AN535

AN535 MOTOROLA | Alldatasheet

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can be approached using the Laplace Transform technique. time response f(t) of a system in the complex domain F(s). Figure 1. Feedback System

1 G(s) H(s)i(s)

1 G(s) H(s)i(s) ( 2 )

Figure 2. Phase Locked Loop Freescale Semiconductor, Inc.

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The phase detector produces a voltage proportional to the phase difference between the signals θi and θo/N. This voltage upon filtering is used as the control signal for the VCO/VCM (VCM – Voltage Controlled Multivibrator). Since the VCO/VCM produces a frequency proportional to its input voltage, any time variant signal appearing on the control signal will frequency modulate the VCO/VCM. The output frequency is fo = N fi ( 3 ) during phase lock. The phase detector, filter, and VCO/VCM compose the feed forward path with the feedback path containing the programmable divider. Removal of the programmable counter produces unity gain in the feedback path (N = 1). As a result, the output frequency is then equal to that of the input. Various types and orders of loops can be constructed depending upon the configuration of the overall loop transfer function. Identification and examples of these loops are contained in the following two sections. TYPE — ORDER These two terms are used somewhat indiscriminately in published literature, and to date there has not been an established standard. However, the most common usage will be identified and used in this article. The type of a system refers to the number of poles of the loop transfer function G(s) H(s) located at the origin. Example: let ( 4 )G(s) H(s) 10 s(s 10) This is a type one system since there is only one pole at the origin. The order of a system refers to the highest degree of the polynomial expression which is termed the Characteristic Equation (C.E.). The roots of the characteristic equation become the closed loop poles of the overall transfer function. Example: ( 6 )G(s) H(s) 10 s(s 10) then

1 G(s) H(s) 1 10

s(s 10) 0 ( 7 ) therefore which is a second order polynomial. Thus, for the given G(s) H(s), we obtain a type 1 second order system. ERROR CONSTANTS Various inputs can be applied to a system. Typically, these include step position, velocity, and acceleration. The response of type 1, 2, and 3 systems will be examined with the various inputs. θe(s) represents the phase error that exists in the phase detector between the incoming reference signal θi(s) and the feedback θo(s)/N. In evaluating a system, θe(s) must be examined in order to determine if the steady state and transient characteristics are optimum and/or satisfactory. The transient response is a function of loop stability and is covered in the next section. The steady state evaluation can be simplified with the use of the final value theorem associated with Laplace. This theorem permits finding the steady state system error θe(s) resulting from the input θi(s) without transforming back to the time domain.3 Lim [θ(t)] = Lim [sθe(s)] ( 10 ) t → s → o Where Simply stated e(s) 1

1 G(s) H(s)i(s) ( 11 )

The input signal θi(s) is characterized as follows: Step position: θi(t) = Cp t ≥ 0 ( 12 ) Or, in Laplace notation: ( 13 )i(s) C p s where Cp is the magnitude of the phase step in radians. This corresponds to shifting the phase of the incoming reference signal by Cp radians: Step velocity: θi(t) = Cvt t ≥ 0 ( 14 ) Or, in Laplace notation: ( 15 )i(s) C v where Cv is the magnitude of the rate of change of phase in radians per second. This corresponds to inputting a frequency that is different than the feedback portion of the VCO frequency. Thus, Cv is the frequency difference in radians per second seen at the phase detector. Step acceleration: θi(t) = Cat2 t ≥ 0 ( 16 ) Or, in Laplace notation: ( 17 )i(s) 2C a C a is the magnitude of the frequency rate of change in radians per second per second. This is characterized by a time variant frequency input. Typical loop G(s) H(s) transfer functions for types 1, 2, and 3 are: Type 1 ( 18 )G(s) H(s) K s(s a) Freescale Sem iconductor, I Freescale Semiconductor, Inc. For More Information On This Product, Go to: www.freescale.com nc...

step phase input is found by using Equations 11 and 13. position (phase) is applied. Table 1. Steady State Phase Errors for Various System two input signals at the phase detector. loop gain and the magnitude of the input step. minimum of type 2 is required. (K =∞ ), where K is loop gain. Where #P (#Z) is the number of poles (zeroes). (Rule 1). The asymptotes can be found according to Rule 3. Freescale Semiconductor, Inc.

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first writing the characteristic equation. zero, then determines the breakaway point. can be plotted as in Figure 3. natural frequency as shown in Figure 3. Figure 3. Type 1 Second Order Root Locus Contour Figure 4. Type 1 Second Order Step Response Freescale Semiconductor, Inc.

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Figure 7. Phase-Locked Loop Circuit Parameters output frequency is shown in Figure 8. Figure 8. MC4324 Input Voltage versus Output Where Kv is the sensitivity in radians per second per volt. the loop transfer function must take the form of Equation 19. Figure 9. Active Filter Design Freescale Semiconductor, Inc.

where A is voltage gain of the amplifier. overall loop characteristics. be applied to Kf in order to properly characterize the function. Kc is found experimentally to be Kc = 0.5. Laplace representation in Figure 10.

0.5 Kp Kv R 2

0.5 Kp Kv

within 5% at ω nt = 4.5. The required lock-up time is 1ms. Figure 10. Laplace Representation of Diagram in Figure 11. Freescale Semiconductor, Inc.

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Figure 11. Circuit Diagram of Type 2 Phase-Locked Loop R 1 is typically selected greater than 1kΩ . PLL can be properly configured. lock-up time and percent overshoot (see Figure 14). Figure 12. Root Locus Variation Freescale Semiconductor, Inc.

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Figure 15. VCM Control Signal Transient Freescale Semiconductor, Inc.

/C0078 11MOTOROLA SEMICONDUCTOR APPLICATION INFORMATION Bibliography 1. Topic: Type Two System Analysis Gardner, F. M., Phase Lock Techniques, Wiley, New York, Second Edition, 1967 2. Topic: Root Locus Techniques Kuo, B. C., Automatic Control Systems, Prentice-Hall, Inc., New Jersey, 1962 3. Topic: Laplace Techniques McCollum, P. and Brown, B., Laplace Transform Tables and Theorems, Holt, New York, 1965 4. Topic: Type One System Analysis Truxal, J. G., Automatic Feedback Control System Synthesis, McGraw-Hill, New York, 1955 5. Topic: Phase Detector Gain Constant DeLaune, Jon, MTTL and MECL Avionics Digital Frequency Synthesizer, AN532 Freescale Sem iconductor, I Freescale Semiconductor, Inc. For More Information On This Product, Go to: www.freescale.com nc...

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Motorola reserves the right to make changes without further notice to any products herein. Motorola makes no warranty, representation or guarantee regarding the suitability of its products for any particular purpose, nor does Motorola assume any liability arising out of the application or use of any product or circuit, and specifically disclaims any and all liability, including without limitation consequential or incidental damages. “Typical” parameters can and do vary in different applications. All operating parameters, including “Typicals” must be validated for each customer application by customer’s technical experts. Motorola does not convey any license under its patent rights nor the rights of others. Motorola products are not designed, intended, or authorized for use as components in systems intended for surgical implant into the body, or other applications intended to support or sustain life, or for any other application in which the failure of the Motorola product could create a situation where personal injury or death may occur. Should Buyer purchase or use Motorola products for any such unintended or unauthorized application, Buyer shall indemnify and hold Motorola and its officers, employees, subsidiaries, affiliates, and distributors harmless against all claims, costs, damages, and expenses, and reasonable attorney fees arising out of, directly or indirectly, any claim of personal injury or death associated with such unintended or unauthorized use, even if such claim alleges that Motorola was negligent regarding the design or manufacture of the part. Motorola and are registered trademarks of Motorola, Inc. Motorola, Inc. is an Equal Opportunity/Affirmative Action Employer. Literature Distribution Centers: USA: Motorola Literature Distribution; P.O. Box 20912; Phoenix, Arizona 85036. EUROPE: Motorola Ltd.; European Literature Centre; 88 Tanners Drive, Blakelands, Milton Keynes, MK14 5BP, England. JAPAN: Nippon Motorola Ltd.; 4-32-1, Nishi-Gotanda, Shinagawa-ku, Tokyo 141 Japan. AN535/D Freescale Sem iconductor, I Freescale Semiconductor, Inc. For More Information On This Product, Go to: www.freescale.com nc...