Alternating Voltages and Currents

1. Chapter 15 Alternating Voltages and Currents • Introduction • Voltage and Current • Reactance of Inductors and Capacitors • Phasor Diagrams • Impedance • Complex Notation

2. 15.1 Introduction • From our earlier discussions we know that where Vp is the peak voltage is the angular frequency  is the phase angle • Since  = 2f it follows that the period T is given by

3. If  is in radians, then a time delay t is given by  / as shown below

4. 15.2 Voltage and Current • Consider the voltages across a resistor, an inductor and a capacitor, with a current of • Resistors • from Ohm’s law we know • therefore if i = Ipsin(t)

5. 15.2 Voltage and Current • Inductors - in an inductor • therefore if i = Ipsin(t) • Capacitors - in a capacitor • therefore if i = Ipsin(t)

7. 15.3 Reactance of Inductors and Capacitors • Let us ignore, for the moment the phase angle and consider the magnitudes of the voltages and currents • Let us compare the peak voltage and peak current • Resistance

8. Inductance • Capacitance

9. Reactance Reactance • The ratio of voltage to current is a measure of how the component opposes the flow of electricity • In a resistor this is termed its resistance • In inductors and capacitors it is termed its reactance • Reactance is given the symbol X • Therefore

10. Since reactance represents the ratio of voltage to current it has units of ohms • The reactance of a component can be used in much the same way as resistance: • for an inductor • for a capacitor

11. Example – see Example 15.3 from course text A sinusoidal voltage of 5 V peak and 100 Hz is applied across an inductor of 25 mH. What will be the peak current? At this frequency, the reactance of the inductor is given by Therefore

12. 15.4 Phasor Diagrams • Sinusoidal signals are characterised by their magnitude, their frequency and their phase • In many circuits the frequency is fixed (perhaps at the frequency of the AC supply) and we are interested in only magnitude and phase • In such cases we often use phasor diagrams which represent magnitude and phase within a single diagram

13. Examples of phasor diagrams (a) here L represents the magnitude and  the phase of a sinusoidal signal (b) shows the voltages across a resistor, an inductor and a capacitor for the same sinusoidal current

14. Phasor diagrams can be used to represent the addition of signals. This gives both the magnitude and phase of the resultant signal

15. Phasor diagrams can also be used to show the subtraction of signals

16. Phasor analysis of an RL circuit • See Example 15.5 in the text for a numerical example

17. Phasor analysis of an RC circuit • See Example 15.6 in the text for a numerical example

18. Phasor analysis of an RLC circuit

19. Phasor analysis of parallel circuits in such circuits the voltage across each of the components is the same and it is the currents that are of interest

20. 15.5 Impedance • In circuits containing only resistive elements the current is related to the applied voltage by the resistance of the arrangement • In circuits containing reactive, as well as resistive elements, the current is related to the applied voltage by the impedance, Z of the arrangement • this reflects not only the magnitude of the current but also its phase • impedance can be used in reactive circuits in a similar manner to the way resistance is used in resistive circuits

21. Consider the following circuit and its phasor diagram

22. From the phasor diagram it is clear that that the magnitude of the voltage across the arrangement V is where • Z is the magnitude of the impedance, so Z =|Z|

23. From the phasor diagram the phase angle of the impedance is given by • This circuit contains an inductor but a similar analysis can be done for circuits containing capacitors • In general and

24. A graphical representation of impedance

25. 15.6 Complex Notation • Phasor diagrams are similar to Argand Diagrams used in complex mathematics • We can also represent impedance using complex notation where • Resistors: ZR = R • Inductors: ZL = jXL = jL • Capacitors: ZC = -jXC =

26. Graphical representation of complex impedance

27. Series and parallelcombinations of impedances • impedances combine in the same way as resistors

28. Manipulating complex impedances • complex impedances can be added, subtracted, multiplied and divided in the same way as other complex quantities • they can also be expressed in a range of forms such as the rectangular, polar and exponential forms • if you are unfamiliar with the manipulation of complex quantities (or would like a little revision on this topic) see Appendix D of the course text which gives a tutorial on this subject

29. Example – see Example 15.7 in the course text Determine the complex impedance of this circuit at a frequency of 50 Hz. At 50Hz, the angular frequency  = 2f = 2 50 = 314 rad/s Therefore

30. Using complex impedance • Example – see Section 15.6.4 in course text Determine the current in this circuit. Since v = 100 sin 250t , then  = 250 Therefore

31. Example (continued) The current is given by v/Z and this is easier to compute in polar form Therefore

32. A further example A more complex task is to find the output voltage of this circuit. The analysis of this circuit, and a numerical example based on it, are given in Section 15.6.4 and Example 15.8 of the course text

33. Key Points • A sinusoidal voltage waveform can be described by the equation • The voltage across a resistor is in phase with the current, the voltage across an inductor leads the current by 90, and the voltage across a capacitor lags the current by 90 • The reactance of an inductor XL = L • The reactance of a capacitor XC = 1/C • The relationship between current and voltage in circuits containing reactance can be described by its impedance • The use of impedance is simplified by the use of complex notation