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Class 4 Systems and Solutions

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1 st Order Systems

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Step Input Ramp Input Harmonic Input

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1 st Order Systems with Step Input

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1 st Order Systems with Step Input Solution by Integration Error Ratio

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1 st Order Systems with Step Input Error Ratio and Excitation Ratio Error: Output deviation from input Excitation: Output deviation from its initial value

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1 st Order Systems with Step Input Solution by Superposition

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1 st Order Systems with Step Input Solution by Laplace Transform

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Transfer Function >> num=1; >> den=[1 1]; >> sys = tf(num,den); >> step(sys) >> grid 1 st Order Systems with Unit Step Input and Unit Time Constant MATLAB Simulation by Transfer Function

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1 st Order Systems with Unit Step Input and Unit Time Constant MATLAB Simulation by Simulink

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1 st Order Systems with Ramp Input

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1 st Order Systems with Ramp Input Solution by Superposition

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1 st Order Systems with Ramp Input Steady State Error and Relative Error

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1 st Order Systems with Ramp Input Relative Input and Relative Excitation Relative Excitation Relative Input 1

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1 st Order Systems with Ramp Input Solution by Laplace Transform

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1 st Order Systems with Unit Ramp Input and Unit Time Constant MATLAB Simulation by Transfer Function MATLAB does not have a ‘ramp’ command to plot the ramp response of the system. However, note that the response, R st (s) of a system with transfer function G(s) to unit step input is R st (s) = G(s)/s, and its response to a unit ramp input is R rmp (s) = G(s)/s 2 = (G(s)/s)/s. Thus, the response of G(s) to unit ramp is equal to the response of H(s) = G(s)/s to unit step. We may use MATLAB’s ‘step’ command to obtain the ramp response of a system G(s) simply by obtaining the step response of H(s) = G(s)/s to unit step. >> num = [0 0 1]; >> den = [1 1 0]; >> t=0:0.1:5; >> sys = step(num,den,t); >> plot(t,sys,'o',t,t,'-') >> grid

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1 st Order Systems with Unit Step Input and Unit Time Constant MATLAB Simulation by Simulink

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1 st Order Systems with Harmonic Input

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1 st Order Systems with Harmonic Input Solution by Superposition ωtωt ϕ τAωτAω A KhKh

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1 st Order Systems with Harmonic Input Amplitude Ratio and Phase f(t) x(t) ϕ rara τωτω r a (τω) ϕ (τω)ϕ (τω)

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1 st Order Systems with Harmonic Input Solution by Complex Exponential ωtωt x A y z = Ae iωt Euler’s Identity ωtωt x A y z = Ae iωt -ωt z* = Ae -iωt Complex Conjugate Multiplication & Division Rules Power Rules

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Second Order Systems In the system shown, the input displacement, x i, will cause a deflection in the spring, and some time will be needed for the output displacement x o to reach the input displacement. m k c xixi xoxo

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Second Order Systems If m/k << 1 s 2 and c/k << 1 s, the system may be approximated as a zero order system with unity gain. If, on the other hand, m/k << 1 s 2, but c/k is not, the system may be approximated by a first order system. Systems with a storage and dissipative capability but negligible inertial may be modeled using a first-order differential equation. m k c xixi xoxo

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Example – Automobile Accelerometer Consider the accelerometer used in seismic and vibration engineering to determine the motion of large bodies to which the accelerometer is attached. The acceleration of the large body places the piezoelectric crystal into compression or tension, causing a surface charge to develop on the crystal. The charge is proportional to the motion. As the large body moves, the mass of the accelerometer will move with an inertial response. The stiffness of the spring, k, provides a restoring force to move the accelerometer mass back to equilibrium while internal frictional damping, c, opposes any displacement away from equilibrium. m k c xixi xoxo Piezoelectric crystal

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Zero-Order systems Can we model the system below as a zero-order system? If the mass, stiffness, and damping coefficient satisfy certain conditions, we may. m k c xixi xoxo

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First Order Systems Measurement systems that contain storage elements do not respond instantaneously to changes in input. The bulb thermometer is a good example. When the ambient temperature changes, the liquid inside the bulb will need to store a certain amount of energy in order for it to reach the temperature of the environment. The temperature of the bulb sensor changes with time until this equilibrium is reached, which accounts physically for its lag in response. In general, systems with a storage or dissipative capability but negligible inertial forces may be modeled using a first-order differential equation.

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1 st Order Systems with Step Input Error Ratio Excitation Ratio Note that the excitation ratio also represents the system response in case of x0=0 and K=1 Excitation ratio may also be called response ratio = current response / desired response

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Example 1 A bulb thermometer with a time constant τ =100 s. is subjected to a step change in the input temperature. Find the time needed for the response ratio to reach 90%

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Example 1 Solution A bulb thermometer with a time constant τ =100 s. is subjected to a step change in the input temperature. Find the time needed for the response ratio to reach 90%

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1 st Order Systems with Ramp Input

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1 st Order Systems with Harmonic Input

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The amplitude ratio, Ar(ω), and the corresponding phase shift, ϕ, are plotted vs. ωτ. The effects of τ and ω on frequency response are shown. For those values of ωτ for which the system responds with Ar near unity, the measurement system transfers all or nearly all of the input signal amplitude to the output and with very little time delay; that is, X will be nearly equal to F in magnitude and ϕ will be near zero degrees.

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1 st Order Systems with Harmonic Input At large values of ωτ the measurement system filters out any frequency information of the input signal by responding with very small amplitudes, which is seen by the small Ar(ω), and by large time delays, as evidenced by increasingly nonzero ϕ.

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1 st Order Systems with Harmonic Input Any equal product of ω and τ produces the same results. If we wanted to measure signals with high- frequency content, then we would need a system having a small τ. On the other hand, systems of large τ may be adequate to measure signals of low-frequency content. Often the trade-offs compete available technology against cost. dB = 20 log Ar(ω)

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1 st Order Systems with Harmonic Input The dynamic error,δ(ω), of a system is defined as δ(ω) = (X(ω) – F)/F δ(ω) = Ar(ω) –1 It is a measure of the inability of a system to adequately reconstruct the amplitude of the input signal for a particular input frequency. We normally want measurement systems to have an amplitude ratio at or near unity over the anticipated frequency band of the input signal to minimize δ(ω). As perfect reproduction of the input signal is not possible, some dynamic error is inevitable. We need some way to quantify this. For a first-order system, we define a frequency bandwidth as the frequency band over which Ar(ω) > 0.707; in terms of the decibel defined as dB = 20 log Ar(ω) This is the band of frequencies within which Ar(ω) remains above 3 dB

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Example 2 A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that a dynamic error of 2% is acceptable.

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Example 2. Solution A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that an absolute value for the dynamic error of 2% is acceptable. Accordingly, a sensor having a time constant of 6.4 ms or less will work.

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Example 2. Solution A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that an absolute value for the dynamic error of 2% is acceptable. Accordingly, a sensor having a time constant of 6.4 ms or less will work.

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2 nd Order Systems Example: Spring – mass damper RLC Circuits Accelerometers Mathematical Model:

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2 nd Order Systems with step input

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2 nd Order Systems with periodic input

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2 nd Order Systems with step input

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