III. Gm-C Filtering - Epublications - Université de Limoges

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Figure 37. First order RC low-pass filter As only one reactive element is present, this is a first order filter. The transfer function of this RC circuit in Laplace domain is given by: 1 H LPF ( s) = s 1+ 2πf c where fc is the cut-off frequency , (II.2) 1 fc = 2π RC (II.3) For an RL low-pass filter, the transfer function has the same form. The difference relies in the cut-off frequency definition which depends on L and R in this case. To obtain a frequency tunable filter, the cut-off frequency has to be made adaptable by modifying the value of the elements of the circuit. 1 It is worth noticing that, for a first order low-pass filter, the transfer function is at 2 fc. Hence, if the filter is centered so that fc = fwanted, then adjacent channels located at lower frequencies are less rejected than the desired channel. This property is true for all low-pass filters. Figure 38 depicts a first order low-pass filter when tuning its cut-off frequency. Figure 39 illustrates its adjacent channels and its harmonics rejections. fc increase Figure 38. Cut-off frequency tunability of a first order low-pass filter - 36 -
Figure 39. H3 and N+5 rejections of a first order low-pass filter A higher order low-pass filter can be obtained increasing the number of poles in the transfer function by the addition of reactive elements in the circuit. Figure 40 presents second order low-pass filters built either cascading two RC first order filters or combining an inductor to a capacitor. Vin R1 C1 R2 C2 - 37 - Vout Vin Figure 40. Second order RC and LC low-pass filters The transfer function of a second order filter is then given by the formulas: α 1 H LPF ( s) = = , 2 2 (II.4) ( s − p1 )( s − p2 ) 1+ β1s + β 2 s α being a constant. Such filter allows getting a steeper slope in order to reject harmonics more strongly as depicted in Figure 41 and Figure 42. For second order (and more) filters, different topologies may be used, as illustrated in Figure 43, corresponding to different sets of coefficients βis in the transfer function [II.2]. These topologies, known as Chebychev, elliptic or Butterworth filters, enable to get a flatter passband or a steeper slope (see Figure 43). Their rejections are plotted in Figure 44. Though elliptic filters show higher rejections, it is worth adding that this kind of filter uses one reactive element more than the other two filters, which creates a transmission zero (notch). For instance, the normalized transfer function of a second order Butterworth low-pass filter is given by: 1 H Butterworth ( s) = . 2 (II.5) 2 1+ 2s + s L C Vout
Page 1: UNIVERSITE DE LIMOGES ECOLE DOCTORA
Page 5 and 6: Acknowledgments En préambule à ce
Page 7 and 8: Summary ACKNOWLEDGMENTS ...........
Page 9 and 10: V. A PERSPECTIVE FOR FUTURE DEVELOP
Page 11 and 12: French Introduction Les signaux TV
Page 13 and 14: I. Introduction to TV tuners and to
Page 15 and 16: I.1.b.ii Digital modulations In dig
Page 17 and 18: makes the covering of a large terri
Page 19 and 20: I.1.c.iii Terrestrial spectrum Firs
Page 21 and 22: I.1.e Broadband Reception Systems F
Page 23 and 24: I.2.b TV Tuner High Performance Spe
Page 25 and 26: I.2.b.ii High linearity Although co
Page 27 and 28: I.3 RF Filter Specifications I.3.a
Page 29 and 30: 27. This leads to the downconversio
Page 31 and 32: These adjacent channel rejection re
Page 33 and 34: Above 870MHz, there are no more ter
Page 35 and 36: Newest silicon tuners generations,
Page 37 and 38: I.6 References [I.1] B. Razavi, RF
Page 39: II. Challenges of RF Selectivity Fo
Page 43 and 44: Figure 43. Second order Low-pass Fi
Page 45 and 46: Figure 46 illustrates the transfer
Page 47 and 48: H HPF s ( s) = s 1 + ω This corres
Page 49 and 50: Topologies comparison for a same Q-
Page 51 and 52: Figure 56. Low-pass to Notch Transf
Page 53 and 54: II.2 Passive LC Second Order Bandpa
Page 55 and 56: A possible solution is to split the
Page 57 and 58: II.2.c Second Order Bandpass Filter
Page 59 and 60: Figure 69. Band Switching Figure 70
Page 61 and 62: Another paper [II.7] provides a FOM
Page 63 and 64: A parallel self resonating circuit
Page 65 and 66: This gives the following values for
Page 67 and 68: Vb - 63 - M1 M2 Zin Figure 76. Anot
Page 69 and 70: when given, reported noise figures
Page 71 and 72: In all publications dedicated to Gm
Page 73 and 74: II.4.c Second Order Bandpass Filter
Page 75 and 76: Figure 87. Biquad Schematic The Gm-
Page 77 and 78: Vin Rm - 73 - C Vout Figure 89. Der
Page 79 and 80: Figure 93 depicts the RF performanc
Page 81 and 82: A second advantage of advanced BiCM
Page 83 and 84: II.7 Conclusion As a conclusion, th
Page 85 and 86: [II.13] C. Andriesei, L. Goras, and
Page 87 and 88: III.I Theoretical Study III. Gm-C F
Page 89 and 90: g V m3 in III.1.b Linearity Conside
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Assuming an ideal input transconduc
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Figure 102. Linearity of the filter
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Furthermore, a Gm-C circuit is not
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In terms of noise, the Flicker nois
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Table 7 summarizes the specificatio
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Idiff (A) levels reached with this
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III.2.e Unbalanced Differential Pai
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Hence, the combination of the expon
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Technique Current Increase Source D
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Using this structure and considerin
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The dimensioning of the transistors
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III.3.c Gm-cells with MGTR III.3.c.
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Furthermore, the high sensitivity t
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Figure 136. Filter in-band IIP3 ver
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III.4 Comparison of the filters III
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III.5 Conclusion Once the Gm-C seco
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IV. Rauch Filtering IV.1 Sallen-Key
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Thus, an equation linking Q and Gai
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IV.1.c.ii Proposed positive feedbac
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For both Rauch and Sallen-Key filte
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The high sensitivity to passive com
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Furthermore, it should also be take
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IV.2.b Innovative Implementation of
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esulting gain K, constituted of the
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Table 25. Sum-up of the OA specific
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Figure 153. OA voltage gain versus
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Filtering in the mirror, by means o
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However, this feedback makes the fi
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Figure 160. OA Gain and phase versu
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IV.3.c Implemented Filter and Test
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Figure 165 depicts the layout of th
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Figure 169. PVT variations of the f
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IV.4 Rauch Filter Performances: Mea
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Figure 178. IIP3 versus input power
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IV.4.c Performances Sum-up The filt
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Table 30. Frequency Limitations of
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IV.7 References [IV.1] Y. Tsividis
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V. A Perspective for Future Develop
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V.2 4-path Filter Simulations V.2.a
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The frequency tunability is ensured
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V.3 State-of-the-Art V.3.a 4-path F
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V.4 Conclusion In this chapter, it
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IEEE International, 2009, pp. 222-2
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RF selectivity challenges In this t
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The positive feedback Rauch filter
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It is worth highlighting that FOM1
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étudiée. Il s’avère que la lin
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It is worth noticing that S0 only d
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A.3 Signal-to-Noise Ratio These dif
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A.5 Friis’ Formula In case of mul
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A.7 References [A.1] Friis, H.T., N
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To estimate the influence of the di
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Now, let’s have a look to the cub
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Figure 200. Third order intercept p
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B.2 RF Filter Linearity Measurement
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B.2.c IIP2 measurement To measure t
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- 199 -
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C.1 Gyrator-C filtering year Ref. f
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C.2 Gm-C Filtering year Ref. f0 (MH
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year C.3 Rm-C filtering Ref. f0 (MH
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C.5 References C.5.a Gyrator-C filt
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[C.23] J. De Lima and C. Dualibe,
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- 211 -
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this gives: I I I md d1 d 2 = I I =
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D.2 MOS Degenerated Common-Source C
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⎛ I s ⎞ Vout = Vin + U T ln ⎜
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Computing, this leads to: V out + v
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- 221 -
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- 223 -
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FIGURE 54. H3 AND N+5 REJECTIONS AC
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FIGURE 168. PVT VARIATIONS OF THE F
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