II ADVANCED APPLICATION OF INDUSTRIAL SERVO DRIVES Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved 7 Background 7.1 INTRODUCTION Part I discussed the basics of industrial servo drives from a hardware point of view. Physical parameters and practical applications were discussed. Part II repeats some of the things in Part I but from a mathematical point of view. The advanced application of industrial servo drives requires the use of differential equations to describe mechanical, electrical, and fluid systems. As applied to servo drives there are numerous academic techniques to analyze these systems (e.g., root locus, Nyquist diagrams, etc.). In working with industrial machinery we live in a sinusoidal world with such things as structural machine resonances. Thus frequency analysis is used in Part II to describe and analyze industrial servo systems. To solve the differential equations describing the physical systems of servo drives, transformation calculus is used to obtain the required transfer functions for the components of servo drives and in analyzing the servo system. There are a multitude of academic textbooks and university courses dealing with feedback control. It is the purpose of Part II to show how the fundamentals of servo drives described in the many academic sources are applied in practice. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved 7.2 PHYSICAL SYSTEM ANALOGS, QUANTITIES, AND VECTORS As a beginning, analogous parameters for an electrical system, a linear mechanical system, and a rotary mechanical system are compared for future reference. In all physical systems there are scalar quantities and vector quantities. Vector quantities can be represented as complex numbers on a complex plane, in polar form, or in exponential form as in Eq. (7.2-1) to (7.2-12). Scalar Quantities (a) Magnitude only (b) Examples: length of a line, mass, volume Vector Quantities (a) Magnitude and direction (b) Examples: force, voltage, weight, velocity Complex Numbers A vector can be represented by its rectilinear components. sin y ¼ F y F cos y ¼ F x F (7.2-1) F y ¼ F sin y F x ¼ F cos y (7.2-2) jFj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðF x Þ 2 þðF y Þ 2 q (7.2-3) y ¼ tan À1 F y F x (7.2-4) ;  FF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðF cos yÞ 2 þðF sin yÞ 2 q ffy ¼ tan À1 F y F x (7.2-5) A vector can also be represented on a complex plane. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved  FF ¼ F x þ jF y (7.2-6)  FF ¼ðjFjcos yÞþjðjFjsin yÞ (7.2-7) j ¼ ffiffiffiffiffiffiffi À1 p (7.2-8) A vector can be represented in polar form.  FF ¼jFjffy (7.2-9)  FF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðF x Þ 2 þðF y Þ 2 q ffy ¼ tan À1 F y F x (7.2-10) A vector can be represented in exponential form.  FF ¼jFje jy y ¼ tan À1 F y F x (7.2-11) Thus: Rectangular Polar Exponential F cos y þjF sin y ¼jFjffy ¼jFje+ jy (7.2-12) 7.3 DIFFERENTIAL EQUATIONS FOR PHYSICAL SYSTEMS The differential equations for physical systems can be written for individual servo components such as motors and amplifiers or for complete multiloop servo drives. In actual practice servo drive block diagrams can be put together with a combination of individual transfer functions representing the differential equation of the separate servo drive components. These individual transfer characteristics can, in general, be represented by single- order or second-order blocks in the overall servo block diagram. A single- order differential equation or transfer characteristic results from a circuit having a single time-varying parameter. Likewise, a second-order transfer characteristic results from a circuit (mechanical, electrical, or fluid) having two time-varying parameters. Most servo drive components can be represented by either a first-order transfer characteristic (or transfer function) or a second-order transfer function. A transfer function is, by definition, the ratio of the Laplace transform of the output to the Laplace transform of the input. In general a transfer function is a shorthand solution for solving differential equations. The derivation of a single-order electrical circuit transfer function having an inductor as a single time-varying parameter is shown in Figure 1. The steady-state equations for the output voltage, based on a sinusoidal Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved input voltage, are 7.3-1 to 7.3-17. Replacing the jo term by the differential operator p or the Laplace transform operator s changes Eq. (7.3 -10) to the transform function of Eq. (7.3-19). This transform function can be represented in the frequency response of Figure 2. To illustrate a second-order circuit, the circuit of Figure 3 with two time-varying parameters has the differential equation of Eq. (7.3-20). The output voltage for the unique case of a sinusoidal input voltage is Eq. (7.3- 25). A second-order mechanical circuit for linear translation is shown in Figure 4. Eq. (7.3-30) is the differential equation for this circuit. Assuming a sinusoidal input, the output displacement is Eq. (7.3-35). Lastly, a rotary mechanical circuit is shown in Figure 5. The output angular motion is Eq. (7.3-44). These examples of single-order and second-order circuits are to illustrate that individual servo drive components can be represented mathematically by differential equations, transfer functions, or the absolute case for a sinusoidal input driving source. The circuit shown in Figure 1 is further described for three cases of absolute, vector, or differential analysis followed by the response of this circuit to a step input and a ramp input. e i ¼ iR e þ iX L (7.3-1) e i ¼ iZ (7.3-2) X L ¼ 2pf ffo L (7.3-3) Zffy ¼ Zðcos y þj sin yÞ (7.3-4) Fig. 1 (a) Inductive/resistive circuit. (b) Circle diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Fig. 2 Single-order frequency response. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Note: a þjb ¼ cffy : Z R e Z þ j X L Z  ¼ R e þ jX L (7.3-5) e i ¼ iðR e þ jX L Þ¼iðR e þ jo L Þ (7.3-6) i ¼ e i R e þ jo L ¼ e i R e 1 þjo L L R e  (7.3-7) Fig. 3 Inductive/capacitive/resistive circuit. Fig. 4 Spring/mass diagram (linear). Fig. 5 Spring/mass diagram (rotary). Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved L R ¼ T 1 (7.3-8) (7.3-8) i ¼ e i R e ð1 þjoT 1 Þ ¼ e i Z (7.3-9) e o ¼ R e i ¼ e i ð1 þjoT 1 Þ (7.3-10) For 1 þjoT 1 ¼ Z (7.3-11) e o e i ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þðoT 1 Þ 2 q ffy ¼ tan À1 oT 1 1 (7.3-12) e o e i ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þðoT 1 Þ 2 q ffÀy ¼ tan À1 oT 1 1 (7.3-13) T 1 ¼ 1 o 1 (7.3-14) e o e i ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ o o 1  2 r ffÀy ¼ tan À1 o o 1 (7.3-15) For o ¼ o 1 ; ;y ¼À45  : Amplitude ratio ¼ 1 ffiffiffi 2 p ¼ 0:707 (7.3-16) ; e o e i ¼ 0:707ffÀ45  (7.3-17) An example of a complex plane plot is: Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved e o ¼ K i ¼ R e e i R e þ pL i ¼ e i 1 þ L i R e p ¼ e i 1 þ L i R e s (7.3-18) e o e i ¼ 1 ð1 þtsÞ (7.3-19) e i ¼ R e i þ di dt L i þ Z idt C a (7.3-20) d dt ¼ p Z ðf Þtdt¼ 1 p (7.3-21) For sinusoidal input, p ¼ jo (7.3-22) e i ¼ iR e þ joL i þ 1 joC a  (7.3-23) i ¼ e i R þjoL i þ 1 joC a ¼ e i joC a ½jRoC a þðjoÞ 2 LC a þ 1 (7.3-24) e o ¼ e i joRC a ½ðjoÞ 2 L i C a þ joRC a þ 1 (7.3-25) Quadratic : ðjoÞ 2 o 2 þ 2d o jo þ 1 (7.3-26) o ¼ ffiffiffiffiffiffiffiffiffiffi 1 L i C a s (7.3-27) SF ¼ M a ¼ F Àkx À c dx dt (7.3-28) M a d 2 x dt 2 ¼ F Àkx Àc dx dt (7.3-29) M a d 2 x dt 2 þ c dx dt þ kx ¼ F (7.3-30) For f ¼F sin ot, the diff erential d dt ¼ jo ¼ p (7.3-31) Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved where p is the differential operator. M a p 2 x þ cpx þkx ¼ F (7.3-32) ðM a p 2 þ cp þ kÞx ¼ F (7.3-33) ½M a ðjoÞ 2 þ cjo þ kx ¼ F (7.3-34) x ¼ F ðjoÞ 2 M a þ joc þ k ¼ F=k ðjoÞ 2 k=M a þ jo c k þ 1 (7.3-35) For quadratic: ðjoÞ 2 o 2 þ 2d o jo þ 1 "# (7.3-36) o n ¼ ffiffiffiffiffiffiffi k M a s 2d o ¼ c=k (7.3-37) d ¼ c k o 2  (7.3-38) ST ¼ Ja ST ¼ J d 2 y 0 dt 2 ¼ G T ðy 1 À y 0 ÞÀb dy 0 dt (7.3-39) d dt ¼ p Differential operator (7.3-40) Jp 2 y þ bpy 0 þ G T y 0 ¼ G T y 1 (7.3-41) ½Jp 2 þ bp þ G T y 0 ¼ G T y 1 (7.3-42) For: y 1 ¼ y 1 sin otp¼ jo ½JðjoÞ 2 þ bjo þ G T y 0 ¼ G T y 1 (7.3-43) y 0 ¼ G T y 1 ðjoÞ 2 J þjob þG T ¼ y 1 ðjoÞ 2 ðG T =JÞ þ b G T jo þ1 (7.3-44) Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved [...]... the components of servo drives were described from a physical point of view These servo- drive components are now described from a mathematical or transfer function point of view In the analysis of electric servo- drive motors, the equations for the motor indicates the presence of two time constants One is a mechanical time constant and the other is an electrical time constant Commercial servo- motor specifications... Ramp response the motor shaft Since these two time constants are part of the motor block diagram used in servo analysis, it is important to know the real value of the time constants under actual load conditions There are two types of servo motors to consider The first is the classical DC servo motor and the second is the AC servo motor often referred to as a brushless DC motor The brushless DC motor is... phase shift of the transport lag y ðdegÞ ¼ ot657:5 o ¼ rad=sec ðdegÞ The significance of the transport lag (dead time in the firing of an SCR) is that each type of SCR amplifier circuit will have an increasing phase lag versus increasing frequency This phase lag adds to the overall phase shift of the servo amplifier, contributing to an unstable servo drive The transport lags for four different SCR servo amplifier... SCR servo amplifier circuits are shown in Figure 11 The relation between the transport lag of the four types of SCR circuits and the phase lag versus frequency is shown in Figure 12 The phase lag of the SCR servo amplifier is a limiting factor in the available frequency response (servo bandwidth) of this type of DC servo drive The transfer function for transport lag does not have any amplitude attenuation... characteristics for SCR circuits SERVO VALVE TRANSFER FUNCTION In servo analysis or in system synthesis it is necessary to have a mathematical representation (transfer function) of the servo valve The usefulness of a linear transfer function to approximate the servo valve response is well established A servo valve is a highly complex device that has high-order nonlinear responses The servo valve torque motor... of performance for the hydraulic resonance that states that this resonance should be oh ¼ 200 rad/sec or larger This index of performance is necessary to provide a stable servo drive The hydraulic resonance is like having a spring inside the servo loop that can cause the servo drive to be unstable The index of performance for hydraulic drives is discussed further in Section 7.9 The damping factor of. .. frequency of 1 rad/sec The integrator is followed by a differentiator represented by an electric tachometer having a transfer function of Ks The gain characteristic increases with frequency at the rate of 20 dB per decade of increasing frequency and has a phase lead of 90 degrees at all frequencies The gain K occurs at a frequency of 1 rad/sec A simple lead is not a usual component of a servo drive,... implied name both time constants are not of constant value Rather, they are both functions of the motor’s operating temperature The electrical resistance of a winding, at a specified temperature, is determined by the length, gauge and composition (i.e., copper, aluminum, etc.) of the wire used to construct the winding The winding in the vast majority of industrial servo motors are constructed using film-coated... frequency of the first stage torque motor is high (about 700 Hz) and can be omitted since it is much higher than the normal frequencies encountered in practice for hydraulic servos In practice the useful bandwidth of a servo valve occurs at a phase lag of 45 degrees for the same reasons as explained in Part I, Section 4.2 For Copyright 2003 by Marcel Dekker, Inc All Rights Reserved industrial hydraulic servo. .. the DC motor, transport lag, hydraulic servo valve, and motor All of these servo components have simple lag-type transfer functions where the output lags the input for a driving input of increasing frequency Most servo- drive Copyright 2003 by Marcel Dekker, Inc All Rights Reserved components can be represented by first- or second-order transfer functions A plot of these transfer functions in the frequency . II ADVANCED APPLICATION OF INDUSTRIAL SERVO DRIVES Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved 7 Background 7.1 INTRODUCTION Part I discussed the basics of industrial servo drives. point of view. Physical parameters and practical applications were discussed. Part II repeats some of the things in Part I but from a mathematical point of view. The advanced application of industrial. systems of servo drives, transformation calculus is used to obtain the required transfer functions for the components of servo drives and in analyzing the servo system. There are a multitude of academic