advertisement

BASICS OF ELECTRICAL CIRCUITS EHB 211 E RL & RC Circuits Asst. Prof. Onur Ferhanoğlu 1 Natural Response of an RL Circuit • t< 0, switch was ON for a long time • All currents & voltages have reached a constant value • Inductor acts as a short circuit before the switch is opened After switch is turned OFF KVL (1st order differential equation) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 2 Natural Response of an RL Circuit Voltage across resistor Asst. Prof. Onur Ferhanoğlu In an inductor: • Change of current is not instantaneous i(0-) = i(0+) • Change of voltage can be instantaneous v(0-) ≠ v(0+) RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 3 Natural Response of an RL Circuit POWER: Asst. Prof. Onur Ferhanoğlu ENERGY (delivered to resistor): RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 4 Natural Response of an RL Circuit – time constant Rate at which current & voltage changes: After 5τ, current & voltages reach their final values Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 5 Natural Response of an RL Circuit – time constant Rate at which current & voltage changes: After 5τ, current & voltages reach their final values Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 6 Exercise The switch in the below circuit has been closed for a long time before it was opened at t = 0. Find a) iL(t) b) io(t) c) vo(t) a) iL(0-) = iL(0+) = 20A (inductor is short at t = 0-) τ = L/Req = 0.2 b) Asst. Prof. Onur Ferhanoğlu c) RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 7 Exercise 2 For the below circuit find: a) i1,, i2 and i3 for t>0 (attention to their initial values) b) Initial stored energy in the parallel inductors c) How much energy is stored in the inductors for t->∞ a) τ = 4/8 = 0.5 v = 8i Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 8 Exercise 2 – continued… For the below circuit find: a) i1,, i2 and i3 for t>0 (attention to their initial values) b) Initial stored energy in the parallel inductors c) How much energy is stored in the inductors for t->∞ b) 0.5L1i2 + 0.5L2i2 c)i1 -> 1.6 A , i2 -> -1.6 A Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 9 Natural response of an RC circuit KCL Similar to RL circuit.. In an capacitor: • Change of voltage is not instantaneous v(0-) = v(0+) • Change of current can be instantaneous i(0-) ≠ i(0+) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 10 Natural response of an RC circuit Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 11 Exercise 3 For the circuit find: a) vc(t), for t> 0 b) vo(t), for t> 0 c) io(t), for t> 0 a) t<0 -> capacitor will charge to 100V (V0 = 100V) b) c) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 12 Exercise 4 For the circuit find: a) v1(t), v2(t), v(t), i(t), for t> 0 b) Calculate the initial energy stored in the capacitors c) Determine the energy stored in the capacitors for t -> ∞ a) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 13 Exercise 4 For the circuit find: a) v1(t), v2(t), v(t), i(t), for t> 0 b) Calculate the initial energy stored in the capacitors c) Determine the energy stored in the capacitors for t -> ∞ b) c) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 14 The Step response of RL circuits Step response: response of the circuit when a source is suddenly applied KVL if I0 = 0 Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 15 The Step response of RL circuits if I0 = 0 Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 16 Exercise 5 For the circuit, find a) i(t) b) v(0+) c) When does v = 24V d) Plot i(t) and v(t) vs. t a) At position a, inductor behaves as a short circuit I0 = -8 A (all the source current passes through the inductor) τ = 200/2 = 100ms = 0.1 s Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 17 Exercise 5 continued.. For the circuit, find a) i(t) b) v(0+) c) When does v = 24V d) Plot i(t) and v(t) vs. t b) i= Alternatively: v = (24 – (-8.2))e-10t Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 18 Exercise 5 continued.. For the circuit, find a) i(t) b) v(0+) c) When does v = 24V d) Plot i(t) and v(t) vs. t c) Asst. Prof. Onur Ferhanoğlu d) RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 19 The Step response of RC circuits KCL Recall from RL circuits therefore i = C.dv/dt Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 20 Exercise 6 For the circuit, find a) vo(t), for t > 0 b) io(t), for t > 0 a) Initial value of vo is 40.(60/80) = 30V (voltage divider) Transform the circuit so that we can take advantage of deduced equations (previous page) NORTON EQUIVALENT for t>0 Inorton = -60 / 40 = -1.5 mA Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 21 Exercise 6 continued.. For the circuit, find a) vo(t), for t > 0 b) io(t), for t > 0 a) b) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 22 General solution for step and natural responses 4 possible 1st order circuits -> For all 4 circuits, the differential equation is of the following form: Because the sources are DC (constant), final value of x (current or voltage) will be constant, dx/dt -> 0 where xf is the final value of the variable (current & voltage) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 23 Exercise 7 For the circuit, find a) Initial value of vC b) Final value of vC c) Time constant when switch is at b d) vC(t) when t>0 e) i(t) when t>0 a) Voltage divider: b) Long after switching from a to b -> capacitor will act as an open circuit vC(∞) = 90V c) d) Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 24 Exercise 7 continued… For the circuit, find a) Initial value of vC b) Final value of vC c) Time constant when switch is at b d) vC(t) when t>0 e) i(t) when t>0 e) Find initial and final values of i(t) i(∞) = 0 (open circuit) i(0+) = (initial value) = current in 400kΩ resistor: Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 25 Exercise 8 For the circuit, the initial charge on the capacitor is 0. Find: a) i(t) for t > 0 b) v(t) for t > 0 a) i(∞) = 0 (open circuit) b) First find vC(t) vC(0) = 0 (given in the question) vC(∞) = 7.5mA * 20k = 150V (0 volts across 30k resistor) v(t) = vC(t) + v30kΩ Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 26 Exercise 9 For the circuit, find: a) v(t) for t > 0 b) i(t) for t > 0 a) i(0) = 20V/4Ω = 5A v(0) = 20V – 5A*1 = 15V v(∞) = 0V (inductor: short circuit) b) i(∞) = 20V / 1 Ω = 20A Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 27 Sequential switching Switching occurs more than once In the circuit, at t=0 the switch is moved from a to b, and at t=15ms, from b to c: (Initial voltage on capacitor = 0) Switch ->b v(0) = 0 v(∞) = 400 V, if the switch remained at b Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS Switch -> c v(15ms) = 310.75V v(∞) = 0 (capacitor: open circuit) 28 Sequential switching Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 29 Integrating amplifier If initial voltage is 0 Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 30 Integrating amplifier + switching The initial voltage on the capacitor is 5V. If switch stays in terminal a for 9ms, when does saturation take place At terminal a At terminal b vo(9ms) = -5 + 9 = 4V Asst. Prof. Onur Ferhanoğlu RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS 31