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First Order Systems: Dynamic Systems ISAT 300 Spring 1999

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Static vs. Dynamic Static means that the system doesnt change with time Dynamic means that the system is changing with time

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Example Static System: Force Transducer

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Example Dynamic System: Cooling of a Cake

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Example Dynamic System: Bacteria Growth

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Mathematics of Bacterial Growth The change in the number of bacteria at any time Is proportional to the number of bacteria present at any time Proportionality Constant The equation is derived from the concept of Conservation of Mass (Dont confuse with K the sensitivity)

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Solve the differential equation for bacteria growth

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Why are 1st order equations important for instrumentation Many instruments exhibit a 1st order response Provides a parameter, called the time constant, for choosing an instrument. This parameter lets you know if the instrument will respond quick enough to capture changes in the system.

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Time Constant The parameter for characterizing the response time of an instrument is the Time Constant,

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Time Constant The Time Constant is the time it takes a first order system to reach 63.3% (0.633) of its final value in response to a step change in the system Time Output

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Time Output System Behavior Instrument Response

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Anatomy of a Differential Equation Homogeneous Diff. Eq. Initial Condition Homogeneous Solution (General Solution)

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Anatomy of a Diff. Eq. Contd. Non Homogeneous Diff. Eq. Forcing Function Initial Condition Complete Solution General+Particular

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Anatomy of a Diff. Eq. Contd. Apply Initial Condition to Complete Solution

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Types of Forcing Functions (inputs) Step Ramp Solution

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Types of Forcing Functions (inputs) Sinusoidal Solution

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