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ML12179-5P Arkusz danych(PDF) 6 Page - LANSDALE Semiconductor Inc. |
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ML12179-5P Arkusz danych(HTML) 6 Page - LANSDALE Semiconductor Inc. |
6 / 11 page www.lansdale.com Page 6 of 11 Issue A LANSDALE Semiconductor, Inc. ML12179 To simplify analysis further a damping factor of 1 will be selected. The normalized closed loop response is illustrated in Figure 7 where the loop bandwidth is 2.5 times the loop natural frequency (the loop natural frequency is the frequency at which the loop would oscillate if it were unstable). Therefore the opti- mum loop bandwidth is15kHz/2.5 or 6kHz (37.7krads) with a damping coefficient, ζ ≈ 1. T(s) is the transfer function of the loop filter. In summary, follow the steps given below: Step 1: Plot the phase noise of crystal reference and the VCO on the same graph. Step 2: Increase the phase noise of the crystal reference by the noise contribution of the loop. Step 3: Convert the divide–by–N to dB (20log 256 – 48 dB) and increase the phase noise of the crystal reference by that amount. Step 4: The point at which the VCO phase noise crosses the amplified phase noise of the Crystal Reference is the point of the optimum loop bandwidth. This is approximately 15 kHz in Figure 6. Step 5: Correlate this loop bandwidth to the loop natural frequency and select components per Figure 8. In this case the 3.0 dB bandwidth for a damping coefficient of 1 is 2.5 times the loop's natural frequency. The relationship between the 3.0 dB loop bandwidth and the loop's “natural” frequency will vary for different values of ζ. Making use of the equations defined above in a math tool or spreadsheet is useful. To aid in the use of such a tool the equations are summarized in Figures 9 through 11. Figure 6. Graphical Analysis of Optimum Bandwidth –150 –140 –130 –120 –110 –100 –90 –80 –70 –60 Crystal Reference 15dB NF of the Noise Contribution from Loop 20*log(256) VCO Optimum Bandwidth 10 100 1k 10k 100k 1M Hz Figure 7. Closed Loop Frequency Response for ζ = 1 –60 –50 –40 –30 –20 –10 0 10 0.1 1k Hz 1 10 100 Natural Frequency 3dB Bandwidth Legacy Applications Information T(s) RoCos1 NCo KpKv s2 RoCos1 2 ωo ζ ζ s1 1 ωo2 s 2 2 ωo s1 NCo KpKv 1 o2 ω o ω o= KpKv NCo Co KpKv N ωo2 RoCo= = == + + ++ ++ → ω → → ζ = → = 2 ζ oRoCo 2 Ro 2 ζ oCo Figure 8. Design Equations for the 2nd Order System ω ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Let: NCo KpKv 1 o2 ω ω ω ω ωω ω ωω ω ω = = == = == == =+ + + ,R == oCo 2 ζ o Figure 9. Loop Parameter Relations Let: Ca aCo ,Cx bCo , A 1 a , and B 1 a b Let: RoCo 1 3 ,RxCx+ 1 4 ,Ro(Ca Cx) 1 5 Let: K33 o ,K44 o ,K55 o |
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