1 / 57

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.

jselma
Télécharger la présentation

EET108 Circuit Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related