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## Chapter 9

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**Chapter 9**RLC Circuits Dr. Iyad Jafar**RLC Circuits**• Analyze circuits containing R, L and C together • Series RLC • Parallel RLC • Similar to RL and RC circuits, RLC circuits has two parts • Source free or natural response • Forced / steady state response • Forced Response a step input causes a step output. • Natural Response Different and more difficult than RL, RC.**Second-order**Differential equation Source-free RLC Circuits We study the natural response by studying source-free RLC circuits. Parallel Source-free RLC Circuit**This second-order differential equation can be solved by**assuming solutions The solution should be in form of If the solution is good, then substitute it into the equation will be true. which means s=??**Use quadratic formula, we got**Both and are solution to the equation Therefore, the complete solution is**From**Define resonant frequency Damping factor Therefore, in which we divide into 3 cases according to the term inside the bracket**Solution to Second-order Differential Equations**• α > ω0 (inside square root is a positive value) Overdamped case • α = ω0 (inside square root is zero) Critical damped case • α < ω0 (inside square root is a negative value) Underdamped case**1. Overdamped case , α > ω0**Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A α > ω0 ,therefore, this is an overdamped case s1 = -1, s2 = -6**Then, we will use initial conditions to find A1, A2**From vc(0) = 0 we substitute t=0 … (1) From KCL At t =0+ … (2)**Solve the equations (1) and (2)**A1 = 84 A2 = -84 and the solution is v(t) t**2. Critical damped case , α = ω0**Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A α = ω0 , this is an critical damped case s1 = s2 = -2.45 The complete solution of this case is in form of**Then, we will use initial conditions to find A1, A2**From vc(0) = 0 we substitute t=0 … (1) Therefore A2 =0 and the solution is reduced to Find A1 from KCL at t=0 … (2)**Solve the equation and we got A1 = 420 and the solution is**v(t) t**3. Underdamped case , α < ω0**from The term inside the bracket will be negative and s will be a complex number define Then and**3. Underdamped case , α < ω0**Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A α < ω0 ,therefore, this is an underdamped case and v(t) is in form where**Then, we will use initial conditions to find B1, B2**From vc(0+) = vc(0-) = 0 we substitute t=0 Therefore B1 =0 and the solution is reduced to Find B2 from KCL at t=0**Series RLC**Overdamped Critical damped Underdamped Solution**Forced RLC Circuits**Similar to RL and RC circuits, the total response is the sum of the transient/natural response and the forced/steady state response X(t) = Xn(t) + Xf(t) Where Xn is one of the three cases Overdamped Critical damped Underdamped And Xf(t) is the value as t ∞ and the constants are found from the initial conditions**Example**Find vc(t) This is overdamped case, so the solution is in form**Consider the circuit we found that the initial conditions**will be • vC(0) = 150 V • iL(0+) = iL(0-) = 5 A • And the steady state values • vC(∞) = 150 V • iL(∞) = 9 A Using vC(0) = 150 V , we got Using vC(∞) = 150 V , we got Therefore, 1. Vf = 150 2. A1+A2 = 0 ------------(1)**Use the initial condition iL(0) = 5 A, we have to change**vc(t) to iL(t) From Next, use KCL on the circle below or**From**Substitute iL(0) = 5A ------------------(2) From (1), (2) A1 = 13.5, A2 = -13.5 Therefore,