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EET108 Circuit Theory

EET108 Circuit Theory. Second-Order Circuits. Second-Order Circuits. Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC. Examples of Second Order RLC circuits.

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EET108 Circuit Theory

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  1. EET108 Circuit Theory Second-Order Circuits

  2. Second-Order Circuits Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC

  3. Examples of Second Order RLC circuits What is a 2nd order circuit? A second-order circuit is characterized by a second-order differential equation.It consists of resistorsand the equivalent of two energy storage elements. RLC series RLC parallel

  4. Second-Order Circuits Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC

  5. Source-Free Series RLC Circuits • The solution of the source-free series RLC circuit is called as the natural responseof the circuit. • The circuit is excitedby the energy initially stored in the capacitor and inductor. • At t=0; • Applying KVL around the loop; and

  6. The 2nd order of expression Source-Free Series RLC Circuits • To eliminate integral, differentiate with respect to t and rearrange the terms; • To solve 2nd order differential equation require the initial value; At t=0

  7. Source-Free Series RLC Circuits The root of the quadratic equation ax2+bx+c=0 are Let i=Aest where A and s are constants to be determined; The two roots are (using quadratic formula);

  8. Source-Free Series RLC Circuits The roots s1 and s2 are called natural frequencies measured in nepers per second (Np/s) ωo is known as the resonant frequency or undamped natural frequency express in radians per second (rad/s) is the neper frequency or the damping factor expressed in neper per second (Np/s)

  9. Source-Free Series RLC Circuits The two values of s (s1 and s2) indicate that there are two possible solutions for i; Thus, the natural response of the series RLC circuit is; ***The constant A1 and A2 are determined from the initial values i(0) and di(0)/dt

  10. where General 2nd order Form Source-Free Series RLC Circuits There are three possible solutions for the following 2nd order differential equation: => • In series RLC circuit, the current is consider because the current is equal for resistor, inductor and capacitor in series circuit • The types of solutions for i(t) depend on the relative values of α and ωo.

  11. 1.If a > wo, over-damped case where 2. If a = wo, critical damped case where 3.If a < wo, under-damped case where Source-Free Series RLC Circuits There are three possible solutions for the following 2nd order differential equation:

  12. Source-Free Series RLC Circuits (5) Example 1 If R = 10 Ω, L = 5 H, and C = 2 mF in 8.8, find α, ω0, s1 and s2. What type of natural response will the circuit have? Ans: α = 1, ω0 = 10, s= -1 ±j9.95 Underdamped

  13. Example 2 In figure below, R=40Ω, L=4H and C=0.25F. Calculate the characteristic roots of the circuit. What is the natural response?? Ans: overdamped, s1=-0.101 s2=-9.899

  14. Source-Free Series RLC Circuits Example 1 The circuit shown below has reached steady state at t = 0-. If the make-before-break switch moves to position b at t = 0, calculate i(t) for t > 0.

  15. Step 2 Find the i(0) and v(0). Refer to the circuit at t<0 Step 1 Draw the circuit at t<0

  16. Step 3 Find by using equation Step 4 Draw the circuit at t>0. Find and

  17. Step 5 If > , the case is over- damped Use general equation for over-damped case. where Step 6 Find the value of A1 and A2.

  18. Source-Free Series RLC Circuits Exercise 1 Determine i(t) at t>0, if • L=1/2H, C=2/3F • L=1/2H, C=1/2F • L=1/2H, C=1/4F Ans

  19. Second-Order Circuits Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC

  20. The 2nd order of expression 8.3 Source-Free Parallel RLC Circuits (1) Assume initial conductor current I0 and initial capacitor voltage V0 inductor capacitor Let v(0) = V0 Apply KCL to the top node: Taking the derivative with respect to t and dividing by C

  21. The 2nd order of expression The root of the quadratic equation ax2+bx+c=0 are Obtain the Characteristic equation by replacing First derivative s Second derivative s2 To find the root of the characteristic equation, use quadratic formula

  22. 1. If a > wo, over-damped case where 2. If a = wo, critical damped case where 3.If a < wo, under-damped case where Source-Free Parallel RLC Circuits There are three possible solutions for the following 2nd order differential equation:

  23. The constant A1 and A2 in each case can be determined from initial condition From KCL equation at the top node, or To find constant A1 and A2: Overdamped Critically damped underdamped Having found the capacitor voltage v(t) for parallel RLC circuit, we can readily obtain:

  24. Source-Free Parallel RLC Circuits Example 2 Refer to the circuit shown below. Find v(t) for t > 0.

  25. Step 2 Find the i(0) and v(0). Refer to the circuit at t<0 Step 1 Draw the circuit at t<0

  26. Step 3 Find by using equation Step 4 Draw the circuit at t>0. Find and

  27. Step 5 If > , the case is over- damped Use general equation for over-damped case. where Step 6 Find the value of A1 and A2.

  28. Source-Free Parallel RLC Circuits Exercise 2 Find v0(t) and i0(t) for t > 0, if • R2 = 0.5Ω • R2 = 1Ω • R2 = 4Ω Ans

  29. Second-Order Circuits Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC

  30. The 2nd order of expression Step Response Series RLC Circuits • The step response is obtained by the sudden application of a dc source. • The above equation has the same form as the equation for source-free series RLC circuit. • The same coefficients (important in determining the frequency parameters). • Different circuit variable in the equation.

  31. Step Response Series RLC Circuits (over-damped) (critically damped) (under-damped) The solution of the equation should have two components: thetransient response vt(t)& the steady-state response vss(t): • The transient response vt is the same as that for source-free case

  32. Step Response Series RLC Circuits • The steady-state response is the final value of v(t) • vss(t) = v(∞) • The values of A1 and A2 are obtained from the initial conditions: • v(0) and dv(0)/dt. In Step Response Series RLC, the solution is v(t), not i(t) like Free Source Series RLC because in series RLC, i(∞) are zero, because of C is act like open circuit. So that, can’t find the vss(t). In Step Response Series RLC, to find dv(t)/dt are using equation:

  33. Step Response Series RLC Circuits Example 3 The switch have been in position for a long times and move to position b at t = 0. Find v(t) and vR(t) for t > 0. Answer: v(t) = {10 + [(–2cos3.464t – 1.1547sin3.464t)e–2t]} V vR(t)= [2.31sin3.464t]e–2t V

  34. Step 1 Draw the circuit at t<0 Step 2 Find the i(0) and v(0). Refer to the circuit at t<0

  35. Step 3 Draw the circuit at t>0. Find using equation , , and v(∞) Under-damped General Equation:

  36. Step 4 Find the value of A1 and A2.

  37. Step 5 Find the value of vR(t). IC is equal with IR, because of series circuit.

  38. Exercise 3 The switch in position a for long time and move to b at t=0. Determine the i(t) at t>0.

  39. Second-Order Circuits Examples of 2nd order RCL circuit The source-free series RLC circuit The source-free parallel RLC circuit Step response of a series RLC circuit Step response of a parallel RLC

  40. Step Response Parallel RLC Circuits • The step response is obtained by the sudden application of a dc source. The 2nd order of expression • It has the same form as the equation for source-free parallel RLC circuit. • The same coefficients (important in determining the frequency parameters). • Different circuit variable in the equation.

  41. Step Response Parallel RLC Circuits (over-damped) (critically damped) (under-damped) The solution of the equation should have two components: thetransient response it(t)& the steady-state response iss(t): • The transient response it is the same as that for source-free case

  42. Step Response Parallel RLC Circuits • The steady-state response is the final value of i(t) • iss(t) = i(∞) • The values of A1 and A2 are obtained from the initial conditions: • i(0) and di(0)/dt. In Step Response Parallel RLC, i(t) at inductor is fined, not v(t) like Free Source Parallel RLC because in series RLC, v(∞) are zero, because of L is act like close circuit. So that, can’t find the iss(t). In Step Response Parallel RLC, to find di(t)/dt are using equation:

  43. Step Response Parallel RLC Circuits Example 4 The switch have been in position for a long times and move to position b at t = 0. Find i(t), iR(t) and v(t) for t > 0.

  44. Step 1 Draw the circuit at t<0 Step 2 Find the i(0) and v(0). Refer to the circuit at t<0

  45. Step 3 Draw the circuit at t>0. Find using equation , , and i(∞) Note: VL=VC Over-damped General Equation:

  46. Step 4 Find the value of A1 and A2.

  47. Step 5 Find the value of v(t). VC is equal with VL, because of parallel circuit.

  48. Step 6 Find the value of IR(t). VR is equal with VC, because of parallel circuit.

  49. Exercise 4 Determine the i(t) at t>0.

  50. Summary

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