{"id":263,"date":"2019-05-12T18:27:21","date_gmt":"2019-05-12T18:27:21","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&#038;p=263"},"modified":"2021-01-12T18:23:43","modified_gmt":"2021-01-12T18:23:43","slug":"1-5-1-5transfer-function-representations-of-simple-physical-systems","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/1-5-1-5transfer-function-representations-of-simple-physical-systems\/","title":{"raw":"1.5 Transfer Function Representations of Simple Physical Systems","rendered":"1.5 Transfer Function Representations of Simple Physical Systems"},"content":{"raw":"<h3 style=\"text-align: justify\"><strong>1.5.1 Electrical Systems<\/strong><\/h3>\r\n<p style=\"text-align: justify\">Modelling of electrical systems is based on:<\/p>\r\n\r\n<ul style=\"text-align: justify\">\r\n \t<li>Ohm's Law<\/li>\r\n \t<li>Kirchhoff's Current Law (KCL)<\/li>\r\n \t<li>Kirchhoff's Voltage Law (KVL)<\/li>\r\n<\/ul>\r\n<table style=\"border-collapse: collapse;width: 100%;height: 30px\" border=\"0\"><caption>\r\n<p style=\"text-align: center\">Table 1\u20113: Basic Equations for Electric Circuits<\/p>\r\n\r\n<\/caption>\r\n<tbody>\r\n<tr style=\"height: 89px\">\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><span class=\"tight\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Resistor.png\" alt=\"\" width=\"60\" height=\"47\" class=\"alignnone wp-image-269\" \/><\/span><\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V=iR[\/latex]<\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i = \\frac{V}{R}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy dissipated through resistance as heat.<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Inductor.png\" alt=\"\" width=\"60\" height=\"46\" class=\"alignnone wp-image-268\" \/><\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V_{L} = L\\frac{di}{dt}[\/latex]<\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i_{L}=\\frac{1}{L}\\int_{-\\infty}^{t} V_{L}dt[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy stored in the magnetic field. No instantaneous change in current.<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Capacitor.png\" alt=\"\" width=\"61\" height=\"53\" class=\"alignnone wp-image-267\" \/><\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V_{C}=\\frac{1}{C}\\int_{-\\infty}^{t} i_{C}dt[\/latex]<\/span><\/td>\r\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i_{C} = C\\frac{dV}{dt}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy stored in the electrostatic field. No instantaneous changes in voltage.<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3 style=\"text-align: justify\"><strong>1.5.2 Mechanical Systems<\/strong><\/h3>\r\n<p style=\"text-align: justify\">Modelling of mechanical translational systems is based on:<\/p>\r\n\r\n<ul style=\"text-align: justify\">\r\n \t<li>Newton's First Law<\/li>\r\n \t<li>Newton's Second Law<\/li>\r\n \t<li>Free Body Diagrams.<\/li>\r\n \t<li>In Table 1-4 below, we have: [latex]F[\/latex]-force, [latex]x[\/latex]- translational displacement, [latex]v[\/latex]- velocity, [latex]a[\/latex]-acceleration<\/li>\r\n<\/ul>\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\"><caption>\r\n<p style=\"text-align: center\">Table 1\u20114: Basic Equations for Mechanical Translational Systems<\/p>\r\n\r\n<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Damper.png\" alt=\"\" width=\"50\" height=\"37\" class=\"alignnone size-full wp-image-290\" \/><\/td>\r\n<td style=\"width: 33.3333%\">[latex]F = Bv = B\\dot{x}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Energy dissipated through viscous damping as heat<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Spring.png\" alt=\"\" width=\"57\" height=\"22\" class=\"alignnone size-full wp-image-289\" \/><\/td>\r\n<td style=\"width: 33.3333%\">[latex]F_{K} = K\\int vdt = Kx[\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Energy stored as kinetic-potential<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Mass.png\" alt=\"\" width=\"72\" height=\"22\" class=\"alignnone size-full wp-image-288\" \/><\/td>\r\n<td style=\"width: 33.3333%\">[latex]F = Ma = M\\dot{v} =M\\ddot{x}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Energy stored as kinetic-potential<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3 style=\"text-align: justify\"><strong>1.5.3 Mechanical Rotational Systems<\/strong><\/h3>\r\n<p style=\"text-align: justify\">Modelling of mechanical rotational systems is based on:<\/p>\r\n\r\n<ul style=\"text-align: justify\">\r\n \t<li>Newton's First Law<\/li>\r\n \t<li>Newton's Second Law<\/li>\r\n \t<li>Free Body Diagrams.<\/li>\r\n \t<li>In Table 1-5 below, we have: [latex]T[\/latex]- torque, [latex]\\omega[\/latex] - angular velocity, [latex]\\theta[\/latex] - angular displacement, [latex]\\epsilon[\/latex] -angular acceleration<\/li>\r\n<\/ul>\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\"><caption>\r\n<p style=\"text-align: center\">Table 1\u20115: Basic Equations for Mechanical Rotational Systems<\/p>\r\n\r\n<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 24.7984%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot1.png\" alt=\"\" width=\"90\" height=\"55\" class=\"alignnone wp-image-299\" \/><\/td>\r\n<td style=\"width: 41.8682%\">[latex]T_{B} = B\\omega = B\\dot{\\theta}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Viscous friction represents a retarding force that dissipates energy as heat<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.7984%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot2.png\" alt=\"\" width=\"90\" height=\"54\" class=\"alignnone wp-image-298\" \/><\/td>\r\n<td style=\"width: 41.8682%\">[latex]T_{K} = K \\int \\omega dt = K\\theta[\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Torsional spring - represents compliance of shaft when subject to torque, stores potential energy of rotational motion<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.7984%\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot3.png\" alt=\"\" width=\"80\" height=\"56\" class=\"alignnone wp-image-297\" \/><\/td>\r\n<td style=\"width: 41.8682%\">[latex]T =J \\epsilon = J \\dot{\\omega} = J \\ddot{\\theta} [\/latex]<\/td>\r\n<td style=\"width: 33.3333%\">Inertia - property of an element that stores the kinetic energy of rotational motion<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3 style=\"text-align: justify\"><strong>1.5.4 Model of Armature Controlled DC Motor<\/strong><\/h3>\r\n<div class=\"textbox shaded\">\u00a0\u00a0\u00a0\u00a0\u00a0<strong> NOTE: This example will help you with your Lab Project # 3. <\/strong><\/div>\r\n<p style=\"text-align: justify\">DC motor is a common actuator in control systems. It directly provides rotary motion and, coupled with wheels or a rack-and-pinion mechanism, can provide transitional motion. The picture to the left in Figure 1\u201112 shows a large industrial DC motor; in control systems applications you're more likely to see a small, lightweight, high-precision geared DC motor, like the one shown on the right. However, their system equations follow the same laws of physics.<\/p>\r\n<p style=\"text-align: justify\">The diagram in Figure 1\u201113 is a representation of the DC armature controlled motor, showing the electric circuit of the armature as well as mechanical parts of the motor, including gears. Small DC motors, such as the one driving the Servo Module in the lab, work most efficiently at high speeds, and therefore they have to be geared for most applications. Direct drives are found in some DC motors with large ratings. Motor equations are shown in Table 1\u20116.<\/p>\r\n<p style=\"text-align: justify\">In the equations, R and L represent the resistance and inductance of the armature winding, [latex]K_t,K_e[\/latex] represent the torque constant and the CEMF constant, respectively, <em>n<\/em> is the gear ratio and [latex]J_{eq},B_{eq}[\/latex] represent the equivalent inertia and viscous friction coefficients of the motor and load combined, as reflected onto the motor side of the gear. Based on the above equations, a block diagram of the DC motor can be built, and is shown in Figure 1\u201114.<\/p>\r\n\r\n\r\n[caption id=\"attachment_1310\" align=\"aligncenter\" width=\"918\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12.png\" alt=\"\" width=\"918\" height=\"376\" class=\"wp-image-1310 size-full\" \/> Figure 1-12: Examples of DC Motors[\/caption]\r\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 213px\" border=\"0\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr style=\"height: 117px\">\r\n<td style=\"width: 337.045px;height: 117px\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC1.png\" alt=\"\" width=\"136\" height=\"97\" class=\"alignnone wp-image-306\" \/><\/td>\r\n<td style=\"width: 337.045px;height: 117px\">Armateur Winding:\r\n\r\n[latex]v_{a} - v_{e} = Ri + L\\frac{di}{dt}[\/latex]\r\n\r\nCounter-electromotive (CEMF) force:\r\n\r\n[latex]v_{e} = K_{e}\\omega[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 96px\">\r\n<td style=\"width: 337.045px;height: 96px\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC2.png\" alt=\"\" width=\"112\" height=\"94\" class=\"alignnone wp-image-305\" \/><\/td>\r\n<td style=\"width: 337.045px;height: 96px\">Energy Conversion: [latex]T=K_{t}i[\/latex]\r\n\r\n[latex]J_{eq}\\dot{\\omega}_{m}=T-B_{eq}\\omega_{m}[\/latex]\r\n\r\n[latex]\\omega=\\dot{\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 100%;height: 168px\" border=\"0\"><caption>Table 1\u20116: Basic Equations for the DC Motor<\/caption>\r\n<tbody>\r\n<tr style=\"height: 168px\">\r\n<td style=\"width: 280.82px;height: 168px\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC3.png\" alt=\"\" width=\"144\" height=\"169\" class=\"aligncenter wp-image-304 \" \/><\/td>\r\n<td style=\"width: 265.82px;height: 168px\">[latex]P_{motor} = P_{load}[\/latex]\r\n\r\n[latex]T_{m}\\omega_{m} = T_{L}\\omega_{L}[\/latex]\r\n\r\n[latex]\\frac{T_{m}}{T_{L}}=\\frac{\\omega_{L}}{\\omega_{m}}=\\frac{1}{n}[\/latex]<\/td>\r\n<td style=\"width: 572.07px;height: 168px\">[latex]J_{m}\\dot{w}_{m} + (J_{L}\\dot{\\omega}_{L})\\cdot \\frac{1}{n}=T -B_{m}\\omega_{m}-(B_{L}\\omega_{L})\\cdot\\frac{1}{n}[\/latex]\r\n\r\n[latex]J_{eq} = J_{m}+\\frac{J_{L}}{n^{2}}[\/latex]\r\n\r\n[latex]B_{eq} = B_{m} + \\frac{B_{L}}{n^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: justify\">The DC motor transfer function, [latex]G_m(s)[\/latex],\u00a0defined as the dynamic ratio of the load position output signal and the armature voltage input signal, is next derived from the above blocks, as shown below:<\/p>\r\n\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 81.2984%\"><span class=\"loose\">[latex]G_{m}(s) = \\frac{\\frac{K_{t}}{(sL+R)(sJ_{eq}+B_{eq})}}{1+\\frac{K_{t}K_{e}}{(sL+R)(sJ_{eq}+B_{eq})}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}=\\frac{K_{t}}{s^{2}LJ_{eq}+s(RJ_{eq}+LB_{eq})+RB_{eq}+K_{t}K_{e}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/span><\/td>\r\n<td style=\"width: 18.7017%;text-align: right\">Equation 1\u201116<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_335\" align=\"aligncenter\" width=\"466\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-300x166.png\" alt=\"Figure 1-13: DC Armature Controlled Motor\" width=\"466\" height=\"258\" class=\"wp-image-335\" \/> Figure 1-13: DC Armature Controlled Motor[\/caption]\r\n\r\n[caption id=\"attachment_334\" align=\"aligncenter\" width=\"532\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-300x98.png\" alt=\"Figure 1-14: Block Diagram of the DC Motor\" width=\"532\" height=\"174\" class=\"wp-image-334\" \/> Figure 1-14: Block Diagram of the DC Motor[\/caption]\r\n<p style=\"text-align: justify\">This is a 3rd order transfer function. For small inductances L (or, L\/R &lt;&lt; J\/B), this can be simplified:<\/p>\r\n\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%\">[latex]G_{m}(s) \\approx \\frac{K_{t}}{sRJ_{eq} + RB_{eq} + K_{t}K_{e}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/td>\r\n<td style=\"width: 50%\">\r\n<p style=\"text-align: right\">Equation 1\u201117<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: justify\">Motor dynamics can now be approximated by a 2nd order transfer function. Two motor parameters are defined: [latex]K_{m}[\/latex], called the motor gain constant, and [latex]\\tau_{m}[\/latex], called the motor time constant.<\/p>\r\n\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 71.3242%\">[latex]K_{m} = \\frac{K_{t}}{RB+K_{t}K_{e}}[\/latex], [latex]\\tau_m=\\frac{RJ}{RB+K_{t}K_{e}}[\/latex]<\/td>\r\n<td style=\"width: 28.6758%\">\r\n<p style=\"text-align: right\">Equation 1\u201118<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: justify\">The motor transfer function can be written as:<\/p>\r\n\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 72.012%\">[latex]G_{m}(s) = \\frac{K_{m}}{s\\tau_{m}+1}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/td>\r\n<td style=\"width: 27.988%\">\r\n<p style=\"text-align: right\">Equation 1\u201119<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: justify\">The block diagram of motor representation shown in Figure 1\u201114 can now be simplified to the one in Figure 1\u201115:<\/p>\r\n\r\n\r\n[caption id=\"attachment_344\" align=\"aligncenter\" width=\"558\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-300x77.png\" alt=\"Figure 1-15: Simplified Block Representation of the Motor\" width=\"558\" height=\"143\" class=\"wp-image-344\" \/> Figure 1-15: Simplified Block Representation of the Motor[\/caption]\r\n<p style=\"text-align: justify\">How accurate is this approximation? Closed loop responses of the accurate servo module model and the model using a 2nd order approximation for the DC motor are shown in Figure 1\u201116 - the responses are practically identical.<\/p>\r\n\r\n\r\n[caption id=\"attachment_345\" align=\"aligncenter\" width=\"502\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-300x202.png\" alt=\"Figure 1-16: DC Motor Responses - Accurate (3rd order) vs. Approximate (2nd order)\" width=\"502\" height=\"338\" class=\"wp-image-345\" \/> Figure 1-16: DC Motor Responses - Accurate (3rd order) vs. Approximate (2nd order)[\/caption]\r\n<h3 style=\"text-align: justify\"><strong>1.5.5 Examples<\/strong><\/h3>\r\n<h4 style=\"text-align: justify\"><strong>1.5.5.1 Example<\/strong><\/h4>\r\n<p style=\"text-align: justify\">Consider a 2nd order filter, with a schematic as shown below:<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-300x195.png\" alt=\"\" width=\"300\" height=\"195\" class=\"size-medium wp-image-349 aligncenter\" \/><\/p>\r\n<p style=\"text-align: justify\">Find the transfer function for this filter, if [latex]V_i[\/latex]is an input to the system and [latex]V_o[\/latex] is an output and the component values are [latex]R_{1}=10\\Omega, R_{2}=5\\Omega , L=2H,[\/latex] and [latex]C=0.5F[\/latex]. What kind of filter is this?<\/p>\r\n\r\n<h4 style=\"text-align: justify\"><strong>1.5.5.2 Example<\/strong><\/h4>\r\n<p style=\"text-align: justify\">Consider a mechanical system shown in the following diagram. Derive the transfer function of this system with Force as an input signal and linear displacement as an output signal. Use MATLAB to simulate system responses for values of mass M = 1, spring flexibility K = 2 and friction B adjustable. Assume the force input to be in a form of an impulse or a very short pulse. Also, check the \u201cMass &amp; Spring\u201d animation\/simulation in Matlab Demos.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18-300x158.png\" alt=\"\" width=\"300\" height=\"158\" class=\"size-medium wp-image-352 aligncenter\" \/><\/p>\r\n\r\n<h4 style=\"text-align: justify\"><strong>1.5.5.3 Example<\/strong><\/h4>\r\n<p style=\"text-align: justify\">Consider the electric circuit below. Find its mechanical analog.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_19.png\" alt=\"\" width=\"275\" height=\"171\" class=\" wp-image-355 aligncenter\" \/><\/p>\r\n\r\n<h4 style=\"text-align: justify\"><strong>1.5.5.4 Example<\/strong><\/h4>\r\n<p style=\"text-align: justify\">Consider an electric circuit, a two-port, shown below, where its components have the following values: [latex]R_{1}=5\\Omega, R_{2}=5\\Omega, C=0.05F,[\/latex] and [latex] L=0.1H.[\/latex]<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-300x159.png\" alt=\"\" width=\"300\" height=\"159\" class=\"size-medium wp-image-354 aligncenter\" \/><\/p>\r\n<p style=\"text-align: justify\">Find the transfer function of the two-port, [latex]G(s) = \\frac{V_{o}(s)}{V_{i}(s)}[\/latex]. Next, find an analytical expression for the two-port step response, [latex]v_{o}(t), t \\geq 0[\/latex].<\/p>","rendered":"<h3 style=\"text-align: justify\"><strong>1.5.1 Electrical Systems<\/strong><\/h3>\n<p style=\"text-align: justify\">Modelling of electrical systems is based on:<\/p>\n<ul style=\"text-align: justify\">\n<li>Ohm&#8217;s Law<\/li>\n<li>Kirchhoff&#8217;s Current Law (KCL)<\/li>\n<li>Kirchhoff&#8217;s Voltage Law (KVL)<\/li>\n<\/ul>\n<table style=\"border-collapse: collapse;width: 100%;height: 30px\">\n<caption>\n<\/caption>\n<tbody>\n<tr style=\"height: 89px\">\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><span class=\"tight\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Resistor.png\" alt=\"\" width=\"60\" height=\"47\" class=\"alignnone wp-image-269\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Resistor.png 110w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Resistor-65x51.png 65w\" sizes=\"auto, (max-width: 60px) 100vw, 60px\" \/><\/span><\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V=iR[\/latex]<\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i = \\frac{V}{R}[\/latex]<\/span><\/td>\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy dissipated through resistance as heat.<\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Inductor.png\" alt=\"\" width=\"60\" height=\"46\" class=\"alignnone wp-image-268\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Inductor.png 112w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Inductor-65x49.png 65w\" sizes=\"auto, (max-width: 60px) 100vw, 60px\" \/><\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V_{L} = L\\frac{di}{dt}[\/latex]<\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i_{L}=\\frac{1}{L}\\int_{-\\infty}^{t} V_{L}dt[\/latex]<\/span><\/td>\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy stored in the magnetic field. No instantaneous change in current.<\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Capacitor.png\" alt=\"\" width=\"61\" height=\"53\" class=\"alignnone wp-image-267\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Capacitor.png 97w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Capacitor-65x57.png 65w\" sizes=\"auto, (max-width: 61px) 100vw, 61px\" \/><\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]V_{C}=\\frac{1}{C}\\int_{-\\infty}^{t} i_{C}dt[\/latex]<\/span><\/td>\n<td style=\"width: 25%;height: 10px\"><span class=\"tight\">[latex]i_{C} = C\\frac{dV}{dt}[\/latex]<\/span><\/td>\n<td style=\"width: 24.9288%;height: 10px\"><span class=\"tight\">Energy stored in the electrostatic field. No instantaneous changes in voltage.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3 style=\"text-align: justify\"><strong>1.5.2 Mechanical Systems<\/strong><\/h3>\n<p style=\"text-align: justify\">Modelling of mechanical translational systems is based on:<\/p>\n<ul style=\"text-align: justify\">\n<li>Newton&#8217;s First Law<\/li>\n<li>Newton&#8217;s Second Law<\/li>\n<li>Free Body Diagrams.<\/li>\n<li>In Table 1-4 below, we have: [latex]F[\/latex]-force, [latex]x[\/latex]&#8211; translational displacement, [latex]v[\/latex]&#8211; velocity, [latex]a[\/latex]-acceleration<\/li>\n<\/ul>\n<table style=\"border-collapse: collapse;width: 100%\">\n<caption>\n<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 33.3333%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Damper.png\" alt=\"\" width=\"50\" height=\"37\" class=\"alignnone size-full wp-image-290\" \/><\/td>\n<td style=\"width: 33.3333%\">[latex]F = Bv = B\\dot{x}[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Energy dissipated through viscous damping as heat<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Spring.png\" alt=\"\" width=\"57\" height=\"22\" class=\"alignnone size-full wp-image-289\" \/><\/td>\n<td style=\"width: 33.3333%\">[latex]F_{K} = K\\int vdt = Kx[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Energy stored as kinetic-potential<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Mass.png\" alt=\"\" width=\"72\" height=\"22\" class=\"alignnone size-full wp-image-288\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Mass.png 72w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Mass-65x20.png 65w\" sizes=\"auto, (max-width: 72px) 100vw, 72px\" \/><\/td>\n<td style=\"width: 33.3333%\">[latex]F = Ma = M\\dot{v} =M\\ddot{x}[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Energy stored as kinetic-potential<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3 style=\"text-align: justify\"><strong>1.5.3 Mechanical Rotational Systems<\/strong><\/h3>\n<p style=\"text-align: justify\">Modelling of mechanical rotational systems is based on:<\/p>\n<ul style=\"text-align: justify\">\n<li>Newton&#8217;s First Law<\/li>\n<li>Newton&#8217;s Second Law<\/li>\n<li>Free Body Diagrams.<\/li>\n<li>In Table 1-5 below, we have: [latex]T[\/latex]&#8211; torque, [latex]\\omega[\/latex] &#8211; angular velocity, [latex]\\theta[\/latex] &#8211; angular displacement, [latex]\\epsilon[\/latex] -angular acceleration<\/li>\n<\/ul>\n<table style=\"border-collapse: collapse;width: 100%\">\n<caption>\n<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 24.7984%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot1.png\" alt=\"\" width=\"90\" height=\"55\" class=\"alignnone wp-image-299\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot1.png 160w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot1-65x40.png 65w\" sizes=\"auto, (max-width: 90px) 100vw, 90px\" \/><\/td>\n<td style=\"width: 41.8682%\">[latex]T_{B} = B\\omega = B\\dot{\\theta}[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Viscous friction represents a retarding force that dissipates energy as heat<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.7984%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot2.png\" alt=\"\" width=\"90\" height=\"54\" class=\"alignnone wp-image-298\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot2.png 146w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot2-65x39.png 65w\" sizes=\"auto, (max-width: 90px) 100vw, 90px\" \/><\/td>\n<td style=\"width: 41.8682%\">[latex]T_{K} = K \\int \\omega dt = K\\theta[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Torsional spring &#8211; represents compliance of shaft when subject to torque, stores potential energy of rotational motion<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.7984%\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot3.png\" alt=\"\" width=\"80\" height=\"56\" class=\"alignnone wp-image-297\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot3.png 146w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_Rot3-65x46.png 65w\" sizes=\"auto, (max-width: 80px) 100vw, 80px\" \/><\/td>\n<td style=\"width: 41.8682%\">[latex]T =J \\epsilon = J \\dot{\\omega} = J \\ddot{\\theta}[\/latex]<\/td>\n<td style=\"width: 33.3333%\">Inertia &#8211; property of an element that stores the kinetic energy of rotational motion<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3 style=\"text-align: justify\"><strong>1.5.4 Model of Armature Controlled DC Motor<\/strong><\/h3>\n<div class=\"textbox shaded\">\u00a0\u00a0\u00a0\u00a0\u00a0<strong> NOTE: This example will help you with your Lab Project # 3. <\/strong><\/div>\n<p style=\"text-align: justify\">DC motor is a common actuator in control systems. It directly provides rotary motion and, coupled with wheels or a rack-and-pinion mechanism, can provide transitional motion. The picture to the left in Figure 1\u201112 shows a large industrial DC motor; in control systems applications you&#8217;re more likely to see a small, lightweight, high-precision geared DC motor, like the one shown on the right. However, their system equations follow the same laws of physics.<\/p>\n<p style=\"text-align: justify\">The diagram in Figure 1\u201113 is a representation of the DC armature controlled motor, showing the electric circuit of the armature as well as mechanical parts of the motor, including gears. Small DC motors, such as the one driving the Servo Module in the lab, work most efficiently at high speeds, and therefore they have to be geared for most applications. Direct drives are found in some DC motors with large ratings. Motor equations are shown in Table 1\u20116.<\/p>\n<p style=\"text-align: justify\">In the equations, R and L represent the resistance and inductance of the armature winding, [latex]K_t,K_e[\/latex] represent the torque constant and the CEMF constant, respectively, <em>n<\/em> is the gear ratio and [latex]J_{eq},B_{eq}[\/latex] represent the equivalent inertia and viscous friction coefficients of the motor and load combined, as reflected onto the motor side of the gear. Based on the above equations, a block diagram of the DC motor can be built, and is shown in Figure 1\u201114.<\/p>\n<figure id=\"attachment_1310\" aria-describedby=\"caption-attachment-1310\" style=\"width: 918px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12.png\" alt=\"\" width=\"918\" height=\"376\" class=\"wp-image-1310 size-full\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12.png 918w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12-300x123.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12-768x315.png 768w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12-65x27.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12-225x92.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/1-12-350x143.png 350w\" sizes=\"auto, (max-width: 918px) 100vw, 918px\" \/><figcaption id=\"caption-attachment-1310\" class=\"wp-caption-text\">Figure 1-12: Examples of DC Motors<\/figcaption><\/figure>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 213px\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr style=\"height: 117px\">\n<td style=\"width: 337.045px;height: 117px\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC1.png\" alt=\"\" width=\"136\" height=\"97\" class=\"alignnone wp-image-306\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC1.png 277w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC1-65x46.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC1-225x160.png 225w\" sizes=\"auto, (max-width: 136px) 100vw, 136px\" \/><\/td>\n<td style=\"width: 337.045px;height: 117px\">Armateur Winding:<\/p>\n<p>[latex]v_{a} - v_{e} = Ri + L\\frac{di}{dt}[\/latex]<\/p>\n<p>Counter-electromotive (CEMF) force:<\/p>\n<p>[latex]v_{e} = K_{e}\\omega[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 96px\">\n<td style=\"width: 337.045px;height: 96px\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC2.png\" alt=\"\" width=\"112\" height=\"94\" class=\"alignnone wp-image-305\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC2.png 210w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC2-65x55.png 65w\" sizes=\"auto, (max-width: 112px) 100vw, 112px\" \/><\/td>\n<td style=\"width: 337.045px;height: 96px\">Energy Conversion: [latex]T=K_{t}i[\/latex]<\/p>\n<p>[latex]J_{eq}\\dot{\\omega}_{m}=T-B_{eq}\\omega_{m}[\/latex]<\/p>\n<p>[latex]\\omega=\\dot{\\theta}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 100%;height: 168px\">\n<caption>Table 1\u20116: Basic Equations for the DC Motor<\/caption>\n<tbody>\n<tr style=\"height: 168px\">\n<td style=\"width: 280.82px;height: 168px\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC3.png\" alt=\"\" width=\"144\" height=\"169\" class=\"aligncenter wp-image-304\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC3.png 199w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_DC3-65x76.png 65w\" sizes=\"auto, (max-width: 144px) 100vw, 144px\" \/><\/td>\n<td style=\"width: 265.82px;height: 168px\">[latex]P_{motor} = P_{load}[\/latex]<\/p>\n<p>[latex]T_{m}\\omega_{m} = T_{L}\\omega_{L}[\/latex]<\/p>\n<p>[latex]\\frac{T_{m}}{T_{L}}=\\frac{\\omega_{L}}{\\omega_{m}}=\\frac{1}{n}[\/latex]<\/td>\n<td style=\"width: 572.07px;height: 168px\">[latex]J_{m}\\dot{w}_{m} + (J_{L}\\dot{\\omega}_{L})\\cdot \\frac{1}{n}=T -B_{m}\\omega_{m}-(B_{L}\\omega_{L})\\cdot\\frac{1}{n}[\/latex]<\/p>\n<p>[latex]J_{eq} = J_{m}+\\frac{J_{L}}{n^{2}}[\/latex]<\/p>\n<p>[latex]B_{eq} = B_{m} + \\frac{B_{L}}{n^{2}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">The DC motor transfer function, [latex]G_m(s)[\/latex],\u00a0defined as the dynamic ratio of the load position output signal and the armature voltage input signal, is next derived from the above blocks, as shown below:<\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 81.2984%\"><span class=\"loose\">[latex]G_{m}(s) = \\frac{\\frac{K_{t}}{(sL+R)(sJ_{eq}+B_{eq})}}{1+\\frac{K_{t}K_{e}}{(sL+R)(sJ_{eq}+B_{eq})}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}=\\frac{K_{t}}{s^{2}LJ_{eq}+s(RJ_{eq}+LB_{eq})+RB_{eq}+K_{t}K_{e}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/span><\/td>\n<td style=\"width: 18.7017%;text-align: right\">Equation 1\u201116<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"attachment_335\" aria-describedby=\"caption-attachment-335\" style=\"width: 466px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-300x166.png\" alt=\"Figure 1-13: DC Armature Controlled Motor\" width=\"466\" height=\"258\" class=\"wp-image-335\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-300x166.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-768x425.png 768w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-65x36.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-225x125.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13-350x194.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_13.png 802w\" sizes=\"auto, (max-width: 466px) 100vw, 466px\" \/><figcaption id=\"caption-attachment-335\" class=\"wp-caption-text\">Figure 1-13: DC Armature Controlled Motor<\/figcaption><\/figure>\n<figure id=\"attachment_334\" aria-describedby=\"caption-attachment-334\" style=\"width: 532px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-300x98.png\" alt=\"Figure 1-14: Block Diagram of the DC Motor\" width=\"532\" height=\"174\" class=\"wp-image-334\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-300x98.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-65x21.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-225x73.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14-350x114.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_14.png 487w\" sizes=\"auto, (max-width: 532px) 100vw, 532px\" \/><figcaption id=\"caption-attachment-334\" class=\"wp-caption-text\">Figure 1-14: Block Diagram of the DC Motor<\/figcaption><\/figure>\n<p style=\"text-align: justify\">This is a 3rd order transfer function. For small inductances L (or, L\/R &lt;&lt; J\/B), this can be simplified:<\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 50%\">[latex]G_{m}(s) \\approx \\frac{K_{t}}{sRJ_{eq} + RB_{eq} + K_{t}K_{e}}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/td>\n<td style=\"width: 50%\">\n<p style=\"text-align: right\">Equation 1\u201117<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">Motor dynamics can now be approximated by a 2nd order transfer function. Two motor parameters are defined: [latex]K_{m}[\/latex], called the motor gain constant, and [latex]\\tau_{m}[\/latex], called the motor time constant.<\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 71.3242%\">[latex]K_{m} = \\frac{K_{t}}{RB+K_{t}K_{e}}[\/latex], [latex]\\tau_m=\\frac{RJ}{RB+K_{t}K_{e}}[\/latex]<\/td>\n<td style=\"width: 28.6758%\">\n<p style=\"text-align: right\">Equation 1\u201118<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">The motor transfer function can be written as:<\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 72.012%\">[latex]G_{m}(s) = \\frac{K_{m}}{s\\tau_{m}+1}\\cdot\\frac{1}{n}\\cdot\\frac{1}{s}[\/latex]<\/td>\n<td style=\"width: 27.988%\">\n<p style=\"text-align: right\">Equation 1\u201119<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">The block diagram of motor representation shown in Figure 1\u201114 can now be simplified to the one in Figure 1\u201115:<\/p>\n<figure id=\"attachment_344\" aria-describedby=\"caption-attachment-344\" style=\"width: 558px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-300x77.png\" alt=\"Figure 1-15: Simplified Block Representation of the Motor\" width=\"558\" height=\"143\" class=\"wp-image-344\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-300x77.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-768x197.png 768w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-65x17.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-225x58.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15-350x90.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_15.png 846w\" sizes=\"auto, (max-width: 558px) 100vw, 558px\" \/><figcaption id=\"caption-attachment-344\" class=\"wp-caption-text\">Figure 1-15: Simplified Block Representation of the Motor<\/figcaption><\/figure>\n<p style=\"text-align: justify\">How accurate is this approximation? Closed loop responses of the accurate servo module model and the model using a 2nd order approximation for the DC motor are shown in Figure 1\u201116 &#8211; the responses are practically identical.<\/p>\n<figure id=\"attachment_345\" aria-describedby=\"caption-attachment-345\" style=\"width: 502px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-300x202.png\" alt=\"Figure 1-16: DC Motor Responses - Accurate (3rd order) vs. Approximate (2nd order)\" width=\"502\" height=\"338\" class=\"wp-image-345\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-300x202.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-65x44.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-225x151.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16-350x236.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_16.png 710w\" sizes=\"auto, (max-width: 502px) 100vw, 502px\" \/><figcaption id=\"caption-attachment-345\" class=\"wp-caption-text\">Figure 1-16: DC Motor Responses &#8211; Accurate (3rd order) vs. Approximate (2nd order)<\/figcaption><\/figure>\n<h3 style=\"text-align: justify\"><strong>1.5.5 Examples<\/strong><\/h3>\n<h4 style=\"text-align: justify\"><strong>1.5.5.1 Example<\/strong><\/h4>\n<p style=\"text-align: justify\">Consider a 2nd order filter, with a schematic as shown below:<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-300x195.png\" alt=\"\" width=\"300\" height=\"195\" class=\"size-medium wp-image-349 aligncenter\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-300x195.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-65x42.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-225x146.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17-350x227.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_17.png 485w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: justify\">Find the transfer function for this filter, if [latex]V_i[\/latex]is an input to the system and [latex]V_o[\/latex] is an output and the component values are [latex]R_{1}=10\\Omega, R_{2}=5\\Omega , L=2H,[\/latex] and [latex]C=0.5F[\/latex]. What kind of filter is this?<\/p>\n<h4 style=\"text-align: justify\"><strong>1.5.5.2 Example<\/strong><\/h4>\n<p style=\"text-align: justify\">Consider a mechanical system shown in the following diagram. Derive the transfer function of this system with Force as an input signal and linear displacement as an output signal. Use MATLAB to simulate system responses for values of mass M = 1, spring flexibility K = 2 and friction B adjustable. Assume the force input to be in a form of an impulse or a very short pulse. Also, check the \u201cMass &amp; Spring\u201d animation\/simulation in Matlab Demos.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18-300x158.png\" alt=\"\" width=\"300\" height=\"158\" class=\"size-medium wp-image-352 aligncenter\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18-300x158.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18-65x34.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18-225x118.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_18.png 302w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h4 style=\"text-align: justify\"><strong>1.5.5.3 Example<\/strong><\/h4>\n<p style=\"text-align: justify\">Consider the electric circuit below. Find its mechanical analog.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_19.png\" alt=\"\" width=\"275\" height=\"171\" class=\"wp-image-355 aligncenter\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_19.png 174w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_19-65x40.png 65w\" sizes=\"auto, (max-width: 275px) 100vw, 275px\" \/><\/p>\n<h4 style=\"text-align: justify\"><strong>1.5.5.4 Example<\/strong><\/h4>\n<p style=\"text-align: justify\">Consider an electric circuit, a two-port, shown below, where its components have the following values: [latex]R_{1}=5\\Omega, R_{2}=5\\Omega, C=0.05F,[\/latex] and [latex]L=0.1H.[\/latex]<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-300x159.png\" alt=\"\" width=\"300\" height=\"159\" class=\"size-medium wp-image-354 aligncenter\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-300x159.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-65x34.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-225x119.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20-350x186.png 350w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/05\/fig1_20.png 458w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: justify\">Find the transfer function of the two-port, [latex]G(s) = \\frac{V_{o}(s)}{V_{i}(s)}[\/latex]. Next, find an analytical expression for the two-port step response, [latex]v_{o}(t), t \\geq 0[\/latex].<\/p>\n","protected":false},"author":118,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-263","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/users\/118"}],"version-history":[{"count":99,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/263\/revisions"}],"predecessor-version":[{"id":2625,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/263\/revisions\/2625"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/263\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=263"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=263"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=263"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}