Noise Analysis for Simple Op-Amp Circuit

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2 Noise Analysis for Simple Op-Amp Circuit R3 1k Noise Sources R2 1k V U1 OPA277 VF2 Op-Amp Voltage Noise Source Op-Amp Current Noise Sources Resistor Noise Sources VG1 + V1 2.5 Calculation Considerations Convert Noise Spectrum to Noise Voltage - External Filter Bandwidth Limit - Op-Amp Closed Loop Bandwidth Noise Gain 17

3 Calculus Reminder dx Height x Width 4 Height Area Width 9 Integral = Area under the curve 18

4 Convert Noise Spectrum to Noise Voltage (Broadband Only Simple Case) Voltage Spectral Density (V/rt-Hz) Power Spectral Density (V 2 /Hz) 5V / rthz 0 Freq (Hz) 10 25V 2 / Hz You can t integrate the Voltage spectral density curve to get noise 10 V_spec_dens 0 df 5 V 10 Hz Hz You integrate the Power spectral density curve to get noise 50 V Hz Hz Wrong 0 Freq (Hz) 10 Correct 19

5 Convert Noise Spectrum to Noise Voltage (Broadband Only Simple Case) You integrate the Power spectral density curve to get noise Correct Noise Power = V 2 Hz * BW (Hz) Noise Voltage = V * BW (Hz) Hz 20

6 Noise Gain for Voltage Noise Source Noise Gain Gain seen by the noise source. Example: Noise_Gain = (R2/R1) + 1 = 2 Signal_Gain = -R2/R1 = -1 Output_Noise = Vn*(Noise_Gain) R2 1k Referred to Output Signal Source Referred to Input + R1 1k Vs + * Vn Noise Source V1 5 V2 5 U1 OPA277 VF1 21

7 Understanding The Spectrum: Total Noise Equation (Current or Voltage) 1/f Noise Region (Pink Noise Region) White Noise Region (Broadband Noise Region) e nt = [(e n1/f ) 2 + (e nbb ) 2 ] where: e nt = Total rms Voltage Noise in volts rms e n1/f = 1/f voltage noise in volts rms e nbb = Broadband voltage noise in volts rms 100k 10k 1k k 10k Frequency (Hz) f L en BB calculation f H en 1/f calculation 22

8 Low Pass Filter Shapes the Spectrum How do we convert this plot to noise? RMS 1/f Region Broadband Region Low pass filter 23

9 Real Filter Correction vs Brickwall Filter where: f P = roll-off frequency of pole or poles f BF = equivalent brickwall filter frequency 24

10 AC Noise Bandwidth Ratios for n th Order Low-Pass Filters BW n = (f H )(K n ) Effective Noise Bandwidth Real Filter Correction vs Brickwall Filter Number of Poles in Filter Kn AC Noise Bandwidth Ratio

11 Broadband Noise Equation e BB BW n = (f H )(K n ) where: BW n = noise bandwidth for a given system f H = upper frequency of frequency range of operation K n = Brickwall filter multiplier to include the skirt effects of a low pass filter en BB = (e BB )( [BW n ]) where: en BB = Broadband voltage noise in volts rms e BB = Broadband voltage noise density ; usually in nv/ Hz BW n = Noise bandwidth for a given system 26

12 1/f Noise Equation (see appendix for derivation) e 1/f@1Hz e 1/f@1Hz = (e 1/f@f )( [f]) where: e 1/f@1Hz = normalized noise at 1Hz (usually in nv) e 1/f@f = voltage noise density at f ; (usually in nv/ Hz) f = a frequency in the 1/f region where noise voltage density is known en 1/f = (e 1/f@1Hz )( [ln(f H /f L )]) where: en 1/f = 1/f voltage noise in volts rms over frequency range of operation e 1/f@1Hz = voltage noise density at 1Hz; (usually in nv) f H = upper frequency of frequency range of operation (Use BW n as an approximation for f H ) f L = lower frequency of frequency range of operation 27

13 Example Noise Calculation R2 1k R1 100k Given: OPA627 Noise Gain of V U1 OPA627/BB VF1 Find (RTI, RTO): Voltage Noise Current Noise Resistor Noise + VG1 V

14 Voltage Noise Spectrum and Noise Bandwidth 50nV/rt-Hz 5nV/rt-Hz Unity Gain Bandwidth = 16MHz Closed Loop Bandwidth = 16MHz / 101 = 158kHz 29

15 Example Voltage Noise Calculation Voltage Noise Calculation: Broadband Voltage Noise Component: BW n (f H )(K n ) (note Kn = 1.57 for single pole) BW n (158kHz)(1.57) =248kHz en BB = (e BB )( BW n ) en BB = (5nV/ Hz)( 248kHz) = 2490nV rms 1/f Voltage Noise Component: e 1/f@1Hz = (e 1/f@f )( f) e 1/f@1Hz = (50nV/ Hz)( 1Hz) = 50nV en 1/f = (e 1/f@1Hz )( [ln(f H /f L )]) Use f H = BW n en 1/f = (50nV)( [ln(248khz/1hz)]) = 176nV rms Total Voltage Noise (referred to the input of the amplifier): en T = [(en 1/f ) 2 + (en BB ) 2 ] en T = [(176nV rms) 2 + (2490nV rms) 2 ] = 2496nV rms 30

16 Example Current Noise Calculation Note: This example amp doesn t have 1/f component for current noise. e n-in = (i n )x(r eq ) e n-out = Gain x (i n )x(r eq ) R1 1k Rf 3k U2 fa * - + IOP1 VF1 Gain Req = R1 Rf * 31

17 Example Current Noise Calculation Broadband Current Noise Component: BWn (f H )(K n ) BWn (158kHz)(1.57) =248kHz i nbb = (i BB )( BW n ) i nbb = (2.5fA/ Hz)( 248kHz) = 1.244pA rms R eq = R f R 1 = 100k 1k = 0.99k e ni = (I n )( R eq ) = (1.244pA)(0.99k) = 1.23nV rms neglect Since the Total Voltage noise is e nvt = 2496nV rms the current noise can be neglected. 32

18 Example Resistor Noise Calculation e nr = (4kT K RΔf) where: R = Req = R1 Rf Δf f = BW n e nr = (4 (1.38E-23)( ( ) (0.99k)(248kHz)) = 2010nV rms e n-in = (4kTRΔf) e n-out = Gain x ( (4kTRΔf) f)) R2 R1 1k U1 * R1 Rf 2k U1 * Gain nv - IOP1 nv VF1 Req = R1 Rf + * 33

19 Total Noise Calculation Voltage Noise From Op-Amp RTI: e nv = 2510nV rms Current Noise From Op-Amp RTI (as a voltage): e ni = 1.24nV rms Resistor Noise RTI: e nr = 2020nV rms Total Noise RTI: e n in = ((2510nV) 2 + ((1.2nV) 2 + ((2010nV) 2 ) = 3216nV rms Total Noise RTO: e n out = e n in x gain = (3216nV)(101) = 325uV rms 34

20 Calculating Noise Vpp from Noise Vrms Relation of Peak-to-Peak Value of AC Noise Voltage to rms Value Peak-to-Peak Amplitude 2 X rms 3 X rms 4 X rms 5 X rms 6 X rms * 6.6 X rms Probability of Having a Larger Amplitude 32% 13% 4.6% 1.2% 0.3% 0.1% *Common Practice is to use: Peak-to-Peak Amplitude = 6 X rms 35

21 Peak to Peak Output For our Example R2 1k R1 100k V VF1 + + U1 OPA627/BB + VG1 V2 15 e n out = 325uV rms e n out p-p = (325uV rms )x6 = 1.95mVp-p 36