System Transfer Functions and Responses: Exercises Solutions
Explore solutions for system transfer function problems including step responses and Laplace transforms with detailed explanations and diagrams.
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Exercise 1 Find the step response for a system whose transfer function is: Solution the step response for a system whose transfer function is: R s =1 Input: unit step ? ? ? = R s ? ? =1 ? 1 Output (System Response): = ? + 4 ? + 8 ? ? + 4)(? + 8 1 =1 ? = (? + 4)? ? ?= 4= Using Partial fraction: ? + 8 4 ? ? ? ? = ? + 4+ ?= 4 ? + 8 1 4 ? + 4 1 4 1 = 1 Using Inverse Laplace Transform: ? ? = ? = (? + 8)? ? ?= 8= ? + 8 ? + 4 4 ? ? =? The time response of the system: ?= 8 ? ?? ? ?? ?
Exercise 2 For the following differential equation, find the solution c(t) if r(t) is a unit step function. Assume all initial conditions are zeros. ) ?2?(? ??2 ?? ) ??(? + 2 + 5? ? = 3?(?) Solution Using Laplace Transform: ?2? ? + 2 ? ? ? + 5? ? = 3?(?) (?2+ 2 ? + 5)? ? = 3?(?) ? ? ? ? 3 3 = ) ? ? = ?2+ 2? + 5 ?(? ?2+ 2? + 5 3 R s =1 = 16 < 0 ??????? ????? Input: unit step Output (System Response): ? ? = ?(?2+ 2? + 5 ) ? ? ? =? ?? + ? (?2+2? + 5) Using Partial fraction: ?+ ? = 1.2 ? = 0.6 ? = 0.6 ? = 1 ??? ? = 2 ?(?2+ 2? + 5 ? ? = ? (?2+2? + 5 + ??2+ ?? = 3 ? + 0.6 = 0 ?2+ 2? + 5 = (? + 1)2+(2)2 ) ) (?2+2? + 5 + ??2+ ?? = 3 ) 0.6 ? + 2 = ? ? + ? + ?? ? = 1 ??? ? = 0.5 ? + 1.2 = 0 (?2+2? + 5)= 0.6? ? + ? + ?? ? + 2 ILT: 0.6 0.6(?? a?????? + ?? a??????) (? + ?)2+(?)2 The time response of the system: ? ? = ?.? ?.?? ?????? + ?.?????? ?(? )
Exercise 3 Consider the following electrical system, Find the transfer function ? ? =?0(?) ??(?) Solution Mesh equation for this circuit is in the above equation. Substitute, the current passing through capacitor Apply Laplace transform on both sides.
Exercise 4 Consider the inverting amplifier, Find the transfer function ?? ?? ?1= 1? ,?1= 1?F, ?2= 10? ,?1= 0.1?F, Solution the transfer function ?1?=?1?1? + 1 1 =? + 1 10 6? ?0(?) ?1(?)=?1+ ?2 ?1= ?1+ ?1? ?1 107 ? + 1 ?2 ?2= ?2?2? + 1= ? + 1 10 6?+107 ? + 1 10 6? (? + 1)2=?2+ 3? + 1 ?0(?) ?1(?)=?1+ ?2 ? ? + 1 = = 1 + (? + 1)2 ?1
Exercise 5 Consider the mechanical system shown in Figure, find: 1. Block diagram of the mechanical system (? ? :?????,?(?):??????). 2. State space representation of the system (state variables: ?1= ? ? ,?2= ? ? ) Solution From the diagram, the system equation (SISO) is: ?1= ? ? ??? ?2= ? ? = ?1 This system is of second order, state variables: ??? The output equation is the outputs of the integrators are state variables. Block diagram of the mechanical system
Exercise 6 Consider the system shown in Figure. Simplify this diagram Solution
