Fall 2011 ME 2305 Network Analysis. Sinusoidal Steady State Analysis of RLC Circuits

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1 Fall 2011 ME 2305 Network Analysis Chapter 4 Sinusoidal Steady State Analysis of RLC Circuits Engr. Humera Rafique Assistant Professor humera.rafique@szabist.edu.pk Faculty of Engineering (Mechatronics) Chapter Contents 2 Sinusoidal forcing functions and Complex exponential forcing function response of RL & RC Circuits for a sinusoidal forcing function The phasor Phasors for R, L & C elements Impedance and Admittance Phasor : Kirchhoff s Laws Mesh & Nodal Network Theorems Phasor diagram 1

2 3 Sinusoidal Forcing Functions Sinusoidal Forcing Functions: σ t x ( t) = X e cos( ωt + θ ) unit s m Sinusoids 4 σ t v ( t) = V e cos( ωt + θ ) V s m σt i ( t) = I e cos( ωt + φ) A s m 2

3 Sinusoidal Forcing Functions: Sinusoids 5 v = V sin( ωt) V i = I sin( ωt) A s m x( t + T ) = x( t) s m 2π f ω = 2 π f = ( rad / s) T Sinusoidal Forcing Functions: Sinusoids 6 3

4 x x /9/2012 Sinusoidal Forcing Functions: Sinusoids σ t x ( t) = X e cos( ωt + θ ) unit s m 7 V m σ ω θ x(t) Type Waveform 0 V m cos (ωt+θ) + sinusoid 0 0 V m cos (ωt) + sinusoid (cos) 0 V m e ±σ t cos (ωt) Exponentially rising/decaying sinusoid (cos) Or Damped sinusoid 0 V m e ±σ t cos (θ) Exponentially rising/decaying 0 0 V m e ±σ t Exponentially rising/decaying V m Pure DC Sinusoids Sinusoidal Sources: Example : The voltage across an element is v = 3 cos 3t V, and associated current through the element is i = -2 sin(3t + 10 O ) A. Determine the phase relationship between the voltage and current. 8 Example : a current has the form i = -6 cos 2t + 8 sin 2t A. Find the current restated in the following form: 2 i = a + b 2 cos( ωt θ ) Exercise to 3-3: 4

5 9 Steady-State Response of RL & RC Circuits Complex Exponential forcing Functions: RL & RC 10 x t X e X t j t unit ± j( ωt+ θ ) s ( ) = m = m[cos( ω + θ ) ± sin( ω + θ )] v( t) = V [cos( ωt + θ ) ± j sin( ωt + θ )] V m i( t)= I [cos( ωt + φ) ± j sin( ωt + φ)] A m Re[ v( t)] = V cos( ωt + θ ) V m Re[ i( t)] = I cos( ωt + φ) A m 5

6 Series RL & RC Circuits responses: Find v s (t) if i(t) = I m e j ω t A. RL & RC 11 jθ jφ V { ( ) me = Ime R + jωl I HT: Find i s (t) if v(t) = V m e j ω t V. jθ jφ 1 Vme = I { me R + j ω C I Parallel RL & RC Circuits responses: Example: Find v s (t) if i(t) = 8 e j3000t ma. R = 500Ω and L = 95 mh. RL & RC 12 Example: Find v s (t) if i(t) = 4 e j800t A. R = 2 Ω and C = 1 mf. Example p. 409: Find the response i of the given RL circuit, if R = 2Ω, L = 1 H, and v s = 10 sin 3t V. Exercise p. 406 Exercise & 2 p

7 13 The Phasor Concept Time to Frequency domain Polar representation: Phasor 14 jφ i( t) = I cos( ωt + φ) A I = I e = I φ m m m jθ v( t) = V cos( ωt + θ ) V V = V e = V θ m m m 7

8 Time to Frequency domain Polar representation: Phasor 15 jφ i( t) = I cos( ωt + φ) A I = I e = I φ m m m jθ v( t) = V cos( ωt + θ ) V V = V e = V θ m m m Phasor 16 Example : Find steady-state voltage v for the RC circuit shown, when, R = 1 Ω, C = 10 mf and ω = 100 rad/s. (Work in time domain) Exercise : Express the current i as a phasor: (a) i = 4 cos (ωt 80 o ) A (b) i = 10 cos (ωt + 20 o ) A (c) i = 8 cos (ωt 20 o ) A Dorf Exercise : Find the steady-state voltage V represented by the phasor: (a) V = o V (b) V = 80 + j 75 V 8

9 Phasor 17 Exercise: : Using time domain method, find the response v, when i s = 10 cos 100t A, R = 1 Ω and C = 1/100 F. Exercise : Using time domain method, find the current i(t) when v s = 4 cos 100t V. 18 Phasor Relationship of R, L & C 9

10 . Phasor R, L & C 19. Phasor R, L & C 20 10

11 . Phasor R, L & C 21 Impedance & Admittance: Phasor R, L & C 22 Element Impedance Admittance Z = V Y = I I V R Z R = R Y R = 1/R L Z L = jωl Y L = 1/jωL C Z C = 1/jωC Y C = jωc 11

12 Phasor 23 HT Exercise & 2(Dorf) Phasor 24 Example 4-8 (Dorf Exercise ): Find: (a) Z R (b) Z C (c) Z L2 (d) V S Example 4-9 (Dorf Exercise ): Find: (a) Z R (b) Z L1 (c) Z L2 (d) I S 12

13 25 KVL: Example : R = 9 Ω, L = 10 mh, C = 1 mf. Find steady-state current i using phasor

14 Voltage and Current Division: 27 Voltage Division: Example : Find steady-state voltage v O (t), if v S (t) = 7.28 cos (4t + 77 O ) V

15 Exercise : Find steady-state voltage v(t). 29 Examples (W Hayt): Practice 10-8 Let ω = 1200 rad/s, IC = oA, IL = 3 53o A. Find: (a) I S, (b) V s, (c) i R (t). 30 Example 10-5: Determine the equivalent impedance of the network: 15

16 Examples (W Hayt): Practice 10-9 Determine the equivalent impedance of the network: 31 Example 10-6: Determine the current i(t): Examples (W Hayt): Practice 10-10: Find I 1, I 2 and I 3 : 32 16

17 33 Nodal and Mesh Analyses Phasor Nodal Analysis Example (p. 429): Find steady-state node voltages, ω = 1000 rad/s and, L = 5mH, C = 100 µf, I m = 10 A, φ = 0 O

18 Phasor Mesh Analysis Dorf Example : Find steady-state current I 1, when v s = 10 2 cos (ωt + 45 O ) V, ω = 100 rad/s and L = 30 mh and C = 5 mf. 35 Example Example : Find steady-state voltage v. 36 Exercise to 3) 18

19 Examples (W Hayt): Example 10-7: Determine the time-domain node voltages: 37 Examples (W Hayt): Practice 10-12: Using nodal find node voltages: 38 Example 10-8: Find the time-domain currents of both loops: 19

20 Examples (W Hayt): Practice 10-13: Use mesh to find the phasor currents: 39 Example 10-8: Find the time-domain currents of both loops: 40 Network Theorems 20

21 Phasor Network Theorems Example : Using Superposition principle, find the steady-state current i, if v S = 10 cos (10t) V, i s = 3 A and L = 1.5 H and C = 10 mf. 41 Source Transformations: Phasor Network Theorems 42 21

22 Phasor Network Theorems Source Transformations: Example : Determine the phasor equivalent current source from v S = 10 cos (100t + 45 O ) V. 43 Thevenin s Equivalent: Example : Determine the Thévenin s equivalent in frequency-domain: Phasor Network Theorems 44 22

23 Norton s Equivalent: Example : Find the Norton s equivalent in frequency-domain: Phasor Network Theorems 45 Norton s Equivalent: Example : Find the Norton s equivalent in frequency-domain: Phasor Network Theorems 46 23

24 Norton s Equivalent: Example : Find the Norton s equivalent in frequency-domain: Phasor Network Theorems 47 Example , Exercise to 3 48 Phasor Diagram 24

25 Phasor Diagram 49 Phasor Diagram: A graphical representation of phasors and their relationship on the complex plane Practice: (Hayt 7e): 1. p. 389 (pracitce 10.9) 2. P. 391 (practice 10.10) 3. p. 393 (example 10.7) 4. p. 393 (practice 10.12) 5. p. 398 (example 10.9 &10) 6. p. 399 (practice 10.14) 7. p. 406 (example 10.12) Practice 50 25

26 References Text 2. Engineering Circuit Analysis 7e (W. Hayt Jr.) 26

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