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Steady-State Sinusoidal Analysis

Steady-State Sinusoidal Analysis . Steady-State Sinusoidal Analysis . 1 . Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2 . Solve steady-state ac circuits using phasors and complex impedances .

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Steady-State Sinusoidal Analysis

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  1. Steady-State Sinusoidal Analysis

  2. Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances. 3. Compute power for steady-state ac circuits. 4. Find Thévenin and Norton equivalent circuits. 5. Determine load impedances for maximum power transfer.

  3. The sinusoidal function v(t) = VM sin  t is plotted (a) versus  t and (b) versus t.

  4. The sine wave VM sin ( t + ) leads VM sin  t by  radian

  5. Frequency Angular frequency

  6. A graphical representation of the two sinusoids v1 and v2. The magnitude of each sine function is represented by the length of the corresponding arrow, and the phase angle by the orientation with respect to the positive x axis. In this diagram, v1 leads v2 by 100o + 30o = 130o, although it could also be argued that v2 leads v1 by 230o. It is customary, however, to express the phase difference by an angle less than or equal to 180oin magnitude.

  7. Euler’s identity In Euler expression, A cos t = Real (A e j t ) A sin t = Im( A e j t) Any sinusoidal function can be expressed as in Euler form. 4-6

  8. The complex forcing function Vm e j ( t + ) produces the complex response Ime j (t + ). It is a matter of concept to make use of the mathematics of complex number for circuit analysis. (Euler Identity)

  9. The sinusoidal forcing function Vm cos ( t + θ) produces the steady-state response Im cos ( t + φ). The imaginary sinusoidal forcing function j Vm sin ( t + θ) produces the imaginary sinusoidal response j Im sin ( t + φ).

  10. Re(Vm e j ( t + ) )  Re(Ime j (t + )) Im(Vm ej ( t + ) )  Im(Ime j (t + ))

  11. Phasor Definition

  12. A phasor diagram showing the sum of V1 = 6 + j8 V and V2 = 3 – j4 V, V1 + V2 = 9 + j4 V = Vs Vs = Ae j θ A = [9 2 + 4 2]1/2 θ= tan-1 (4/9) Vs = 9.8524.0o V.

  13. Adding Sinusoids Using Phasors Step 1: Determine the phasor for each term. Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function.

  14. Using Phasors to Add Sinusoids

  15. = + V V V s 1 2 = Ð - + Ð - o o 20 45 10 30 = - + - 14 . 14 j 14 . 14 8 . 660 j 5 = - 23 . 06 j 19 . 14 = Ð - o 29 . 97 39 . 7

  16. Phase Relationships To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise. Then when standing at a fixed point, if V1 arrives first followed by V2 after a rotation of θ , we say that V1 leads V2 by θ . Alternatively, we could say that V2 lags V1 by θ . (Usually, we take θ as the smaller angle between the two phasors.)

  17. To determine phase relationships between sinusoids from their plots versus time, find the shortest time interval tpbetween positive peaks of the two waveforms. Then, the phase angle isθ = (tp/T )× 360°. If the peak of v1(t)occurs first, we say that v1(t)leads v2(t)or that v2(t)lags v1(t).

  18. COMPLEX IMPEDANCES

  19. (a) (b) In the phasor domain, (a) a resistor R is represented by an impedance of the same value; (b) a capacitor C is represented by an impedance 1/jC; (c) an inductor L is represented by an impedance jL. (c)

  20. Zc is defined as the impedance of a capacitor The impedance of a capacitor is 1/jC. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the capacitor and the current flowing through the capacitor.

  21. ZL is defined as the impedance of an inductor The impedance of a inductor is jL. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the inductor and the current flowing through the inductor.

  22. Complex Impedance in Phasor Notation

  23. Kirchhoff’s Laws in Phasor Form We can apply KVL directly to phasors. The sum of the phasor voltages equals zero for any closed path. The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving.

  24. Circuit Analysis Using Phasors and Impedances • Replace the time descriptions of the voltage and current sources with the corresponding phasors. • (All of the sources must have the same frequency.) • 2.Replace inductances by their complex impedances • ZL= jωL. • Replace capacitances by their complex impedances • ZC = 1/(jωC). • Resistances remain the same as their resistances.

  25. 3. Analyze the circuit using any of the techniques studied earlier performing the calculations with complex arithmetic.

  26. Solve by nodal analysis

  27. Vs= - j10, ZL=jL=j(0.5×500)=j250 Use mesh analysis,

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