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MC13110A Datasheet(PDF) 40 Page - Motorola, Inc |
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MC13110A Datasheet(HTML) 40 Page - Motorola, Inc |
40 / 68 page ![]() MC13110A/B MC13111A/B 40 MOTOROLA ANALOG IC DEVICE DATA Loop Filter Characteristics Lets consider the following discussion on loop filters. The fundamental loop characteristics, such as capture range, loop bandwidth, lock–up time, and transient response are controlled externally by loop filtering. Figure 96 is the general model for a Phase Lock Loop (PLL). Phase Detector (Kpd) Filter (Kf) VCO (Ko) fo Divider (Kn) fi Figure 96. PLL Model Where: Kpd = Phase Detector Gain Constant Kf = Loop Filter Transfer Function Ko = VCO Gain Constant Kn = Divide Ratio (1/N) fi = Input frequency fo = Output frequency fo/N = Feedback frequency divided by N From control theory the loop transfer function can be represented as follows: A = Kpd Kf Ko Kn Open loop gain Kpd can be either expressed as being 2.5 V/4.0 π or 1.0 mA/2.0 π for the CT–0 circuits. More details about performance of different type PLL loops, refer to Motorola application note AN535. The loop filter can take the form of a simple low pass filter. A current output, type 2 filter will be used in this discussion since it has the advantage of improved step response, velocity, and acceleration. The type 2 low pass filter discussed here is represented as follows: From Phase Detector To VCO R2 C2 C1 Figure 97. Loop Filter with Additional Integrating Element From Figure 97, capacitor C1 forms an additional integrator, providing the type 2 response, and filters the discrete current steps from the phase detector output. The function of the additional components R2 and C2 is to create a pole and a zero (together with C1) around the 0 dB point of the open loop gain. This will create sufficient phase margin for stable loop operation. In Figure 98, the open loop gain and the phase is displayed in the form of a Bode plot. Since there are two integrating functions in the loop, originating from the loopfilter and the VCO gain, the open loop gain response follows a second order slope (–40 dB/dec) creating a phase of –180 degrees at the lower and higher frequencies. The filter characteristic needs to be determined such that it is adding a pole and a zero around the 0 dB point to guarantee sufficient phase margin in this design (Qp in Figure 98). Phase Figure 98. Bode Plot of Gain and Phase in Open Loop Condition wp Open Loop Gain Qp –180 –90 0 0 The open loop gain including the filter response can be expressed as: A openloop + K pd Ko(1 ) jw(R2C2)) jwKn jw 1 ) jw R2C1C2 C1 )C2 (1) The two time constants creating the pole and the zero in the Bode plot can now be defined as: T1 + R2C1C2 C1 ) C2 T2 + R2C2 (2) By substituting equation (2) into (1), it follows: A openloop + K pd KoT1 w2C1KnT2 1 ) jwT2 1 ) jwT1 (3) The phase margin (phase + 180) is thus determined by: Qp + arctan(wT2)–arctan(wT1) (4) At w=wp, the derivative of the phase margin may be set to zero in order to assure maximum phase margin occurs at wp (see also Figure 98). This provides an expression for wp: dQp dw + 0 + T2 1 ) (wT2) 2 – T1 1 ) (wT1) 2 (5) w + wp + 1 T2T1 (6) Or rewritten: T1 + 1 wp 2T2 (7) |
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