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Unformatted text preview: ME375 Handouts Electrical Systems • Basic Modeling Elements • Interconnection Relationships • Derive Input/Output Models School of Mechanical Engineering Purdue University ME375 Electrical - 1 Variables • • • • • • • • q : charge [C] (Coulomb) i : current [A] e : voltage [V] R : resistance [Ω] C : capacitance [Farad] L : inductance [H] (Henry) p : power [Watt] w : work ( energy ) [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 to the through current. eR = R i + ⇔ eR i= – Charge collected is proportional 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. 1 2 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 through current. eL = L + i F d iI H dt K eL − e(t) + − • Current Source L – 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. – Maintain specified current, regardless of the required voltage. 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 zero. ∑e j =0 • Kirchhoff’s Current Law Kirchhoff’ – The algebraic sum of the currents at any node in the 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 • 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. 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 In Class Exercise 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. + + eR − R + i Element Laws: eL − L + C ei(t) No. of Unknowns: eC _ _ Simplify: School of Mechanical Engineering Purdue University ME375 Electrical - 7 Loop Approach vs. Node Approach • Loop Approach: – Identify pertinent current loops and label the currents – Write elemental voltages in terms of these currents • Node Approach: – Identify pertinent voltage nodes and label these voltages – Identify and label all distinct currents – Write elemental equations in terms of these voltages and currents School of Mechanical Engineering Purdue University ME375 Electrical - 8 4 ME375 Handouts Example – Loop Approach Obtain the I/O model for the following circuit. The input is the voltage ei(t) of the voltage source and the through current of the inductor is the output. + + eL − eR1 − + R1 + L eR2 R2 ei(t) i + C eC _ _ _ Input: Input: Output: Output: School of Mechanical Engineering Purdue University ME375 Electrical - 9 Example – Node Approach Obtain the I/O model for the following circuit. The input is the voltage ei(t) of the voltage source and the through current of the inductor is the output. + ei(t) R1 L R2 i C _ Input: Input: Output: Output: School of Mechanical Engineering Purdue University ME375 Electrical - 10 5 ME375 Handouts 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 current of the inductor is the output. + ei(t) R1 L R2 i C _ Input: Input: Output: Output: School of Mechanical Engineering Purdue University ME375 Electrical - 11 6 ...
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