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ELECTRICAL CIRCUITS ET 201

ELECTRICAL CIRCUITS ET 201. Define, analyze and calculate the response of RLC (resistance R and reactance X ). Define and analyze the impedance and impedance diagram for RLC elements. 14.2 The Derivative.

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ELECTRICAL CIRCUITS ET 201

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  1. ELECTRICAL CIRCUITS ET 201 Define, analyze and calculate the response of RLC (resistance R and reactance X). Define and analyze the impedance and impedance diagram for RLC elements.

  2. 14.2 The Derivative • To understand the response of the basic R, L and C elements to a sinusoidal signal, you need to examine the concept of the derivative. • The derivativedx/dt is defined as the rate of change of x with respect to time. • If x fails to change at a particular instant, dx = 0, and the derivative is zero. • The derivative dx/dt is actually the slope of the graph at any instant of time. 2

  3. 14.2 The Derivative • The derivative dx/dt is zero only at the positive and negative peaks (ωt = π/2 and 3π/2) since x fails to change at these instants of time. • For the sinusoidal waveform, the greatest change in x will occur at the instants ωt = 0, π, and 2π. The derivative is therefore a maximum at these points. 3

  4. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Resistor For power-line frequencies and frequencies up to a few hundred kilohertz, resistance is, for all practical purposes, unaffected by the frequency of the applied sinusoidal voltage or current. For this frequency region, the resistor R can be treated as a constant.

  5. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Resistor Given Where; For a given Where; 5

  6. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current • Resistor • For a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law.

  7. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Inductor The inductive voltage is directly related to the frequency f (or more specifically, the angular velocity of the sinusoidal ac current through the coil) and the inductance of the coil L. The voltage across an inductor is directly related to the rate of change of current through the coil.

  8. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Inductor Given Or; Where; Hence; XL : Inductive reactance

  9. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current • Inductor • For an inductor, vL leads iL by 90°, or iL lags vL by 90°.

  10. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Capacitor For a particular capacitance, the greater the rate of change of voltage across the capacitor, the greater the capacitive current. The capacitive current is directly related to the rate of change of the voltage across the capacitor.

  11. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Capacitor Given Or; Where; Hence; XC : Capacitive reactance

  12. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current • Capacitor • For a capacitor, iC leads vC by 90°, or vC lags iC by 90°.

  13. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current • If the source current leads the applied voltage, the network is predominantly capacitive. • If the applied voltage leads the source current, the network is predominantly inductive. 13

  14. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Resistor: Inductor: Capacitor: vR and iR in phase vL leads iL by 90° iC leads vC by 90° 14

  15. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Relationship between differential, integral operation in phasor listed as follow:

  16. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current 16

  17. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) The voltage across a 10  resistor is given by the expression; Find the expression for the current i through the resistor and sketch the curves for v and i.

  18. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) – solution and Hence;

  19. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) – solution (cont’d) The alternative waveform;

  20. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) The voltage across a 10  resistor is given by the expression; Find the expression for the current i through the resistor and sketch the curves for v and i.

  21. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) – solution and Hence;

  22. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) – solution (cont’d) The alternative waveform;

  23. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) The current a 0.1 H coil is given by the expression; Find the expression for the voltage v across the coil and sketch the curves for v and i.

  24. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution vL leads iL by 90°, Hence, if the current is; the voltage will be; where; and;

  25. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) From the expression; Hence; Hence the expression for the voltage is;

  26. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) The waveforms;

  27. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) The current a 0.1 H coil is given by the expression; Find the expression for the voltage v across the coil and sketch the curves for v and i.

  28. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution vL leads iL by 90°, Hence, if the current is; the voltage will be; where; and;

  29. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) From the expression; Hence; Hence the expression for the voltage is; 29

  30. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution (cont’d) The waveforms;

  31. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 The voltage across a 1 Fcapacitor is given by the expression; Find the expression for the current i through the capacitor and sketch the curves for v and i.

  32. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution iC leads vC by 90°, Hence, if the voltage is; the current will be; where; and;

  33. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution (cont’d) From the expression; Hence; Hence the expression for the current is;

  34. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution (cont’d) The waveforms;

  35. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 The current through a 100 Fcapacitor is given by the expression; Find the expression for the voltage v across the capacitor.

  36. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 – solution iC leads vC by 90°, Hence, if the current is; the voltage will be; where; and;

  37. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 – solution (cont’d) From the expression; Hence; Hence the expression for the voltage is;

  38. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(a) Determine the type of element in the box (C, L or R) and calculate its value if; and Solution The voltage and current are in phase . The element is a resistor (R).

  39. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(b) Determine the type of element in the box (C, L or R) and calculate its value if; and Solution The voltage leads the current by 90 The element is an inductor (L).

  40. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(c) Determine the type of element in the box (C, L or R) and calculate its value if; and Solution The voltage lags the currentby 90 The element is a capacitor (C).

  41. 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(d) Determine the type of element in the box (C, L or R) and calculate its value if; and Solution The voltage and current are in phase . The element is a resistor (R).

  42. Impedance Z It is a ratio of the phasor voltage V to the phasor currentI. Unit in ohms (Ω). 15.2 Impedance and the Phasor diagram [Ω] 42

  43. 15.2 Impedance and the Phasor diagram • Impedance Z • Impedance Z has two components: • Real component (ZRe) : Resistance, R • Imaginary component (ZIm) : Reactance, X • Reactance can be inductor, L and capacitance, C. • Positive X is for L and negative X is for C. [Ω] 43

  44. 15.2 Impedance and the Phasor diagram • Admittance Y • It is the reciprocal of impedance Z. • Unit in siemens (S). [S] 44 44

  45. 15.2 Impedance and the Phasor diagram Impedance for Resistor, R v and i are in phase; Or; In phasor form; Where;

  46. 15.2 Impedance and the Phasor diagram Impedance for Resistor, R Applying Ohm’s law and phasor algebra; Since v and i are in phase; Hence; Therefore;

  47. 15.2 Impedance and the Phasor diagram Impedance for Resistor, R • The boldface Roman quantity ZR, having both magnitude and an associate angle, is referred to as the impedance of a resistive element. • ZR is not a phasor since it does not vary with time. • Even though the format R0° is very similar to the phasor notation for sinusoidal current and voltage, R and its associated angle of 0° are fixed, non-varying quantities.

  48. 15.2 Impedance and the Phasor diagram Example 15.1 Find i in the figure. Solution Applying Ohm’s law;

  49. 15.2 Impedance and the Phasor diagram Example 15.1 – solution (cont’d) Inverse-transform; vR and iR in phase

  50. 15.2 Impedance and the Phasor diagram Example 15.2 Find v in the figure Solution Applying Ohm’s law;

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