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Reading Assignment: Chapter 8 in Electric Circuits, 9 th Ed. by Nilsson 1 Chapter 8 EGR 272 – Circuit Theory II 2 nd -order circuits have 2 independent energy storage elements (inductors and/or capacitors) Analysis of a 2 nd -order circuit yields a 2 nd -order differential equation (DE) A 2nd-order differential equation has the form: Solution of a 2 nd -order differential equation requires two initial conditions: x(0) and x(0) All higher order circuits (3 rd, 4 th, etc) have the same types of responses as seen in 1 st -order and 2 nd -order circuits

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Series RLC and Parallel RLC Circuits Since 2nd-order circuits have two energy-storage types, the circuits can have the following forms: 1) Two capacitors 2) Two inductors 3) One capacitor and one inductor A) Series RLC circuit * B) Parallel RLC circuit * C) Others * The textbook focuses on these two types of 2nd-order circuits 2 Chapter 8 EGR 272 – Circuit Theory II Series RLC circuitParallel RLC circuit

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Form of the solution to differential equations As seen with 1 st -order circuits in Chapter 7, the general solution to a differential equation has two parts: where x h or x n is due to the initial conditions in the circuit and x p or x f is due to the forcing functions (independent voltage and current sources for t > 0). The forced response The forced response is due to the independent sources in the circuit for t > 0. Since the natural response will die out once the circuit reaches steady-state, the forced response can be found by analyzing the circuit at t =. In particular, x f = x( ) 3 Chapter 8 EGR 272 – Circuit Theory II x(t) = x h + x p = homogeneous solution + particular solution or x(t) = x n + x f = natural solution + forced solution

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The natural response A 2 nd -order differential equation has the form: where x(t) is a voltage v(t) or a current i(t). To find the natural response, set the forcing function f(t) (the right-hand side of the DE) to zero. Substituting the general form of the solution Ae st yields the characteristic equation: s 2 + a 1 s + a o = 0 Finding the roots of this quadratic (called the characteristic roots or natural frequencies) yields: 4 Chapter 8 EGR 272 – Circuit Theory II

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Characteristic Roots The roots of the characteristic equation may be real and distinct, repeated, or complex. Thus, the natural response to a 2 nd -order circuit has 3 possible forms: 1) Overdamped response Roots are real and distinct [ (a 1 ) 2 > 4a o ] Solution has the form: Sketch the form of the solution. Discuss the concept of the dominant root. 5 Chapter 8 EGR 272 – Circuit Theory II

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2) Critically damped response Roots are repeated [ (a 1 ) 2 = 4a o ] so s 1 = s 2 = s = -a 1 /2 Solution has the form: Sketch the form of the solution. 6 Chapter 8 EGR 272 – Circuit Theory II

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3) Underdamped response Roots are complex [ (a 1 ) 2 < 4a o ] so s 1, s 2 = j Show that the solution has the form: Sketch the form of the solution. Discuss the concept of the exponential envelope. Sketch x n if A 1 = 0, A 2 = 10, =-1, and =. Sketch x n if A 1 = 0, A 2 = 10, =-10, and = 100. 7 Chapter 8 EGR 272 – Circuit Theory II

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Illustration: The transient response to a 2 nd -order circuit must follow one of the forms indicated above (overdamped, critically damped, or underdamped). Consider the circuit shown below. v(t) is 0V for t < 0 and the steady-state value of v(t) is 10V. How does it get from 0 to 10V? Discuss the possible responses for v(t) Define the terms damping, rise time, ringing, and % overshoot 8 Chapter 8 EGR 272 – Circuit Theory II

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Examples: When is each of the 3 types of responses desired? Discuss the following cases: An elevator A cruise-control circuit The output of a logic gate The start up voltage waveform for a DC power supply 9 Chapter 8 EGR 272 – Circuit Theory II

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Series and Parallel RLC Circuits Two common second-order circuits are now considered: series RLC circuits parallel RLC circuits. Relationships for these circuits can be easily developed such that the characteristic equation can be determined directly from component values without writing a differential equation for each example. A general 2nd-order differential equation has the form: A general 2nd-order characteristic equation has the form: where = damping coefficient w o = resonant frequency 10 Chapter 8 EGR 272 – Circuit Theory II s 2 + 2 s + w o 2 = 0

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Series RLC Circuit - develop expressions for and w o 11 Chapter 8 EGR 272 – Circuit Theory II

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Parallel RLC Circuit - develop expressions for and w o 12 Chapter 8 EGR 272 – Circuit Theory II

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Procedure for analyzing 2 nd -order circuits (series RLC and parallel RLC) 1.Find the characteristic equation and the natural response A)Is the circuit a series RLC or parallel RLC? (for t > 0 with independent sources killed) B)Find and w o 2 and use these values in the characteristic equation: s 2 + 2 s + w o 2. C)Find the roots of the characteristic equation (characteristic roots or natural frequencies). D)Determine the form of the natural response based on the type of characteristic roots: a)Overdamped: Real, distinct roots (s 1 and s 2 ): b)Underdamped: Complex roots (s 1,s 2 = j ): c)Critically damped: Repeated roots (s=s 1 =s 2 ): 2.Find the forced response - Analyze the circuit at t = to find x f = x( ). 3.Find the initial conditions, x(0) and x(0). A)Find x(0) by analyzing the circuit at t = 0 - (find all capacitor voltages and inductor currents) B)Analyze the circuit at t = 0 + (using v C (0) and i L (0) from step 3B) and find: 4.Find the complete response A)Find the total response, x(t) = x n + x f. B)Use the two initial conditions to solve for the two unknowns in the total response. 13 Chapter 8 EGR 272 – Circuit Theory II

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Example: Determine v(t) in the circuit shown below for t > 0 if : A) R = 7 B) R = 2 C) R = 14 Chapter 8 EGR 272 – Circuit Theory II

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Example: B) Continued with R = 2 15 Chapter 8 EGR 272 – Circuit Theory II

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Example: C) Continued with R = 16 Chapter 8 EGR 272 – Circuit Theory II

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Example: Determine v(t) in the circuit shown below for t > 0. 17 Chapter 8 EGR 272 – Circuit Theory II

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Example: Determine v(t) in the circuit shown below for t > 0. Note: In determining if a circuit is a series RLC or parallel RLC circuit, consider the circuit for t > 0 with all independent sources killed. 18 Chapter 8 EGR 272 – Circuit Theory II

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Example: Determine i(t) in the circuit shown below for t > 0. 20H 19 Chapter 8 EGR 272 – Circuit Theory II

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