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Second-Order Circuits (7.3) Dr. Holbert April 19, 2006 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) + – C L • 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) + – + vc(t) C – vl(t) + – L KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) ECE201 Lect-21 6 Differential Equation t di (t ) 1 Ri (t ) L i ( x)dx vs (t ) dt C 2 d i(t ) R di(t ) 1 1 dvs (t ) i(t ) 2 dt L dt LC L dt 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: 2 d i(t ) di(t ) 2 2 0 0 i(t ) f (t ) 2 dt dt ECE201 Lect-21 8 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 2 st st 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 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 ECE201 Lect-21 s2t 14 Important Concepts • The differential equation • Forced and homogeneous solutions • The natural 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. ic (t ) K1e 2 1 t 0 0 K 2e 2 1 t 0 0 • This solution is overdamped. ECE201 Lect-21 18 1 0.8 0.8 0.6 0.6 i(t) i(t) Overdamped 0.4 0.4 0.2 0.2 0 -1.00E-06 -0.2 0 -1.00E-06 t 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 • This solution is underdamped. ECE201 Lect-21 20 Underdamped 1 0.8 0.6 i(t) 0.4 0.2 0 -1.00E-05 -0.2 1.00E-05 3.00E-05 -0.4 -0.6 -0.8 -1 t ECE201 Lect-21 21 Real Equal Roots • If = 1, s1 and s2 are real and equal. ic (t ) K1e 0t K 2 te 0t • This solution is critically damped. ECE201 Lect-21 22 Example i (t) 10W vs(t) + – 769pF 159mH • This is one possible implementation of the filter portion of the IF amplifier. ECE201 Lect-21 23 More of the Example 2 d i(t ) R di(t ) 1 1 dvs (t ) i(t ) 2 dt L dt LC L dt 2 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 Even More Example • = 0.011 • 0 = 2p455000 • 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 i(t) 0.4 0.2 0 -1.00E-05 -0.2 1.00E-05 3.00E-05 -0.4 -0.6 -0.8 -1 t ECE201 Lect-21 26 Slightly Different Example i (t) 1kW vs(t) + – 769pF 159mH • Increase the resistor to 1kW • What are and 0? ECE201 Lect-21 27 More Different Example • = 2.2 • 0 = 2p455000 • 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 i(t) 0.8 0.6 0.4 0.2 0 -1.00E-06 t ECE201 Lect-21 29 Damping Summary ζ Roots (s1, s2) Damping ζ>1 Real and unequal Overdamped ζ=1 Real and equal 0<ζ<1 Complex Critically damped Underdamped ECE201 Lect-21 30 Class Example • Learning Extension E7.9 ECE201 Lect-21 31