Maddy, S L,"Phase-Locked Loop The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Maddy, S.L., “Phase-Locked Loop” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
76 Phase-Locked Loop 76.1 Introduction 76.2 Loop Filter 76.3 Noise 76.4 PLL Design Proced Steven L. Maddy 76.5 Components 76.6 Applications 76.1 Introduction a phase-locked loop(PLL)is a system that uses feedback to maintain an output signal in a specific phase relationship with a reference signal. PLLs are used in many areas of electronics to control the frequency and/or phase of a signal. These applications include frequency synthesizers, analog and digital modulators and demod- ulators, and clock recovery circuits. Figure 76.1 shows the block diagram of a basic PLL system. The phase etector consists of a device that produces an output voltage proportional to the phase difference of the two input signals. The VCO (voltage-controlled oscillator) is a circuit that produces an ac output signal whose frequency is proportional to the input control voltage. The divide by n is a device that produces an output ignal whose frequency is an integer(denoted by n) division of the input signal frequency. The loop filter is a circuit that is used to control the Pll dynamics and therefore the performance of the system. The Rs)term is used to denote the Laplace transfer function of this filter. Servo theory can now be used to derive the equations for the output signal phase relative to the reference input signal phase. Because the VCO control voltage sets the frequency of the oscillation(rather than the phase) this will produce a pure integration when writing this expression. Several of the components of the PLl have a fixed gain associated with them. These are the VCo control voltage to output frequency conversion gain(k), he phase detector input signal phase difference to output voltage conversion gain(K,), and the feedback division ratio (N). These gains can be combined into a single factor called the loop gain(k). This loop gain is calculated using Eq (76. 1)and is then used in the following equations to calculate the loop transfer function (76.1) The closed-loop transfer function [H(s)] can now be written and is shown in Eq (76.2). This function is typically used to examine the frequency or time-domain response of a PLL and defines the relationship of the phase of the VCO output signal (0) to the phase of the reference input(0 ) It also describes the relationship of a change in the output frequency to a change in the input frequency. This function is low-pass in nature. H(s)= 0(s) KF(s) (76.2) (s) s+ kF(s) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 76 Phase-Locked Loop 76.1 Introduction 76.2 Loop Filter 76.3 Noise 76.4 PLL Design Procedures 76.5 Components 76.6 Applications 76.1 Introduction A phase-locked loop (PLL) is a system that uses feedback to maintain an output signal in a specific phase relationship with a reference signal. PLLs are used in many areas of electronics to control the frequency and/or phase of a signal. These applications include frequency synthesizers, analog and digital modulators and demodulators, and clock recovery circuits. Figure 76.1 shows the block diagram of a basic PLL system. The phase detector consists of a device that produces an output voltage proportional to the phase difference of the two input signals. The VCO (voltage-controlled oscillator) is a circuit that produces an ac output signal whose frequency is proportional to the input control voltage. The divide by N is a device that produces an output signal whose frequency is an integer (denoted by N) division of the input signal frequency. The loop filter is a circuit that is used to control the PLL dynamics and therefore the performance of the system. The F(s) term is used to denote the Laplace transfer function of this filter. Servo theory can now be used to derive the equations for the output signal phase relative to the reference input signal phase. Because the VCO control voltage sets the frequency of the oscillation (rather than the phase), this will produce a pure integration when writing this expression. Several of the components of the PLL have a fixed gain associated with them. These are the VCO control voltage to output frequency conversion gain (Kv), the phase detector input signal phase difference to output voltage conversion gain (Kf), and the feedback division ratio (N). These gains can be combined into a single factor called the loop gain (K). This loop gain is calculated using Eq. (76.1) and is then used in the following equations to calculate the loop transfer function. (76.1) The closed-loop transfer function [H(s)] can now be written and is shown in Eq. (76.2). This function is typically used to examine the frequency or time-domain response of a PLL and defines the relationship of the phase of the VCO output signal (uo) to the phase of the reference input (ui ). It also describes the relationship of a change in the output frequency to a change in the input frequency. This function is low-pass in nature. (76.2) K K K N v = f ¥ H s s s KF s s KF s o i ( ) ( ) ( ) ( ) ( ) = = + q q Steven L. Maddy RLM Research
FIGURE 76.1 PLL block diagram. The loop error function, shown in Eq (76.3), describes the difference between the vco phase and the reference phase and is typically used to examine the performance of PLls that are modulated. This function 6(s)-6(s)6(s) (76.3) 6;(s) 6;(s)+KF(s) The open-loop transfer function [G(s)] is shown in Eq (76.4). This function describes the operation of the loop before the feedback path gn of the system in determining the and phase margin of the PLL. These are indications of the stability of a PlL when the feedback loop is connected. KF(s) (76.4) These functions describe the performance of the basic PLL and can The synthesis equations will be used to calculate circuit components that will give a desired performance characteristic. These characteristics usually involve the low-pass corner frequency and shape of the closed-loop response characteristic [Eq (76.2)] and determine such things as the loop lock-up time, the ability to track the input signal, and the output signal noise characteristics. 76.2 Loop Filter The loop filter is used to shape the overall response of the Pll to meet the design goals of the system. There re two implementations of the loop filter that are used in the vast majority of PLLs: the passive lag circuit shown in Fig. 76.2 and the active circuit shown in Fig. 76.3. These two circuits both produce a PLL with a second-order response characteristic. The transfer functions of these loop filter circuits may now be derived and are shown in Eqs. (76.5)for the ssive circuit(Fig. 76.2)and(76.6)for the active circuit(Fig. 76.3) F(S= SCiR,+ 1 (76.5) s(R1+R2C1+1 FIGURE 76.2 Passive loop filter. FIGURE 76.3 Active loop filter. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The loop error function, shown in Eq. (76.3), describes the difference between the VCO phase and the reference phase and is typically used to examine the performance of PLLs that are modulated. This function is high-pass in nature. (76.3) The open-loop transfer function [G(s)] is shown in Eq. (76.4). This function describes the operation of the loop before the feedback path is completed. It is useful during the design of the system in determining the gain and phase margin of the PLL. These are indications of the stability of a PLL when the feedback loop is connected. (76.4) These functions describe the performance of the basic PLL and can now be used to derive synthesis equations. The synthesis equations will be used to calculate circuit components that will give a desired performance characteristic. These characteristics usually involve the low-pass corner frequency and shape of the closed-loop response characteristic [Eq. (76.2)] and determine such things as the loop lock-up time, the ability to track the input signal, and the output signal noise characteristics. 76.2 Loop Filter The loop filter is used to shape the overall response of the PLL to meet the design goals of the system. There are two implementations of the loop filter that are used in the vast majority of PLLs: the passive lag circuit shown in Fig. 76.2 and the active circuit shown in Fig. 76.3. These two circuits both produce a PLL with a second-order response characteristic. The transfer functions of these loop filter circuits may now be derived and are shown in Eqs. (76.5) for the passive circuit (Fig. 76.2) and (76.6) for the active circuit (Fig. 76.3). (76.5) FIGURE 76.1 PLL block diagram. FIGURE 76.2 Passive loop filter. FIGURE 76.3 Active loop filter. q q q q q i o i e i s s s s s s s KF s ( ) ( ) ( ) ( ) ( ) ( ) - = = + G s KF s s ( ) ( ) = F s sC R s R R C p ( ) ( ) = + + + 1 2 1 2 1 1 1
E(s)=SR2C+I (76.6) rC These loop filter equations may now be substituted into Eq (76.2)to form the closed-loop transfer functions of the PLL. These are shown as Eqs.(76.7)for the case of the passive filter and (76.8 )for the active. H(S= R1+R2(R1+R2)C1 (76.7) (R1+R2)C1R1+R2」(R1+R2)C H()= R RC (768) s2+sKR2,K RI RCI These closed-loop equations can also be written in the forms shown below to place the function in terms of the damping factor(4)and the loop natural frequency(n). It will be shown later that these are very useful parameters in specifying PLL performance. Equation(76.9)is the form used for the PLL with a passive loop filter, and Eq. (76.10)is used for the active loop filter case. H6)=92(01-(0kK (76.9 H(s)= (76.10) s4+s25o,+o Solving Eqs. (76. 7)and(76.9)for R, and R2 in terms of the loop parameters s and o, we now obtain the synthesis equations for a Pll with a passive loop filter. These are shown as Eqs. (76.11)and(76.12) (76.11) OC KC K R (76.12) To maintain resistor values that are positive the passive loop filter PLL must meet the constraint shown in > (76.13) 2K e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (76.6) These loop filter equations may now be substituted into Eq. (76.2) to form the closed-loop transfer functions of the PLL. These are shown as Eqs. (76.7) for the case of the passive filter and (76.8) for the active. (76.7) (76.8) These closed-loop equations can also be written in the forms shown below to place the function in terms of the damping factor (z) and the loop natural frequency (vn). It will be shown later that these are very useful parameters in specifying PLL performance. Equation (76.9) is the form used for the PLL with a passive loop filter, and Eq. (76.10) is used for the active loop filter case. (76.9) (76.10) Solving Eqs. (76.7) and (76.9) for R1 and R2 in terms of the loop parameters z and vn, we now obtain the synthesis equations for a PLL with a passive loop filter. These are shown as Eqs. (76.11) and (76.12). (76.11) (76.12) To maintain resistor values that are positive the passive loop filter PLL must meet the constraint shown in Eq. (76.13). (76.13) F s sR C sR C a ( ) = 2 1 + 1 1 1 H s s KR R R K R R C s s R R C KR R R K R R C p ( ) = 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 + + + + + + + È Î Í Í ˘ ˚ ˙ ˙ + + ( ) ( ) ( ) H s s KR R K R C s s KR R K R C a ( ) = 2 1 1 1 2 2 1 1 1 + + + H s s K s s p n n n n n ( ) = / 2 2 2 [2 ( )] 2 2 zw w w zw w - + + + H s s s s a n n n n ( ) = 2 2 2 2 2 zw w zw w + + + R nC KC 2 = 2z 1 w - R K C R n 1 = w2 2 - z w > n 2K
3 t+ iN .0 tNin 20.0 Frequency(Hz) FIGURE 76.4 Closed-loop second-order type-2 PLL error response for various damping factors. For the active loop filter case Eqs. (76. 8)and (76. 10)are solved and yield the synthesis equations shown in Eqs.(76.14)and(76. 15). It can be seen that no constraints on the loop damping factor exist in this case. K RI 0-C A typical design procedure for these loop filters would be, first, to select the loop damping factor ar frequency based on the system requirements. Next, all the loop gain parameters are determined. A capacitor value may then be selected. The remaining resistors can now be computed from the synthesis Figure 76.4 shows the closed-loop frequency response of a PLL with an active loop filter [Eq.76.10)]for various values of damping factor. The loop natural frequency has been normalized to 1 Hz for all cases. Substituting Eq (76.6)into(76. 3)will give the loop error response in terms of damping factor. This function is shown plotted in Fig. 76.5. These plots may be used to select the Pll performance parameters that will give a desired frequency response shap The time response of a PLL with an active loop filter to a step in input phase was also computed and is shown plotted in Fig. 76.6 76.3 noise An impe aspect of a PlL is the noise content of the output. The dominant resultant noise will appear as Gitter)on the output signal from the VCO. Due to the dynamics of the Pll these noise sources will be filtered by the loop transfer function [Eq (76.2)] that is a low-pass characteristic. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC For the active loop filter case Eqs. (76.8) and (76.10) are solved and yield the synthesis equations shown in Eqs. (76.14) and (76.15). It can be seen that no constraints on the loop damping factor exist in this case. (76.14) (76.15) A typical design procedure for these loop filters would be, first, to select the loop damping factor and natural frequency based on the system requirements. Next, all the loop gain parameters are determined. A convenient capacitor value may then be selected. The remaining resistors can now be computed from the synthesis equations presented above. Figure 76.4 shows the closed-loop frequency response of a PLL with an active loop filter [Eq. (76.10)] for various values of damping factor. The loop natural frequency has been normalized to 1 Hz for all cases. Substituting Eq. (76.6) into (76.3) will give the loop error response in terms of damping factor. This function is shown plotted in Fig. 76.5. These plots may be used to select the PLL performance parameters that will give a desired frequency response shape. The time response of a PLL with an active loop filter to a step in input phase was also computed and is shown plotted in Fig. 76.6. 76.3 Noise An important design aspect of a PLL is the noise content of the output. The dominant resultant noise will appear as phase noise (jitter) on the output signal from the VCO. Due to the dynamics of the PLL some of these noise sources will be filtered by the loop transfer function [Eq. (76.2)] that is a low-pass characteristic. FIGURE 76.4 Closed-loop second-order type-2 PLL error response for various damping factors. R K nC 1 2 = w R nC 2 = 2z w