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ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001

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ECE201 Lect-232 2nd Order Circuits Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2. Any voltage or current in such a circuit is the solution to a 2nd order differential equation.

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ECE201 Lect-233 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

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ECE201 Lect-234 A 2nd Order RLC Circuit The source and resistor may be equivalent to a circuit with many resistors and sources. R Cv s (t) i (t) L +–+–

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ECE201 Lect-235 Applications Modeled by a 2nd Order RLC Circuit Filters –A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.

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ECE201 Lect-236 The Differential Equation KVL around the loop: v r (t) + v c (t) + v l (t) = v s (t) R Cv s (t) + – v c (t) + – v r (t) L +– v l (t) i (t) +–+–

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ECE201 Lect-237 Differential Equation

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ECE201 Lect-238 The Differential Equation Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:

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ECE201 Lect-239 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

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ECE201 Lect-2310 The Particular Solution The particular (or forced) solution i p (t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then i p (t) is constant. If f(t) is sinusoidal, then i p (t) is sinusoidal.

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ECE201 Lect-2311 The Complementary Solution The complementary (homogeneous) solution has the following form: K is a constant determined by initial conditions. s is a constant determined by the coefficients of the differential equation.

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ECE201 Lect-2312 Complementary Solution

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ECE201 Lect-2313 Characteristic Equation To find the complementary solution, we need to solve the characteristic equation: The characteristic equation has two roots- call them s 1 and s 2.

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ECE201 Lect-2314 Complementary Solution Each root (s 1 and s 2 ) contributes a term to the complementary solution. The complementary solution is (usually)

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ECE201 Lect-2315 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

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ECE201 Lect-2316 Damping Ratio ( ) and Natural Frequency ( 0 ) The damping ratio is. The damping ratio determines what type of solution we will get: –Exponentially decreasing ( >1) –Exponentially decreasing sinusoid ( < 1) The natural frequency is 0 –It determines how fast sinusoids wiggle.

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ECE201 Lect-2317 Roots of the Characteristic Equation The roots of the characteristic equation determine whether the complementary solution wiggles.

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ECE201 Lect-2318 Real Unequal Roots If > 1, s 1 and s 2 are real and not equal. This solution is overdamped.

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ECE201 Lect-2319 Overdamped

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ECE201 Lect-2320 Complex Roots If < 1, s 1 and s 2 are complex. Define the following constants: This solution is underdamped.

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ECE201 Lect-2321 Underdamped

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ECE201 Lect-2322 Real Equal Roots If = 1, s 1 and s 2 are real and equal. This solution is critically damped.

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ECE201 Lect-2323 Example This is one possible implementation of the filter portion of the IF amplifier. 10 769pFv s (t) i (t) 159 H +–+–

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ECE201 Lect-2324 More of the Example For the example, what are and 0 ?

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ECE201 Lect-2325 Even More Example = 0.011 0 = 2 455000 Is this system over damped, under damped, or critically damped? What will the current look like?

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ECE201 Lect-2326 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current.

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ECE201 Lect-2327 Slightly Different Example Increase the resistor to 1k What are and 0 ? 1k 769pFv s (t) i (t) 159 H +–+–

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ECE201 Lect-2328 More Different Example = 2.2 0 = 2 455000 Is this system over damped, under damped, or critically damped? What will the current look like?

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ECE201 Lect-2329 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current.

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ECE201 Lect-2330 Damping Summary

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ECE201 Lect-2331 Class Examples Learning Extension E6.9 Learning Extension E6.10

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