. Amplifier model example in SimRF

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Concept of a noisy operational amplifier

Resistor noise model:

Amplifier noise model:

B – circuit bandwidth in Hz (bandwidth over which white noise is collected)

[V]4kTRBeR

[A][V], nNnN iBieBe

Op-amp datasheet reports:

]Hz[pA/ ],Hz[nV/ nn ie

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Two basic op-amp circuits (inv. and non-inv. configurations)

Inverting configuration :

Non- inverting configuration:

noiseinput equivalent

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21

1

2 )()()()(

Nie

NSNS

Sout tiRRte

RRRR

RRRt

noiseinput equivalent

21

21

1

2 )()()(1)(

Nie

NSNNout tiRtiRRRRte

RRt

0)( tS

0)( tS

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Non-inverting configuration:

Equivalent input noiseInverting configuration:

221

22

2

212 )(

NSN

SNi iRRe

RRRR

e222

2

21

2122

NSNNNi iRiRRRR

ee

Example 1: Find noise voltage added by two operational amplifier circuits (inv. and non-inv. op-amps) given that

1. HzpA/8.1,HznV/14 Nn ie (LM7171 of National Semiconductor); 2. k100,k1,k1 21 RRRS ; 3. The circuit bandwidth B is 20 kHz in every case; 4. The voltage noise of resistors is neglected.

Solution: Equivalent input noise becomes

μV01.2,μV08.2 NiNi ee for the inverting and non-inverting amplifier, respectively. At the output to the amplifier, these values are transformed (through gain multiplication) to

mV203.0,mV104.0 NoutNout ee .

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Non-inverting configuration:

Extra contribution of noisy resistors R1, R2

Inverting configuration:

22

2

2

121

2R

SRNi e

RRR

ee

2

2

2

21

121

2

21

22RRNi e

RRR

eRR

Re

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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 k100,k1 21 RR is additionally taken into account? Assume room temperature, T= 298K. 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

μV59.01032.11029.3 1413 Nie

at the input to the inverting amplifier. Similarly, one has

μV57.01023.31023.3 1513 Nie

for the non-inverting amplifier.

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Generic model of an op-amp circuit with noise – noise figure

Noise factor: Noise figure:in

a

NN

NF 1

in

adB N

NNF 1log10 10

Added noise: ][V 22Nia eN Input (reference) noise:

Inf. input res.: Matched input res.:(MATLAB SimRF)

][V4 22 BkTReN SRSin ][V4

22

BkTRe

N SRS

in

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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

21 , RR . It is given that

1. HzpA/8.1,HznV/14 nn ie (LM7171 high-speed op-amp); 2. k5,50,50 21 RRRS ; 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

μV02.2101.711010.4 1412 Nie

The added noise power referenced to amplifier’s input is given by ][V1094.3 2122 Nia eN

Input or reference noise ( kHz20B ): 50],[V1011.4 215

SSi RBkTRN Therefore,

dB301096.91log101log10 21010

in

adB N

NNF !

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… but it decreases with a higher source resistance (causing a higher input noise)

Example 4: Solve example 3 when k100,k1,1k 21 RRRS . Solution:

dB18dBNF

…or when a better IC chip is used

Example 5: Solve example 3 for LMH6624 op-amp from National Semiconductor with HznV/9.0ne . Solution:

dB10dBNF

Question: Why use an op-amp then? Answer: One major advantage is a high gain; another advantage is matching flexibility

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MATLAB script for finding the noise figure using the previous analysis:

clear all; k = 1.38066e-23; % Boltzmann constant [J/K] T = 298; % temperature [K] VT = 4*k*T; % temperature constant [W/Hz] B = 2e4; % 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

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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

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Test of op-amp model

Theoretical prediction (based on Example 3): For an inverting amplifier at room temperature with

1. HzpA/8.1,HznV/14 Nn ie (LM7171 high-speed op-amp); 2. k5,50,50 21 RRRS ; 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:

μV02.2Nie Total output noise to the amplifier is the input noise times the voltage gain: μV101 NiNout eGe

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

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SimRF set upParameters: 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:

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Theory vs. simulations

The calculated rms output noise voltage is:

The output from the experimental setup is a close match:

The experimental setup calculates a running rms over a finite time window - this could be a source of error:

VeNout 101

VeNout 7.98

%3.2%100101

7.98101

Error

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Amplifier model example in SimRF: basic RF power detector/AM radio