If you're going to study process control, the first thing you must understand is how to solve a second-order differential equation. Why is that? Because most of the basic mechanical and electrical processes that we are trying to control have their mathematical basis in these equations. If you went to engineering school, you were required to study this stuff, but people forget stuff once they get out of school. For the undergraduates that I teach, they forget stuff once the final is over. Who needs graduation? So let's do a rapid review of second-order differential equations, particularly for oscillatory behavior. This occurs in a spring-mass damper system and a series resistor, capacitor, inductor circuit. For the spring-mass damper system, the inertial force is proportional to the acceleration d squared, x/dt squared of the mass. The damping force is proportional to the velocity dx/dt, and the spring force is proportional to the displacement. The respective proportional constants or the mass M, the viscous damping coefficient b, and the spring constant k. For the RLC circuit, the change in the inductive voltage with respect to time is proportional to the second derivative of the current d squared i dt squared. The change in resistive voltage drop is proportional to the first derivative of the current di dt, and the change in capacity voltage is proportional to the current. The respective proportional constants are the inductance L, the resistance R, and one over the capacitance C. We let displacement x of t equals capital Xe_ st, and the current i of t equals capital Ie_st, where e is Euler's number. Recall that the number e has a convenient attribute in calculus. The derivative of e_x is e_x. So when you substitute for x of t in our differential equation for the natural response and take the Laplace transform, you get a quadratic equation for the spring-mass damper system, ms squared plus bs plus k equals 0. When you substitute for i of t in our differential equation for the natural response of the current, you get for the series RLC circuit, s squared plus R over LS plus 1 over LC equals 0. Now, let's go back to algebra. I know, we're going back to your teenage years, a time you feel you have a right to forget. The quadratic equation has a standard formula for finding the two roots. For the equation ax squared plus bx plus c equals 0, the values of x which are the solutions of the equation are given by x equals minus b plus or minus square root of b squared minus 4ac over 2a. Please don't get mixed up between the b in that formula and the b we are using for the viscous damping coefficient. The idea here is to apply the standard formula to the coefficients in the quadratic equations that we're using. In our spring-mass damper case, the b of the standard quadratic formula corresponds exactly to the b for the viscous damping coefficient, lucky for us. The a corresponds to the mass m, and the c corresponds to the spring constant k. In our series RLC circuit, a equals 1, b is R over L, and C equals 1 over LC. So plugging in our constants to the quadratic equation, we get the formula for the two roots of the two equations shown in the bottom left and right corners of the slide. For the spring-mass damper system, S_1 equals minus b over 2m plus square root of b squared minus 4mk over 2m, and S_2 equals minus b over 2m minus the square root of b squared minus 4mk over 2m. For the series RLC circuit, S_1 equals a minus R over 2L, plus the square root of R over 2L squared, minus 1 over LC, and S_2 equals minus R over 2L, minus the square root of R over 2L quantity squared, minus 1 over LC. The behavior for the solution depends on whether we have a positive, negative, or zero value under the square root sign for the poles S_1 and S_2. The location of the poles in the real complex a plane for the under-damped solution depends on the damping ratio in the natural frequency. Let's discuss four cases as shown in the lower left and lower right corners of the slide. We continue to show the spring mass a damper equations on the left side of this slide, and the series RLC equations on the right side. For the spring-mass damper system, the damping ratio Zeta is given by c divide by 2 times the square root of km. The undamped natural frequency Omega n is given by the square root of k over m. This represents the frequency of the oscillating mass for the undamped case, where b the viscous damping coefficient is zero and the mass with theoretically oscillate around the zero-position point forever. The damped natural frequency Omega d is given by Omega n times the square root of 1 minus Zeta squared, and this represents the frequency of the oscillating mass for the underdamped case, where the damping ratio is somewhere between zero and one. In this case, the mass would oscillate around the zero-position point but the oscillations would slowly decay to zero. If the quantity b squared minus a 4mk is less than 0, we would be taking the square root of a negative number. Recall that the square root of negative one is the imaginary number i, sometimes shown as imaginary number j. This would position the poles of the quadratic equation on the left side of the imaginary axis shown in the graph in the center of the slide. They would exist at the points marked with the letter x, where the radius of the distance from the zero-zero-point is Omega n, the coordinate on the imaginary axis is Omega d, and the coordinate on the real axis is minus Zeta Omega n. If the quantity b squared minus 4mk equals 0, we have critical damping. In this case, the position of the mass would drop back to zero without oscillating. The poles are equal to each other, located on the real axis at a value of minus b over 2m. If the quantity b squared minus 4mk is greater than 0, we have overdamping. In that case, both poles are located on the real axis but at different places. The position of the mass will still drop back to zero without oscillating, but it will take more time to do this so than for critical damping. We have a similar situation for the series RLC circuit when evaluating the current. For the series RLC circuit, the damping coefficient Alpha is given by Alpha equals R over 2L. The undamped natural frequency Omega n is given by 1 over the square root of LC, this represents the frequency of the oscillating circuit for the undamped case where the resistance R is zero, and the current would theoretically oscillate around the zero-point forever.. The damped natural frequency Omega d is given by Omega d it's the square root over Omega n squared minus Alpha squared. This represents the frequency of the oscillating current for the underdamped case. In this case, the current would oscillate around the zero-point but the oscillations would slowly decay to zero. If the quantity R over 2L quantity squared minus 1 over LC is less than 0, we would again be taking the square root of a negative number. This would position the poles of the quadratic equation on the left side of the imaginary axis shown in the graph in the center of the slide. They would exist at the points marked with the letter X, where the radius of the distance from the point (0,0) is Omega n, the coordinate on the imaginary axis Omega d, and the coordinate on the real axis is minus Alpha. If the quantity R over 2L quantity squared minus 1 over LC equals 0, we have critical damping. In this case, the current would drop to zero without oscillating. The Poles would be equal to each other located on the real axis at a value of minus R over 2L. If the quantity R over 2L squared minus 1 over LC is greater than 0, we have overdamping. In that case, both poles are located on the real axis but at different places. The current will still drop to zero without oscillating, but it will take more time to do so than for critical damping. Let's study this solution for the underdamped case. This is the most complex one for PID control methods, so it is worth looking at. For the spring-mass-damper system, let the location of pole S_1 in the real complex plane be given by minus Sigma plus j Omega d, where Sigma equals minus Zeta Omega n, the coordinate on the real axis. The location of pole S_2 is minus Sigma minus j Omega d. Then the solution for the position x of t is given by either the negative Sigma t times the quantity Alpha cosine Omega d times t plus Beta sine times Omega d times t, where Alpha is X nought the initial position, Beta equals v nought plus Sigma x nought quantity divided by Omega d, v nought being the initial velocity and the initial position velocity for the mass or the boundary conditions that would allow us to solve for the constants Alpha and Beta. For the series RLC circuit, let the location of pole S_1 in the real complex plane be given by negative Alpha plus j Omega d, where Alpha is the damping coefficient, the coordinate on the real axis. The location of pole S_2 is minus Alpha minus j Omega d. Then the solution for the current i of t is given by e to the minus Alpha t times quantity B_1 cosine Omega d times t plus B_2 times sine Omega d times t. Where B_1 equals i nought the initial current and B_2 equals di nought/dt plus Alpha i nought quantity divided by Omega d, and di nought is the initial slope of the current. So the initial current, initial derivative of the current are the boundary conditions that allow us to solve for the constants B_1 and B_2. I went through both the mechanical and electrical derivations of second-order solutions because most of the sensors we cover in this course are by nature electromechanical devices. It would be nice if the mechanical and electrical engineering professors of the world could get together and standardized their nomenclature and equations. But that will never happen. Each side goes their own way and acts like the other side does not exist. If you don't believe me, try googling solutions for the second-order equations. You will see consistency amongst the mechanical engineers and consistency amongst the electrical engineers but no consistency between the two. No wonder that these guys get into the workforce, put up brick walls between their departments, and don't cooperate on projects. Here is a graph of response time for a step function of 10 volts. It illustrates the variances in response time for the overdamped, critically damped, and underdamped cases. The overdamped cases never oscillate, but they are very slow to get to 10 volts. The critically damped case gets to the 10 volt level as fast as possible without overshooting 10 volts. The underdamped cases hit the 10 volt level quickly but they also overshoot it, and take several oscillations for them to decay to the 10 volt level. In practice, significant overshoot of a set point in a manufacturing process is not desired. The same can be said for robots. It is usually considered worse than a slower response. Can you imagine the robotic your local spacebar serving you a drink. He overshoots the set point and punches you in the mouth, that wouldn't be too cool, would it? So solutions at critical damping or just barely underdamped tend to be the ones commonly used. This slide shows how the decay rate of the underdamped solution depends on both the system natural frequency and the damping ratio. The graph on the left side shows the case for a constant damping ratio and varying natural frequency. In this case, the higher the natural frequency, the higher the system stabilizes to zero output in response to a disturbance. The graph on the right side shows the case for a constant natural frequency and varying damping ratios. In this case, the higher the damping ratio, the less the system will overshoot the set point of one, but the longer it takes to initially get to one. Let's recap. Now, I've reminded you of the pain you went through in math classes, somewhere between algebra and differential equations. Sorry about that, I had to do it. This is a course on sensors and process control. In the next video, we will discuss the particulars of PID control, where sensors are used to keep production processes and machines operating in control.