1. Alternating Current Circuits
AC Circuits
Alternating Current Circuits

2. Why AC power and not DC?
In the earliest days Direct Current (DC), where the power flows in one direction like a water hose, was the standard for delivering electrical power.  Now Alternating Current (AC), where the power flow is constantly alternating direction, is the standard for delivering electrical power.
The standard for delivering electrical power changed from Direct Current (DC) to Alternating Current (AC) because Alternating Current (AC) delivers electrical power more efficiently over long distances.
In the US, 60 Hertz (cycles per second) is the Alternating Current (AC) frequency.
In some countries, 50 Hertz (cycles per second) is the Alternating Current (AC) frequency.

3. Alternating Voltages and Currents
Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance.
A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts.
A circuit breaker is similar, except that it is a bimetallic strip that bends enough to break the connection when it becomes too hot. When it cools, it can be reset.

4. RMS voltage or current
The amount of AC power that produces the same heating effect as an equivalent DC power
The RMS value is the square root of the mean (average) value of the squared function of the instantaneous values. The symbols used for defining an RMS value are VRMS or IRMS.
The term RMS, ONLY refers to time-varying sinusoidal voltages, currents or complex waveforms where the magnitude of the waveform changes over time and is not used in DC circuit analysis or calculations where the magnitude is always constant. When used to compare the equivalent RMS voltage value of an alternating sinusoidal waveform that supplies the same electrical power to a given load as an equivalent DC circuit, the RMS value is called the “effective value” and is generally presented as: Veff or Ieff.

5. RMS value continued
In other words, the effective value is an equivalent DC value which tells you how many volts or amps of DC that a time-varying sinusoidal waveform is equal to in terms of its ability to produce the same power. For example, the domestic mains supply in the United Kingdom is 240Vac. This value is assumed to indicate an effective value of “240 Volts rms”. This means then that the sinusoidal rms voltage from the wall sockets of a UK home is capable of producing the same average positive power as 240 volts of steady DC voltage as shown below.

6. RMS value continued
RMS Voltage Equivalent

7. RMS Voltage Analytical Method
A periodic sinusoidal voltage is constant and can be defined as V(t) = Vm.cos(ωt) with a period of T. Then we can calculate the root-mean- square (rms) value of a sinusoidal voltage (V(t)) as:

8. Integrating through with limits taken from 0 to 360o or “T”, the period gives:
Dividing through further as ω = 2π/T, the complex equation above eventually reduces down too:

9. RMS Voltage Equation

10. Average Voltage
The process used to find the Average Voltage of an alternating waveform is very similar to that for finding its RMS value, the difference this time is that the instantaneous values are not squared and we do not find the square root of the summed mean.
The average voltage (or current) of a periodic waveform whether it is a sine wave, square wave or triangular waveform is defined as: “the quotient of the area under the waveform with respect to time”. In other words, the averaging of all the instantaneous values along time axis with time being one full period, (T).

11. For a periodic waveform, the area above the horizontal axis is positive while the area below the horizontal axis is negative. The result is that the average or mean value of a symmetrical alternating quantity is zero because the area above the horizontal axis (the positive half cycle) is the same as the area below the axis (the negative half cycle) and cancel each other out in the sum of the two areas as a negative cancels a positive producing zero average voltage.

12. Then the average or mean value of a symmetrical alternating quantity, such as a sine wave, is the average value measured over only half a cycle since over a complete cycle the average value is zero regardless of the peak amplitude.
The electrical terms Average Voltage and Mean Voltage or even average current, can be used in both an AC and DC circuit analysis or calculations. The symbols used for representing an average value are defined as: VAV or IAV.

13. Average Voltage Analytical Method
As said previously, the average voltage of a periodic waveform whose two halves are exactly similar, either sinusoidal or non-sinusoidal, will be zero over one complete cycle. Then the average value is obtained by adding the instantaneous values of voltage over one half cycle only. But in the case of an non-symmetrical or complex wave, the average voltage (or current) must be taken over the whole periodic cycle mathematically.

14. Approximation of the Area

15. The area under the curve can be found by various approximation methods such as the trapezoidal rule, the mid-ordinate rule or Simpson’s rule. Then the mathematical area under the positive half cycle of the periodic wave which is defined as V(t) = Vp.cos(ωt) with a period of T using integration is given as:

16. For a sinusoid

17. Average Voltage Equation

18. Alternating Voltages and Currents
Wall sockets provide current and voltage that vary sinusoidally with time.
Here is a simple ac circuit:

19. Resistors in an AC Circuit
Consider a circuit consisting of an AC source and a resistor.
The AC source is symbolized by
ΔvR = DVmax= Vmax sin wt
ΔvR is the instantaneous voltage across the resistor.

20. Alternating Voltages and Currents
The voltage as a function of time is:

21. Alternating Voltages and Currents
Since this circuit has only a resistor, the current is given by:
Here, the current and voltage have peaks at the same time – they are in phase.

22. Phasor Diagram
To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram can be used.
A phasor is a vector whose length is proportional to the maximum value of the variable it represents.
The vector rotates counterclockwise at an angular speed equal to the angular frequency associated with the variable.
The projection of the phasor onto the vertical axis represents the instantaneous value of the quantity it represents.

23. A Phasor is Like a Graph
An alternating voltage can be presented in different representations.
One graphical representation is using rectangular coordinates.
The voltage is on the vertical axis.
Time is on the horizontal axis.
The phase space in which the phasor is drawn is similar to polar coordinate graph paper.
The radial coordinate represents the amplitude of the voltage.
The angular coordinate is the phase angle.
The vertical axis coordinate of the tip of the phasor represents the instantaneous value of the voltage.
The horizontal coordinate does not represent anything.
Alternating currents can also be represented by phasors.

24. Resistors in an AC Circuit
The graph shows the current through and the voltage across the resistor.
The current and the voltage reach their maximum values at the same time.
The current and the voltage are said to be in phase.
For a sinusoidal applied voltage, the current in a resistor is always in phase with the voltage across the resistor.
The direction of the current has no effect on the behavior of the resistor.
Resistors behave essentially the same way in both DC and AC circuits.

25. Inductors in an AC Circuit
Kirchhoff’s loop rule can be applied and gives:

26. Current in an Inductor
The equation obtained from Kirchhoff's loop rule can be solved for the current
This shows that the instantaneous current iL in the inductor and the instantaneous voltage ΔvL across the inductor are out of phase by (p/2) rad = 90o.

27. Phase Relationship of Inductors in an AC Circuit
The current is a maximum when the voltage across the inductor is zero.
The current is momentarily not changing
For a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90° (π/2).

28. Phasor Diagram for an Inductor
The phasors are at 90o with respect to each other.
This represents the phase difference between the current and voltage.
Specifically, the current lags behind the voltage by 90o.

29. Inductive Reactance
The factor ωL has the same units as resistance and is related to current and voltage in the same way as resistance.
Because ωL depends on the frequency, it reacts differently, in terms of offering resistance to current, for different frequencies.
The factor is the inductive reactance and is given by:
XL = ωL

30. Inductive Reactance, cont.
Current can be expressed in terms of the inductive reactance:
As the frequency increases, the inductive reactance increases
This is consistent with Faraday’s Law:
The larger the rate of change of the current in the inductor, the larger the back emf, giving an increase in the reactance and a decrease in the current.

31. Voltage Across the Inductor
The instantaneous voltage across the inductor is

32. Inductors in AC Circuits
The voltage across an inductor leads the current by 90°.

33. Inductors in AC Circuits
The power factor for an RL circuit is:
Currents in resistors, capacitors, and inductors as a function of frequency:

34. Capacitors in an AC Circuit
The circuit contains a capacitor and an AC source.
Kirchhoff’s loop rule gives:
Δv + Δvc = 0 and so
Δv = ΔvC = ΔVmax sin ωt
Δvc is the instantaneous voltage across the capacitor.

35. Capacitors in an AC Circuit, cont.
The charge is q = CΔVmax sin ωt
The instantaneous current is given by
The current is p/2 rad = 90o out of phase with the voltage

36. More About Capacitors in an AC Circuit
The current reaches its maximum value one quarter of a cycle sooner than the voltage reaches its maximum value.
The current leads the voltage by 90o.

37. Phasor Diagram for Capacitor
The phasor diagram shows that for a sinusoidally applied voltage, the current always leads the voltage across a capacitor by 90o.

38. Capacitive Reactance
The maximum current in the circuit occurs at cos ωt = 1 which gives
The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by

39. Voltage Across a Capacitor
The instantaneous voltage across the capacitor can be written as ΔvC = ΔVmax sin ωt = Imax XC sin ωt.
As the frequency of the voltage source increases, the capacitive reactance decreases and the maximum current increases.
As the frequency approaches zero, XC approaches infinity and the current approaches zero.
This would act like a DC voltage and the capacitor would act as an open circuit.

40. RC Circuits
In an RC circuit, the current across the resistor and the current across the capacitor are not in phase. This means that the maximum current is not the sum of the maximum resistor current and the maximum capacitor current; they do not peak at the same time.

41. RC Circuits
This phasor diagram illustrates the phase relationships. The voltages across the capacitor and across the resistor are at 90° in the diagram; if they are added as vectors, we find the maximum.

42. RC Circuits
This has the exact same form as V = IR if we define the impedance, Z:

43. RC Circuits
There is a phase angle between the voltage and the current, as seen in the diagram.

44. RC Circuits The power in the circuit is given by:
Because of this, the factor cos φ is called the power factor.

45. The RLC Series Circuit
The resistor, inductor, and capacitor can be combined in a circuit.
The current and the voltage in the circuit vary sinusoidally with time.

46. The RLC Series Circuit, cont.
The instantaneous voltage would be given by Δv = ΔVmax sin ωt.
The instantaneous current would be given by i = Imax sin (ωt - φ).
φ is the phase angle between the current and the applied voltage.
Since the elements are in series, the current at all points in the circuit has the same amplitude and phase.

47. i and v Phase Relationships – Equations
The instantaneous voltage across each of the three circuit elements can be expressed as

48. More About Voltage in RLC Circuits
ΔVR is the maximum voltage across the resistor and ΔVR = ImaxR.
ΔVL is the maximum voltage across the inductor and ΔVL = ImaxXL.
ΔVC is the maximum voltage across the capacitor and ΔVC = ImaxXC.
The sum of these voltages must equal the voltage from the AC source.
Because of the different phase relationships with the current, they cannot be added directly.

49. Vector Addition of the Phasor Diagram
Vector addition is used to combine the voltage phasors.
ΔVL and ΔVC are in opposite directions, so they can be combined.
Their resultant is perpendicular to ΔVR.
The resultant of all the individual voltages across the individual elements is Δvmax.
This resultant makes an angle of φ with the current phasor Imax.

50. Total Voltage in RLC Circuits
From the vector diagram, ΔVmax can be calculated

51. Impedance The current in an RLC circuit is
Z is called the impedance of the circuit and it plays the role of resistance in the circuit, where
Impedance has units of ohms

52. Phase Angle
The right triangle in the phasor diagram can be used to find the phase angle, φ.
The phase angle can be positive or negative and determines the nature of the circuit.

53. Determining the Nature of the Circuit
If f is positive
XL> XC (which occurs at high frequencies)
The current lags the applied voltage.
The circuit is more inductive than capacitive.
If f is negative
XL< XC (which occurs at low frequencies)
The current leads the applied voltage.
The circuit is more capacitive than inductive.
If f is zero
XL= XC
The circuit is purely resistive.

54. Resonance in Electrical Circuits
The rms voltages across the capacitor and inductor must be the same; therefore, we can calculate the resonant frequency.

55. Resonance in Electrical Circuits
In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency:

56. Resonance in Electrical Circuits
The smaller the resistance, the larger the resonant current: