Presentation on theme: "ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001."— Presentation transcript:
ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001
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.
ECE201 Lect-233 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio
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 +–+–
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.
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) +–+–
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:
ECE201 Lect-239 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio
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.
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.
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.
ECE201 Lect-2314 Complementary Solution Each root (s 1 and s 2 ) contributes a term to the complementary solution. The complementary solution is (usually)
ECE201 Lect-2315 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio
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.
ECE201 Lect-2317 Roots of the Characteristic Equation The roots of the characteristic equation determine whether the complementary solution wiggles.
ECE201 Lect-2318 Real Unequal Roots If > 1, s 1 and s 2 are real and not equal. This solution is overdamped.