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

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and As discussed in the previous chapter, in Figure 98, it is assumed that 1 r s = . r g g 0 m1 m2 A first computation step gives: g V ⎛ ⎞ ⎜ 1 1 = V + ⎟ out + jC ω ⎜ b ⎟ ⎝ r0 jLeff ω + rs ⎠ - 84 - Ca L eff = g g m1 m2 m3 in (III.1) Hence, the filter transfer function can be deduced: V H ( jω) = V out in = r r s 0 g ω + 1+ j r 0 m3 ( jL Such a filter resonates at a central frequency with a quality-factor set to f 0 Q = r 0 1 = 2π g eff ω + r ) 2 ( Leff + rs r0C b ) −ω Leff Cb m1 a 1 gm 2 + 2 r0 C C a b b ⎛ 1 CaCb ⎜ + g 2 ⎝ r0 C + C m1 g m2 s ⎞ ⎟ ⎠ . (III.2) (III.3) (III.4) It is worth noticing that tunable capacitors Ca correspond to an equivalent tunable inductor Leff. While passive inductor – varying capacitor resonator leads to a central frequency proportional to1 / C , here it is possible to tune both Leff and C provided that Ca and Cb are proportional. f0 then becomes proportional to1 / C . This is a very interesting property for a tunable filter since it results in a larger tuning range for a same capacitance ratio: f max Cmax ∝ . (III.5) f C min min That is why in the following, it is assumed Ca = Cb = C . Moreover, when only tuning Cb, it leads to a filter gain increasing with frequency, because Q increases as well, whereas tuning both capacitors makes the gain remain constant. This structure also permits to achieve a constant-Q frequency sweep because Q then becomes constant, independent from C since: r 1 = g . (III.6) 0 Q + 2 2 r0 m1g m2 As previously mentioned, this allows rejecting harmonic frequencies in a constant way.
g V m3 in III.1.b Linearity Considerations The fully-differential equivalent topology of this circuit is considered. This configuration allows neglecting second order distortion as a first approximation. Furthermore, for each transconductor Gm,i, the output differential current Iout,i is given by: I = g V + g ′′ V . (III.7) out, i mi in mi 3 in The transfer function of the schematic can be computed, since the type of solution expected is: V = α V + βV , (III.8) out in These α and β coefficients can then be obtained. The differential current in the gyrator IinL gives the following relation: I + g′ ′ V ⎛ ⎜ ⎝ r 3 in ( ) ( ) 3 3 ⎞ ⎛ r0 3 ⎞ g V + g′ ′ V ⎟ + g ′′ ⎜ g V + g ′′ V 0 inL = gm 2⎜ m1 inL m1 inL 2 1 1 1 + ⎟ m ⎜ m inL m inL jr0C aω 1 + jr0C aω 3 m3 in Limiting to the third order terms, Equation (III.9) becomes: I inL gm1g m2r0 = V 1 + jr C ω 0 a inL ⎛ + ⎜ g ⎜ ⎝ m2 ⎠ ⎝ 3 g′ ′ m1r0 ⎛ gm1r0 ⎞ ⎞ + g ′′ ⎟ m2 V 1 jr0C ⎜ a 1 jr0C ⎟ + ω ⎝ + ⎟ aω ⎠ ⎠ - 85 - 3 inL ⎟ ⎠ (III.9) , (III.10) Keeping first and third order terms, the current of the gyrator can be written as: I Now, assuming this results in = I inL + V out g g r g′ ′ r m1 m2 0 m1 0 3 inL = VinL + gm 2 VinL , (III.11) 1 + jr0C aω 1 + jr0C aω V V = , (III.12) out inL 3 1+ jr0C bω g 1g 2r ⎛ 0 g 1r0 g 1r ⎞ m m 0 3 1 jr0C V ⎜ ′′ m ⎛ m ⎞ + bω = out + g m2 + g ′′ ⎟ m2 Vout + Vout r0 1 jr0C aω ⎜ 1 jr0C aω ⎜ 1 jr0C aω ⎟ + + ⎟ r0 ⎝ ⎝ + ⎠ ⎠ (III.13) Since the solution is of the type: V = α V + βV . (III.14) out in 3 in
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UNIVERSITE DE LIMOGES ECOLE DOCTORA
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Acknowledgments En préambule à ce
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Summary ACKNOWLEDGMENTS ...........
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V. A PERSPECTIVE FOR FUTURE DEVELOP
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French Introduction Les signaux TV
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I. Introduction to TV tuners and to
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I.1.b.ii Digital modulations In dig
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makes the covering of a large terri
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I.1.c.iii Terrestrial spectrum Firs
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I.1.e Broadband Reception Systems F
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I.2.b TV Tuner High Performance Spe
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I.2.b.ii High linearity Although co
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I.3 RF Filter Specifications I.3.a
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27. This leads to the downconversio
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These adjacent channel rejection re
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Above 870MHz, there are no more ter
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Newest silicon tuners generations,
Page 37 and 38: I.6 References [I.1] B. Razavi, RF
Page 39 and 40: II. Challenges of RF Selectivity Fo
Page 41 and 42: Figure 39. H3 and N+5 rejections of
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: III.I Theoretical Study III. Gm-C F
Page 91 and 92: Assuming an ideal input transconduc
Page 93 and 94: Figure 102. Linearity of the filter
Page 95 and 96: Furthermore, a Gm-C circuit is not
Page 97 and 98: In terms of noise, the Flicker nois
Page 99 and 100: Table 7 summarizes the specificatio
Page 101 and 102: Idiff (A) levels reached with this
Page 103 and 104: III.2.e Unbalanced Differential Pai
Page 105 and 106: Hence, the combination of the expon
Page 107 and 108: Technique Current Increase Source D
Page 109 and 110: Using this structure and considerin
Page 111 and 112: The dimensioning of the transistors
Page 113 and 114: III.3.c Gm-cells with MGTR III.3.c.
Page 115 and 116: Furthermore, the high sensitivity t
Page 117 and 118: Figure 136. Filter in-band IIP3 ver
Page 119 and 120: III.4 Comparison of the filters III
Page 121 and 122: III.5 Conclusion Once the Gm-C seco
Page 123 and 124: IV. Rauch Filtering IV.1 Sallen-Key
Page 125 and 126: Thus, an equation linking Q and Gai
Page 127 and 128: IV.1.c.ii Proposed positive feedbac
Page 129 and 130: For both Rauch and Sallen-Key filte
Page 131 and 132: The high sensitivity to passive com
Page 133 and 134: Furthermore, it should also be take
Page 135 and 136: IV.2.b Innovative Implementation of
Page 137 and 138: 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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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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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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FIGURE 54. H3 AND N+5 REJECTIONS AC
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FIGURE 168. PVT VARIATIONS OF THE F
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