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### ECE201Lect-21

Course: ECE 201, Spring 2011
School: ASU
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Word Count: 826

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Circuits Second-Order (7.3) ECE201 Lect-21 1 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. ECE201 Lect-21 2 Important Concepts The differential equation Forced and homogeneous solutions The...

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Circuits Second-Order (7.3) ECE201 Lect-21 1 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. ECE201 Lect-21 2 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-21 3 A 2nd Order RLC Circuit i (t) R vs(t) + L C The source and resistor may be equivalent to a circuit with many resistors and sources. ECE201 Lect-21 4 Applications Modeled by a 2nd Order RLC Circuit Filters A lowpass filter with a sharper cutoff than can be obtained with an RC circuit. ECE201 Lect-21 5 The Differential Equation i (t) + vr(t) R vs(t) + C vl(t) + L + vc(t) KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) ECE201 Lect-21 6 Differential Equation di (t ) 1 Ri (t ) + L + i ( x)dx = vs (t ) dt C - d i (t ) R di (t ) 1 1 dvs (t ) + + i (t ) = 2 dt L dt LC L dt 2 t ECE201 Lect-21 7 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: d i (t ) di (t ) 2 + 2 0 + 0 i (t ) = f (t ) 2 dt dt ECE201 Lect-21 8 2 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-21 9 The Particular Solution The particular (or forced) solution ip(t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then ip(t) is constant. If f(t) is sinusoidal, then ip(t) is sinusoidal. ECE201 Lect-21 10 The Complementary Solution The complementary (homogeneous) solution has the following form: ic (t ) = Ke st K is a constant determined by initial conditions. s is a constant determined by the coefficients of the differential equation. ECE201 Lect-21 11 Complementary Solution d Ke dKe 2 st + 2 0 + 0 Ke = 0 2 dt dt s Ke + 2 0 sKe + Ke = 0 2 st st 2 0 st 2 st st s + 2 0 s + = 0 2 2 0 ECE201 Lect-21 12 Characteristic Equation To find the complementary solution, we need to solve the characteristic equation: s + 2 0 s + = 0 2 2 0 The characteristic equation has two rootscall them s1 and s2. ECE201 Lect-21 13 Complementary Solution Each root (s1 and s2) contributes a term to the complementary solution. The complementary solution is (usually) ic (t ) = K1e + K 2 e s1t s2t ECE201 Lect-21 14 Important Concepts The differential equation Forced and homogeneous solutions natural The frequency and the damping ratio ECE201 Lect-21 15 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. ECE201 Lect-21 16 Roots of the Characteristic Equation The roots of the characteristic equation determine whether the complementary solution wiggles. s1 = - 0 + 0 - 1 2 s2 = - 0 - 0 - 1 2 ECE201 Lect-21 17 Real Unequal Roots If > 1, s1 and s2 are real and not equal. - + 2 -1 t 0 0 - - 2 -1 t 0 0 ic (t ) = K 1e + K 2e This solution is overdamped. ECE201 Lect-21 18 Overdamped 1 0.8 0.8 0.6 i(t) 0.4 0.2 0 -1.00E-06 i(t) t 0.6 0.4 0.2 0 -1.00E-06 -0.2 t ECE201 Lect-21 19 Complex Roots If < 1, s1 and s2 are complex. Define the following constants: = 0 d = 0 1 - ic (t ) = e -t 2 ( A1 cos d t + A2 sin d t ) ECE201 Lect-21 20 This solution is underdamped. Underdamped 1 0.8 0.6 0.4 0.2 0 -1.00E-05 -0.2 -0.4 -0.6 -0.8 -1 1.00E-05 3.00E-05 i(t) t ECE201 Lect-21 21 Real Equal Roots If = 1, s1 and s2 are real and equal. ic (t ) = K 1e - 0t + K 2 te - 0 t This solution is critically damped. ECE201 Lect-21 22 Example i (t) 10 vs(t) + 769pF 159H This is one possible implementation of the filter portion of the IF amplifier. ECE201 Lect-21 23 More of the Example d i (t ) R di (t ) 1 1 dvs (t ) + + i (t ) = 2 dt L dt LC L dt d i (t ) di (t ) 2 + 2 0 + 0 i (t ) = f (t ) 2 dt dt For the example, what are and 0? ECE201 Lect-21 24 2 2 Even More Example = 0.011 0 = 2455000 Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-21 25 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. 1 0.8 0.6 0.4 0.2 0 -1.00E-05 -0.2 -0.4 -0.6 -0.8 -1 1.00E-05 3.00E-05 i(t) t ECE201 Lect-21 26 Slightly Different Example i (t) 1k vs(t) + 769pF 159H Increase the resistor to 1k What are and 0? ECE201 Lect-21 27 More Different Example = 2.2 0 = 2455000 Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-21 28 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. 1 0.8 i(t) 0.6 0.4 0.2 0 -1.00E-06 t ECE201 Lect-21 29 Damping Summary >1 =1 Roots (s1, s2) Damping Real and unequal Overdamped Real and equal Critically damped Underdamped ECE201 Lect-21 30 0<<1 Complex Class Example Learning Extension E7.9 ECE201 Lect-21 31
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