Electrical System

Electrical System - ME375 Handouts Electrical Systems •...

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Unformatted text preview: ME375 Handouts Electrical Systems • Basic Modeling Elements Modeling Elements • Interconnection Relationships • Derive Input/Output Models School of Mechanical Engineering Purdue University ME375 Electrical - 1 Variables • • • • • • • • q : charge [C] (Coulomb) [C] i : current [A] [A] e : voltage [V] [V] R : resistance [Ω] C : capacitance [Farad] [Farad] L : inductance [H] (Henry) [H] p : power [Watt] power [Watt] w : work ( energy ) [J] [J] 1 [J] (Joule) = 1 [V-A-sec] [V- d q=i dt q (t1 ) = q (t0 ) + p = e⋅i w (t1 ) = w (t0 ) + School of Mechanical Engineering Purdue University = w ( t0 ) + z z z t1 t0 i(t ) dt t1 t0 t1 t0 p(t ) dt ( e ⋅ i ) dt ME375 Electrical - 2 1 ME375 Handouts Basic Modeling Elements • Resistor • Capacitor – Ohms Law Voltage across is proportional Voltage across is proportional to to the through current. eR = R i + ⇔ eR i= − – Charge collected is proportional to the voltage across to the voltage across. – Current is proportional to the rate of change of the voltage across. 1 eR R q = C eC i=C Fde I H dt K C + eC − R i ⇔ i – Dissipates energy through heat. 12 p= Ri = e R C – Energy supplied is stored in its electric field and can affect future circuit response. – Analogous to in mechanical systems. 2 – Analogous to friction elements in mechanical systems, e.g. dampers School of Mechanical Engineering Purdue University ME375 Electrical - 3 Basic Modeling Elements • Inductor • Voltage Source – Voltage across is proportional to the rate of the change of the the rate of the change of the through through current. eL = L + i F d iI H dt K eL − e(t) + − • Current Source L – Maintain specified current, regardless of the required voltage. – Energy supplied is stored in its magnetic field. w= – Maintain specified voltage across two points, regardless of the required current. 1 L i2 2 – Analogous to mechanical systems. in i(t) School of Mechanical Engineering Purdue University ME375 Electrical - 4 2 ME375 Handouts Interconnection Laws • Kirchhoff's Voltage Law – The total voltage drop along any closed loop in the circuit is th zero. ∑e j =0 • Kirchhoff’s Current Law – The algebraic sum of the currents at any node in the th circuit is zero. Closed Loop ∑i j =0 Any Node School of Mechanical Engineering Purdue University ME375 Electrical - 5 Modeling Steps • Understand System Function and Identify Input/Output Variables Input/Output Variables • Draw Simplified Schematics Using Basic Elements • Develop Mathematical Model – – – – Label Each Element and the Corresponding Voltages. Label Each Node and the Corresponding Currents. Apply Interconnection Laws. Interconnection Laws Check that the Number of Unknown Variables equals the Number of Equations – Eliminate Intermediate Variables to Obtain Standard Forms. School of Mechanical Engineering Purdue University ME375 Electrical - 6 3 ME375 Handouts Example I Derive the I/O model for the following circuit. Let voltage ei(t) be the input and the voltage across the capacitor be the output. + + ei(t) _ eR − R + i Element Laws: eL − L + C EOM: eC _ School of Mechanical Engineering Purdue University ME375 Electrical - 7 Complex Impedance School of Mechanical Engineering Purdue University ME375 Electrical - 8 4 ME375 Handouts Example II Obtain the I/O model for the following circuit. The input is the voltage ei(t) of the voltage source and the through the voltage source and the through current current of the inductor is the output. + R1 L R2 ei(t) i C _ Input: Output: School of Mechanical Engineering Purdue University ME375 Electrical - 9 Example – Complex Impedance Obtain the I/O model for the following circuit. The input is the voltage ei(t) of the voltage source and the through the voltage source and the through current current of the inductor is the output. + ei(t) R1 L R2 i C _ Input: Output: School of Mechanical Engineering Purdue University ME375 Electrical - 10 5 ME375 Handouts Parallel Connection of Elements School of Mechanical Engineering Purdue University ME375 Electrical - 11 Series Connection of Elements School of Mechanical Engineering Purdue University ME375 Electrical - 12 6 ...
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