HO7_315aSP09_RC_nonidealities

HO7_315aSP09_RC_nonidealities - RC Integrator Nonidealities...

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Unformatted text preview: RC Integrator Nonidealities Boris Murmann Stanford University murmann@stanford.edu Copyright © 2009 by Boris Murmann EE315A ― HO #7 B. Murmann Corrections: Slide 9: 1/sqrt(wp1*wp2) sqrt Slide 11: αa αi, βa βi Slide 15: Replaced content entirely Slide 16: new Slide 39 (38): ω0 ωx 5/17/09: Slides 39-41: fixed algebraic errors in derivation (same end result) Slide 44: Modified plot to show only amp noise contribution 1 Outline • Impact of finite amplifier bandwidth and g p p gain – Integrator – Filter • Thermal noise – Passive filters • RC • RLC • Ladder – Active filters • Active RC • Biquad B. Murmann EE315A ― HO #7 2 Amplifier Model with First Order Nonidealities 0 u noise p a -a0 1 ωp = noiseless a a(s ) = − a0 1+ ≅− s a0ωp s ωp ≅− ωu for s 1 RaCa ωu ≅ a0 ⋅ωp ω >> ωp EE315A ― HO #7 B. Murmann 3 RC Integrator with Nonideal Amplifier • Using return ratio analysis, we can write A∞ = − ω 1 =− 0 sRC s A(s ) = Vout (s ) T (s ) = A∞ Vin (s ) 1 + T (s ) s T (s ) = −a(s ) R R+ • 1 sC = ω0 a0 1+ s ωp 1+ s ω0 ωp ≅ ωu a0 As long as T(s) is large, the transfer function A(s) is close to the desired ideal transfer function (A∞) B. Murmann EE315A ― HO #7 4 a0→∞, ωu=100ω0 M Magnitude [d dB] 100 A 50 ∞ T(s) A(s) 0 -50 -100 -6 10 10 -4 10 -2 10 0 10 2 10 4 Phas [deg] se ω /ω 0 A 150 “phase lag” 100 ∞ A(s) 50 0 -6 10 10 -4 10 -2 10 0 10 2 10 4 ω /ω 0 EE315A ― HO #7 B. Murmann 5 a0=10,000, ωu→∞ Magnitude [d dB] 100 A 50 ∞ T(s) A(s) 0 -50 -100 -6 10 10 -4 10 -2 10 0 10 2 10 4 Phase [deg] ω /ω 0 A 150 100 ∞ A(s) “phase lead” 50 0 -6 10 10 -4 10 -2 10 0 10 2 10 4 ω /ω 0 B. Murmann EE315A ― HO #7 6 M Magnitude [d dB] a0=10,000, ωu=100ω0 100 A 50 ∞ T(s) A(s) 0 -50 -100 -6 10 10 -4 10 -2 10 0 10 2 10 4 Phas [deg] se ω /ω 0 A 150 ∞ A(s) 100 50 0 -6 10 10 -4 10 -2 10 0 10 2 10 4 ω /ω 0 EE315A ― HO #7 B. Murmann 7 RC Integrator with Finite ωu • Ignoring finite DC gain for the time being, i.e. using a(s ) = − • ωu s The equations from slide 4 y q yield for this case Aactual (s ) = − ωo s A ideal 1 1+ ω0 ωu Magnitude error • 1 1+ s ωo + ωu Undesired pole (Magnitude and phase error) The first error term modifies only the magnitude, and effectively alters the integration time constant (RC = 1/ω0) B. Murmann EE315A ― HO #7 8 Significance of ω0 (Biquad Example) Ci1 = H(s) = − • 1 2 1+ sRC + s LC L 2 Rx Ci 2 = C 1 =− 1+ 1 1 = = ω01 ⋅ ω02 LC RxCi 2RxCi 2 ωP = s s2 + 2 ωPQP ωP 2 Rx Ci 2 Rx ω* 1 L 01 = = 2 R C R ω02 R Ci 1 QP = Integrator ω0 typically not too far off from pole frequency ωP EE315A ― HO #7 B. Murmann 9 Baseline Requirement for ωu Aactual (s ) = − ωo s A ideal 1 1+ 1 ω0 ωu Magnitude error 1+ s ωo + ωu Undesired pole (Magnitude and phase error) • High-Q filters will be sensitive to variations and uncertainty in the “effective” value of ω0 • In a practical design, we therefore require ωu >> ω0, typically ωu = 10…50⋅ω0 • Assuming that we comply with this guideline, we are left with Aactual (s ) ≅ − ωo 1 s 1+ s ωu B. Murmann EE315A ― HO #7 10 Effect on Filter Response • For a filter that uses ideal integrators, we know that Hfilter_ideal (s ) • s = pideal →∞ For the case with finite ωu ith ⎛ ⎛ s H filt t l ⎜ s ⎜ 1 + filter_actual ⎜ ⎝ ⎝ ωu ⎞⎞ →∞ ⎟⎟ ⎟ ⎠ ⎠ s = pactual and therefore pideal = α i ± j β i ⎛ ⎞ p pideal = pactual ⎜ 1 + actual ⎟ ωu ⎠ ⎝ pactual = α a ± j βa EE315A ― HO #7 B. Murmann 11 Solving for pactual (1) ⎛ α i + j β i = (α a + j β a ) ⎜ 1 + ⎝ • Equating real and imaginary parts, we find ⎛ 2 ⎛ α a ⎞ βa αi = αa ⎜1 + ⎟ − ⎝ ωu ⎠ ωu • α a + j βa ⎞ ⎟ ωu ⎠ βi = βa ⎜ 1 + ⎝ αa ⎞ ⎟ ωu ⎠ To proceed, it makes sense to customize the analysis for high Q proceed high-Q poles, which should represent the most critical case QPideal = − QPactual = − B. Murmann 1 ωPideal >> 1 2 αi 1 ωPactual >> 1 2 αa ωPideal = α i2 + β i2 ≅ β i 2 ωPactual = α a + βa2 ≅ βa EE315A ― HO #7 12 Solving for pactual (2) ⎛ αi = αa ⎜1 + ⎝ • αa ωu ⎛ 2 ⎞ βa − ⎟ ⎠ ωu βi = βa ⎜ 1 + ⎝ αa ⎞ ⎟ ωu ⎠ We can now simplify using α a << ωPactual ≅ ω0 << ωu to obtain αi ≅ αa − β i ≅ βa αa ≅ αi + • βa2 ωu βa2 ωu βa ≅ βi Negligible change in the pole’s imaginary part; real part affected by finite ωu y EE315A ― HO #7 B. Murmann 13 Effect on Pole Locations jω αa ≅ αi + QPactual = − αi αa σ 1 ωPactual 1 ωPideal 1 ωPideal 1 ≅− =− 2 2 β ω ω2 2 2 2 αi αi + a α i + Pideal 1 + Pideal ωu ωu 1 = QPideal 1 − 2QPideal • βa2 ωu ωPideal ωu α i ωu ⎛ ⎞ ω ≅ QPideal ⎜ 1 + 2QPideal Pideal ⎟ ωu ⎠ ⎝ Example for QPideal = 30, <2% (0.17dB) increase in QPactual p , ( ) 2 ⋅ 30 ⋅ B. Murmann ωPideal < 2% ωu ωu > 3000 (!) ωPideal EE315A ― HO #7 14 Corresponding Phase Error • ωu = 3000ωPideal For ωPideal ≅ ω0 and we can estimate the phase error of the integrator at ω0 using ti t th h f th i t t t i ⎞ ⎟ ⎟≅ ⎟ ⎟ ⎠ ωPideal ⎞ ωPideal ⎞ ⎛ 1 1 180 −1 ⎛ =− ⋅ = −0 02 0.02 ⎜1− j ⎟ = tan ⎜ − ⎟≅− ωu ⎠ 3000 3000 π ⎝ ⎝ ωu ⎠ 100 Magnitude [dB] [ φerr = ⎛ ⎜ 1 ⎜ ⎜ 1 + j ω0 ⎜ ωu ⎝ A 50 ∞ T(s) A(s) 0 -50 -100 -6 10 10 -4 10 -2 10 0 10 2 10 4 Phase [deg] ω /ω 0 A 150 ∞ A(s) 100 Phase = 89.98º 50 0 -6 10 10 -4 10 -2 10 0 10 2 10 4 ω /ω 0 EE315A ― HO #7 B. Murmann 15 Cascaded Biquad Example 10 H1 Magnitude [dB] 5 0 -5 -10 10 H2 5 0 -5 -10 -15 -15 -20 4 10 -20 4 10 10 f [Hz] 6 • H3 H1H2H3 6 B. Murmann • The same is true for H2, but this biquad is much less sensitive to finite ωu – Since error in QP is p p proportional to QP itself 0 -10 -15 10 f [Hz] 6 -20 4 10 10 f [Hz] 6 0 -5 -10 4 10 A 0.17 dB (2%) change ( ) g in the QP of biquad 3 (H3), will cause ~0.17 dB increase in the passband ripple -5 5 Magnitu [dB] ude Magnitude [dB] 5 Magnitude [dB] 10 5 10 f [Hz] 10 6 EE315A ― HO #7 16 Pole-Zero Cancellation (1) H (s ) = − = 1 + sRzC ⎛ s ⎞ s sRC ⎜ 1 + (1 + sRzC ) ⎟+ ⎝ ωu ⎠ ωu − 1 sRC Ideal response 1 + sRzC 1 1+ + (1 + sRzC ) ωu ωu RC s Should be 1 =− ωo =− ωo B. Murmann s 1 + sRzC ⎛ Rz ⎞ ⎜1+ R ⎟ ⎝ ⎠ ω s 1+ o + ωu ωu 1 ω s 1+ o ωu 1 + sRzC ⎛ Rz ⎞ ⎜1+ R ⎟ ⎠ 1+ s ⎝ ⎛ ωo ⎞ ωu ⎜ 1 + ⎟ ⎝ ωu ⎠ EE315A ― HO #7 17 Pole-Zero Cancellation (2) Rz ⎞ ⎛ ⎜1 + R ⎟ ⎝ ⎠ =RC z ⎛ ωo ⎞ ωu ⎜ 1 + ⎟ ⎝ ωu ⎠ • We can achieve “ideal” operation by letting ideal • Assuming ωu >> ω0, this is accomplished for • In hi h I high-speed filt d filters, this t i k t i ll h l reduce th amplifier thi trick typically helps d the lifi bandwidth requirements by more than an order of magnitude Rz ≅ 1 ωuC − Note that the requirements do not drop to “zero” because we still need t maintain ωu >> ω0 till d to i t i − Practicality issue: How to ensure that RZ tracks variations in ωu and C, for both global variations and random mismatch errors B. Murmann EE315A ― HO #7 18 Auxiliary Amplifiers A∞ = − T (s ) ≅ • Rf Rs A(s ) = − ωu Rs s Rs + Rf R2 R1 1 s ⎛ Rf ⎞ 1+ ⎜1+ ⎟ ωu ⎝ Rs ⎠ No (good) way to cancel error from inverting or summing amplifiers – But these amplifiers also contribute to the overall p p phase shift B. Murmann EE315A ― HO #7 19 “Tweaking” a Tow-Thomas Biquad L. C. Thomas, “The Biquad: Part I -Some Practical Design Considerations,” IEEE Trans. Ckt. Theory, Vol. CT-l, No. 3, May 1971. • May be able to cancel the phase error from all stages by introducing a “strategically” tuned zero – Practicality questionable B. Murmann EE315A ― HO #7 20 RC Integrator with Finite Gain a(s ) = −a0 T (s ) = a0 Aactual (s ) = − R R+ 1 sC = a0 sRC RC sRC + 1 1 1 1 1 1 =− = −ω0 sRC 1 + 1 sRC 1 + 1 + sRC ⎛ 1⎞ ω s ⎜1+ ⎟ + 0 T (s ) a0sRC ⎝ a0 ⎠ a0 EE315A ― HO #7 B. Murmann 21 Effect on Filter Response • For the case of finite gain, we can therefore write ⎛ 1⎞ ω pideal = pactual ⎜ 1 + ⎟ + 0 a0 ⎠ a0 ⎝ • For the case of high-Q poles, it can then be shown that high Q QPactual ≅ QPideal • ⎛ ⎞ Q 1 ≅ QPideal ⎜ 1 − 2 Pideal ⎟ Q a0 ⎠ ⎝ 1 + 2 Pideal a0 Example for Qpideal = 30, <2% (0.17dB) decrease in QPactual 2⋅ B. Murmann 30 < 2% a0 a0 > 3000 EE315A ― HO #7 22 Summary • Finite amplifier bandwidth leads to QP enhancement – Typically seen as excess peaking in the filter’s magnitude response • Finite amplifier gain leads to QP degradation – Typically seen as droop in the filter’s magnitude response • Wait! – Can’t we cancel the QP enhancement against the QP degradation? B. Murmann EE315A ― HO #7 23 Q-Tuning V. Gopinathan et al., “Design Considerations for High-Frequency Continuous-Time Filters and Implementation of an Anti-aliasing Filter for Digital Video,” IEEE JSSC, Vol. 25, No. 6, Dec. 1990. B. Murmann EE315A ― HO #7 24 Noise Filter Signal Si l Noise SNR = Psignal g Pnoise 1 2 ˆ v out =f 2 2 2 v out ∫ Δf ⋅ df f 1 EE315A ― HO #7 B. Murmann 25 RC Lowpass Filter 2 vout 2 f2 1 = ∫ 4kTR ⋅ df 1 + j 2πf ⋅ RC f 1 f2 = 4kTR ∫ f1 1 + B. Murmann d df ( 2πfRC ) 2 ; EE315A ― HO #7 du ∫ 1 + u 2 = ttan −1 u 26 Integration Bandwidth • Over which bandwidth should we integrate the noise? • Two interesting cases g – The output is observed by a system with finite bandwidth • E.g. human ear or another circuit with finite bandwidth • Use frequency range of that system as integration limits • Applies on a case by case basis – Total integrated noise • Integrate noise from zero to “infinite” frequency infinite • Applies to cases when there is no significant band limiting in the circuit that processes the output (e.g. a sampler) • Yields the noise you would measure at the output using an “RMS voltmeter” • We’ll use the total integrated noise in our analyses – Anal ticall “nice” and conser ati e Analytically conservative B. Murmann EE315A ― HO #7 27 Total Integrated Noise (RC Filter) ∞ 2 v out ,tot = ∫ 4kTR ⋅ 0 2 1 df 1 + j 2πf ⋅ RC ∞ = 4kTR ∫ df 0 1+ = 4kTR ⋅ = B. Murmann ( 2πfRC ) 2 ; du ∫ 1 + u 2 = tan 1 4RC kT C EE315A ― HO #7 28 −1 u Effect of Varying R Increasing R increases th noise i the i power spectral density, but also decreases the bandwidth – R drops out in the end result For C=1pF (example to the right), the total integrated noise is approximately 64μVrms B. Murmann EE315A ― HO #7 29 MDS and DR • Minimum detectable signal (MDS) – Quantifies the signal level that yields SNR=1 • I.e. noise power = signal power • Dynamic range (DR) is defined as DR = Psignal ,max MDS If the noise level in the circuit is independent of the signal level (which is often, but not always the case), it follows that the DR is equal to the "peak SNR," i.e. the SNR with the maximum signal applied B. Murmann EE315A ― HO #7 30 Significance of Thermal Noise • SNR of an RC filter – Carrying a sinusoidal output signal of 1-Vpeak – Considering the total integrated noise (kT/C) SNR [dB] 20 40 60 80 100 120 140 B. Murmann C [pF] 0.00000083 0.000083 0.0083 0.83 83 8300 830000 Hard to H d t make such small capacitors… k h ll it Designer will be concerned about thermal noise A difficult battle with thermal noise … EE315A ― HO #7 31 Interpretation • Even though these numbers were found using a simple example, they provide a good initial feel for the difficulty of achieving l hi i low noise i – As we will see later, the noise of more complicated circuits is often a multiple of kT/C, unfortunately making the situation for hi h f high SNR l k even worse look • Rules of thumb – Up to SNR ~ 30-40dB, integrated circuits are often not p , g limited by thermal noise – Achieving SNR >100dB is extremely difficult • Must usually rely on external components, or reduce bandwidth and remove noise by a succeeding filter • See e.g. oversampling ADCs in EE315B B. Murmann EE315A ― HO #7 32 LC Lowpass Filter (1) ωP = 2 ∞ 2 vout ,tot = ∫ 4kTR ⋅ 0 B. Murmann 1 s s2 1+ + 2 ωPQP ωP 1 LC QP = 1 L R C df EE315A ― HO #7 33 Useful Integrals A. Dastgheib and B. Murmann, "Calculation of total integrated noise in analog circuits," IEEE Trans. on Circuits and Systems I, Vol. 55, pp. 2988-2993, Nov. 2008. B. Murmann EE315A ― HO #7 34 LC Lowpass Filter (2) 2 ∞ 2 vout ,tot = ∫ 4kTR ⋅ , 0 = 4kTR 1+ 2 s s + 2 ωPQP ωP df ωPQP 4 ωP = 1 LC QP = 1 1 L R C ωPQP = kT = C 1 RC EE315A ― HO #7 B. Murmann 35 Interesting Theorem • Consider the parallel connection of a resistor and an arbitrary (passive) reactive network with port impedance Z(jω) 1 = lim j ωZ ( j ω ) C ω →∞ • • 2 ⇒ vtot = kT C For a proof see – Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed., pp. 352, McGraw Hill. Example 2 ⇒ vtot = B. Murmann EE 315 Lecture 5 kT C 36 LC Ladder Rs C~43pF kT = 9.7 μVrms C Rt 10k Ctot = 3.2 C Ctot~139pF **** the results of the sqrt of integral (v**2 / freq) **** total output noise voltage = 9.6683u volts **** the results of the sqrt of integral (v**2 / freq) **** total output noise voltage = 9.6683u volts 10 Rs=∞ -16 10 -15 Rs=10k -16 2 Noise PSD [V ] P 10 -15 2 Noise PSD [V ] P 10 10 10 10 -17 -18 -19 10 4 5 10 10 Frequency [Hz] 6 10 7 10 10 10 -17 -18 -19 10 4 5 10 10 Frequency [Hz] EE315A ― HO #7 B. Murmann 6 10 7 37 Active RC Lowpass R C Rs Vin Vout 2 ∞ ⎛1 1 ⎞ R 2 vout ,res = ∫ 4kT ⎜ + df ⎟⋅ ⎝ R Rs ⎠ 1 + j 2πf ⋅ RC 0 2 ⎛ ⎡ R ⎤⎞ 1 = ∫ 4kT ⎜ R ⎢1 + df ⎥⎟ ⋅ ⎜ ⎟ ⎝ ⎣ Rs ⎦ ⎠ 1 + j 2πf ⋅ RC 0 ∞ = B. Murmann kT C ⎛ R ⎞ ⎜1+ ⎟ Rs ⎠ ⎝ (noise due to resistors only) EE315A ― HO #7 38 Amplifier Noise Analysis (1) 2 v n = 4kTRnoise Δf vx vout = − t ω0 = s 1+ ω0 Rs vx 1 = = 1 R s vout R + 1+ 1+ s 1 Rs ωx + sC R ωu (v + v n ) s x 1 RC ωx = 1 RxC Rx = R Rs EE315A ― HO #7 B. Murmann 39 Amplifier Noise Analysis (2) • Solving for vout/vn yields vout R =− vn Rx 1+ s ωx ⎛ 1 R 1 ⎞ s2 1+ s ⎜ + + ⎟ ⎝ ωu Rx ω0 ⎠ ωu ω0 2 2 vout ,amp 2 ∞ s 1+ ωx ⎛ R ⎞ = ∫ 4kTRnoise ⋅ ⎜ ⎟ 2 ⎝ Rx ⎠ 1 + s ⎛ 1 R + 1 ⎞ + s 0 ⎜ ⎟ ⎝ ωu Rx ω0 ⎠ ωu ω0 df 2 s ∞ 1+ z ∫ s s2 0 1+ + 2 ωnQ ωn B. Murmann ω Q⎛ ω2 ⎞ df = n ⎜ 1 + n ⎟ 2 4 ⎜ z ⎟ ⎝ ⎠ EE315A ― HO #7 ⎛ 1 R 1 ⎞ ωnQ = ⎜ + ⎟ ⎝ ωu Rx ω0 ⎠ 2 ωn = ωu ω0 −1 z = −ωx ω 40 Amplifier Noise Analysis (3) 2 v out ,amp ⎛ ⎞ 2 2 2 ⎞ ⎟⎛ ω ω ⎞ ⎛ R ⎞ ωnQ ⎛ ⎛ R ⎞ ⎜ ω 1 ⎜ 1 + n ⎟ = kTRnoise ⎜ ⎜ ⎟ ⎜1 + u 0 ⎟ = 4kTRnoise ⎜ ⎟ ⎟ 2⎟ 1 R 1 ⎟⎜ ⎜ ω2 ⎟ z ⎠ ⎝ Rx ⎠ 4 ⎝ ⎝ Rx ⎠ ⎜ x ⎠ ⎝ ⎜ω R +ω ⎟ 0 ⎠ ⎝ u x 2 ⎛ ω ⎡R ⎤ ⎞ ⎜ 1+ u ⎢ x ⎥ ⎟ ω0 ⎣ R ⎦ ⎟ = kTRnoise ωu ⎜ ≅ kTRnoise ωu ⎜ R ⎛ ω R ⎞⎟ x 1+ u x ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎝ R ⎝ ω0 R ⎠ ⎠ ≅ B. Murmann Note: The same result can be obtained by approximating vout/ n as i ti /v a single pole response before carrying out the integral. kT Rnoise ωu C R ω0 EE315A ― HO #7 41 Total Noise for Active RC Filter 2 2 2 vout ,tot = vout ,res + vout ,amp ≅ kT ⎛ R Rnoise ωu ⎞ + ⎜1+ ⎟ C ⎝ Rs R ω0 ⎠ • Amplifier noise contribution is large for large ωu/ω0 – But, unfortunately, we need ωu >> ω0 to maintain an accurate transfer function • Given that we need ωu >> ω0 , the only option we have is to choose Rnoise << R to minimize amplifier noise – I a transistor-level i l In t i t l l implementation, thi requires l t ti this i large gm (and large IBIAS), since Rnoise ~ 1/gm B. Murmann EE315A ― HO #7 42 Frequency Response with Finite ωu M Magnitude [dB] [ 0 ω u=1000 ω 0 -50 Phase [deg] -100 100 ω u=10ω 0 10 -5 10 ω /ω 0 150 10 5 0 10 5 ω u=1000 ω 0 100 ω u=10ω 0 50 0 0 10 -5 10 ω /ω 0 EE315A ― HO #7 B. Murmann 43 10 Integra al/(kT/C) 2 PSD/ /(4kTRnoise*( (R/Rx) ) Amplifier Noise Contribution (Rnoise=0.1R) 0 ω u=100ω 0 ω u=10ω 0 10 10 10 -2 10 0 10 0 ω /ω 0 10 2 10 4 10 6 10 2 10 4 10 6 ω u=100ω 0 8 ω u=10ω 0 6 4 2 10 B. Murmann -4 -4 10 -2 ω /ω 0 EE315A ― HO #7 44 Active Second Order Lowpass in 2 = 4kT 1 Δf Rx 1 H( s ) = 1+ s s2 + 2 ωPQP ωP ωP = QP = ⎛ 2 R ⎞ in1 = 4kT ⎜ + 2 ⎟ Δf ⎜ ⎟ ⎝ Rx Rx ⎠ vout = in1 Rx 1+ vout = RQP in 2 2 s s + 2 ωPQP ωP s ωP 1+ 2 s s + 2 ωPQP ωP 1 LC 1 L R C ωPQP = 1 RC (after some algebra) EE315A ― HO #7 B. Murmann 45 Analysis 2 ∞ ⎛ 2 R ⎞ 2 vout ,1 = ∫ 4kT ⎜ + 2 ⎟⋅ ⎜R ⎟ ⎝ x Rx ⎠ 1 + 0 = 4kT ( 2Rx + R ) = kT C ∞ 2 s s + 2 ωPQP ωP df 2 vout ,2 = ∫ 4kT ωPQP 4 • 0 = 4kT kT 2Rx + R C R = Rx 2 = Rx ⎞ ⎛ ⎜1+ 2 R ⎟ ⎝ ⎠ 2 QP R 2 ⋅ Rx s ωP s s2 1+ + 2 ωPQP ωP 2 QP ωPQP Rx 4 kT R 2 QP C Rx For high QP, we definitely need to make R << Rx B. Murmann EE315A ― HO #7 46 df Optimum 2 N = (1 + 2k ) + QP 1 k k= Rx R dN 2 1 = 2 − QP 2 = 0 dk k kopt = 2 2 2 vout = vout ,1 + vout ,1 = kT C QP 2 ⎛ ⎡ 2 ⎞ kT ⎤ + 2 ⎥ QP ⎟ = 1 + 2 2QP ⎜1+ ⎢ ⎦ ⎝ ⎣ 2 ⎠ C ( ) • In a properly designed filter (and for large QP,) the noise will be roughly proportional to QP • For a poorly designed filter, the noise can be p p p y g , proportional to QP2 B. Murmann EE315A ― HO #7 47 Tow-Thomas Noise Example B. Murmann EE315A ― HO #7 48 Frequency Response (BP Output) M Mag [dB] QP=7 QP=30 Ph hase [deg] R1 = R4 = 42kΩ 10kΩ QP = 30 7 900 1100 Frequency [Hz] F [H ] EE315A ― HO #7 B. Murmann 49 Noise versus QP (Noiseless Amplifier) [V V/rt-Hz], [Vrm ] ms Noise drops by √ 30/7 2.8μVrms 1.2μVrms 10 B. Murmann 1k Frequency [ ] q y [Hz] EE315A ― HO #7 100M 10G 50 Noisy Amplifiers Unfortunately the amplifiers add significant noise at hi h f i ifi t i t high frequency 20.6μV [V/r rt-Hz], [Vrms] 2.8μV Noise from the passband dominates this integral 10 B. Murmann 1k Frequency [Hz] 100M EE315A ― HO #7 10G 51 Adding an RC Filter 1kΩ / 5nF RC LPF Corner at 32kHz 0.9μVrms noise from 5nF is negligible B. Murmann EE315A ― HO #7 52 Frequency Response with RC Filter M Mag [dB] Without RC Ph hase [deg] RC provides negligible attenuation. p g g But that’s not the point. Let’s look at the noise … 10 1k Frequency [Hz] 100M 10G EE315A ― HO #7 B. Murmann 53 [V/r rt-Hz], [Vrms] Noise after RC Filter RC filter reduces total noise from 20μVrms to 5μVrms (With noiseless amplifier ~3μVrms) 10 B. Murmann 1k Frequency [Hz] EE315A ― HO #7 100M 10G 54 Summary • Thermal noise is a fundamental issue in electronic circuits • The total integrated thermal noise is related to capacitor size – Multiple of kT/C • In filters, noise is proportional to the filter order, QP, and strongly dependent on the implementation • Amplifiers can contribute significantly to (if not dominate the) overall filter noise – Minimizing the amplifier noise contribution costs power • Need small Rnoise, i e large gm (IBIAS) i.e. • More later… B. Murmann EE315A ― HO #7 55 ...
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