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What are the general forms of first and second order dynamic systems? 
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#1
Apr1114, 05:15 PM

P: 198

What is the general form of a first and second order dynamic system, in laplace and time space?
I have some of the information, but a lot of it is missing and I can't find a clear answer on the web. The general form of a second order dynamic system is: $$\frac{d^2x(t)}{dt^2}+2\zeta\omega_n\frac{dx(t)}{dt}+\omega_n^2x(t)=f( t)\:\:\:\:\:\:\:[1]$$ where x(t)  is the output, e.g current f(t)  is the input, e.g a voltage signal zeta  is the damping coefficient wn  is the natural frequency An example of a second order dynamic system is a RLC circuit. If a resistor, capacitor, and inductor are connected to a power source with voltage at time t equal to f(t), the summed voltage across the three components always equals f(t) at any time t. $$Ri(t)+\frac{1}{C}\int_0^ti(t)dt+L\frac{d\,i(t)}{d\,t}=f(t)\:\:\:\:\:\ :[2]$$ So for example if f(t) is a step input from 0 to 6V, the equation will be: $$Ri(t)+\frac{1}{C}\int_0^ti(t)dt+L\frac{d\,i(t)}{d\,t}=6\:\:\:\:\:\:\:[3]$$ If this equation is rearranged so that it is in the general form shown in equation [1], then you will be able to find out the natural frequency of the system (omega_n) and the damping coefficient of the system (zeta). The general form of a first order dynamic system in laplace space is: $$G(s)=\frac{k(s+a)}{s+b}\:\:\:\:\:\:\:[4]$$ where G(s)  is the transfer function (output/input) k  is the "gain" of the system a  is the zero of the system, which partially determines transient behavior b  is the pole of the system, which determines stability and settling time An example of a first order dynamic system is a RC circuit. The voltage across the resistor plus the voltage across the capacitor equal the voltage of the source f(t) at any time t. $$Ri(t)+\int_0^ti(t)dt=f(t)\:\:\:\:\:\:\:[5]$$ So again, for a step input from 0 to 6V, the equation will be: $$Ri(t)+\int_0^ti(t)dt=6\:\:\:\:\:\:\:[6]$$ In laplace space this is $$RI(s)+\frac{1}{Cs}I(s)=\frac{6}{s}\:\:\:\:\:\:\:[7]$$ rearrange to general form: $$I(s)*(Rs+\frac{1}{c})=6\:\:\:\:\:\:\:[8]$$ $$I(s) =\frac{6}{Rs+\frac{1}{c}}\:\:\:\:\:\:\:[9]$$ $$I(s) =\frac{\frac{6}{R}}{s+\frac{1}{Rc}}\:\:\:\:\:\:\:[10]$$ Since the input was the step input 6, you can see that the steady state gain is equal to 6/R, the pole is equal to 1/RC, and there are no zeros. You could do the inverse laplace transform of this equation to find out the instantaneous current i(t) at any time t. Now my questions:  I have shown the general form of a first order dynamic system in laplace space, and the general form of a second order dynamic system in time space, but what is the general form of a first order system in time space, and a second order system in laplace space? I have definitely seen these written somewhere but I can't find them in my notes or online.  How would you go from equation [3], to the general form (time space) as shown in equation [1]? Any answer is much appreciated, even if it only answers one of the questions, or partially answers a question. Thanks for reading! 


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