MINISTRY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF TECHNICAL AND VOCATIONAL EDUCATION

1. MINISTRY OF SCIENCE AND TECHNOLOGYDEPARTMENT OFTECHNICAL AND VOCATIONAL EDUCATION EP 3032 ELECTRICAL TRANSIENT B.Tech (First Year) Electrical Power Engineering

2. Electrical Transient • Source Free RL and RC Circuits • The Application of the Unit-Step Forcing Function • The RLC Circuit • Complex Frequency • Frequency Response • State-Variable Analysis • Fourier Analysis • Fourier Transforms • Laplace Transform Techniques

3. Source Free RL and RC Circuits • Inductor Short Circuit to dc (at t = 0-) Current Source at t = 0+ iL (0-) = iL (0+) vL = L L= Li2 iL L + vL - Figure 1. Inductor

4. i(t) - + vR R L vL + - Figure 2 A Series RL Circuit

5. 2 Capacitor Open circuit to dc (t = 0-) Voltage source at (t = 0+) vc(0+) = vC (0-) iC (t) = C c = Cv2 iC C + vC - Figure 3 Capacitor

6. i(t) + C v R Figure 4. A parallel RC Circuit

7. 3. Response (Natural Response) i(t) = i(0+) e –t/ ,  = v(t) = v(0+) e-t/ ,  = RC 0  t(s) Figure 5. Time Versus i/I0 i/I0

8. 5 Switch Operations t = 0 just before the switch changes t < 0 close just after the switch changes t > 0 open t = 0 just before the switch changes t < 0 open just after the switch changes t > 0 close

9. The Application of the Unit-Step Forcing Function 1.Introduction A complete response contains two parts. • Forced response • Natural response v(t) = vf + vn i(t) = if + in

10. 1 ON t OFF 0 1 ON t ON t 0 2.The Unit-Step Forcing Function (u) u(t) u(t) = u(t) u (-t) = 0 OFF (t + t0) u 1 u (t + t0) = OFF

11. u ON t 0 OFF u (t0 - t ) ON OFF t t0 0 (t - t0) u (t – t0) = u (t0- t) =

12. 50u(t) V 2 - + i(t) 3. Determine i(t) for all values of time in the circuit of Figure. 50V 6 3H At t<0 2 iL (0- ) = iL (0+) =50/2 = 25A iL(0-) 50V 6 At t >0 2 Reqt=2 // 6  = L/R 50V iL 50V 6 3H -+ = 3/1.5 = 2S = 1.5 • i(t) = in + if = A e -t/ + i()

13. At t=  2 -+ iL 50V 50V 6 i(t),A • i(t) = in + if = Ae -t/ + 50 i(0+) = 50+A = 25 • A = -25 •  i(t) = 50 -25e -0.5t (t>0) 50 25 t(s) A sketch of the complete response of i(t)

14. The RLC Circuit (1) Over damped ( >0) (2) Under damped (< 0) (3) Critically damped ( = 0) Series RLC Circuit  = R /2L where  = exponential damping coefficient (s-1) 0 = resonant frequency (rad/s) 0 = Parallel RLC Circuit  = 0 =

15. (1) Overdamped ( > 0) i(t) or v(t) = A1 s1,2 = -  (2) Under damped (  < 0) i(t) or v(t) = e-t (A1 cos dt + A2 sin dt) d = where, d = natural resonant frequency (rad/s) (3) Critically damped v(t) (or) i(t) = e-t (A1t + A2)

16. Complex Frequency Complex Frequency (s) s =  + j  where,  = neper frequency (neper per second)  = radian frequency (radian per second) Time Domain Frequency Domain v = Ri = R v = L =j L v = = Critical Frequency (1) Pole frequency () (2) Zero frequency ()

17. Euler's Identity  e j(t + ) = cos (t + ) + j sin (t + ) Impedance Admittance ZR = R YR = 1/R ZL = sL YL = 1/sL ZC =1/sC YC= sC Gain = where, = Transfer function If = Forced response = Forcing function