Derivation of the Dc motor differential equations Under typical operating conditions, the torque generated by a DC motor is directly proportional to the cumrent passing through it T-kai where kis a...

Extracted text: Derivation of the Dc motor differential equations Under typical operating conditions, the torque generated by a DC motor is directly proportional to the cumrent passing through it T-kai where kis a proportionality constant determined by the characteristics of the specific motor. This relationship fails for large values of i. When the armature ratates, z generates a back electromotive force (back EMF) that opposes the applied valtage. In the circuit diagram, this is represented by a mator symbol. The voltage drop dua to the back EMF, denoted e, is proportional to the rotational velocity af the armature e-kp where k, is a constant of proportionality. Ta derive the relationship between the current i and the motor shaftt angular position 0. yau can apply KirchhofT's voltage law and Nowton's second law in terms of torques acting on the motor shaft. Kirchhoff's voltage law implies that the sum at the vatage drops across the circuit must equal tha applied voltage: iR + (1) where v is the electric potential applied to the motor. In the absence af other applied torques, the equation aft motion can be derived by balancing the inertia of the motor shaft (Jõ) with the applied torque (7) and mictional resistance (k Jõ-T-R Using the linear relationship batwoen torque and curent, this is equivalent to: Jõ + Bồ -i (2) Exerolce. In this axercise, you will compute transfer functions of the DC motor system. Assume that the system starts at rest (i-i-0-0-0). Also, assume that the applied voltage is a function of time v-ve). (a) Compute the Laplace transform af the voltage conservation equation (1) and apply the initial condiions Type your result below in terms af the defined symbolic variables. Pro-tip: When you write a symbolic equatian, you assign the whole aquation to a MATLAB variable using the assignment aperator"-" (singie equals). The equals sign in the symbolic equation is written using "--" (double equals). For exampie: syms y b lineartay * use these sysbolic variables syns LRke Bkn syms Theta ISV * write your laplace transform equation here * Constants * Laplace donaln varlables %checkEgn1(egn1) (b) Compute the Laplace transform of the equation of motion (2) and apply the initial condions. Type your answer below in tems of the defined symbolic variables. * write your laplace transform equation here egn2 - NaN, IcheckEgn2(egn2) (0) Salve for the angular displacement transfer funcion Gs) es using your results from (a) and (D). * write your transfer function here Thetaoverv - Na, XcheckThetaoverv( Thetadverv) (d) Solve for tha angular velocity transter function lis where 2 is the Laplace transform of the angular velocity a) -). * write your transfer function here Onegaoverv - NaN; Xcheckonegaoverv(Onegaoverv)