AC Theory
Article : Andy Collinson
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In electronics, ac sources are widely used. A dc (direct current) source, current flows only in one direction, but an ac (alternating current) source both voltage and current continually change direction. The rate of change is called the angular velocity, symbol ω and is measured in radians.

One complete cycle of an ac waveform, starts at 0V amplitude, rises to a positive peak and then decreases to a negative trough before ending at 0 volts again. This is shown in Figure 1 below:


Figure 1

The shape of the waveform is sinusoidal, and this is the most common waveform used in power supplies and signal generators. The time taken for 1 complete cycle shown in figure 1 is 1ms. The frequency of the waveform is 1/periodic time or 1 kHz in this example. The waveform has a peak of 10Volts and its peak to peak value is total of the positive and negative half cycle.

If viewed on an oscilloscope you will see the peak to peak value of the waveform. If measured on a multimeter set to AC you will read a different vale known as the RMS value. The RMS (root mean square) value is 0.7071 x the peak value as indicated by the blue horizontal line. The RMS value is very import in power supply design. Another value is the Average value, this is 0.637 * the peak value, indicated in yellow.



Relationship between RMS, Peak and Average Values.


Figure 2

To quickly convert between Peak, Average or RMS values use the multipliers in Figure 2. So, for example take an AC voltage source of 12V RMS. To convert from RMS to its peak value (light blue on the pie chart) multiply by 1.41. E.g. 1.41 * 12 = 16.92 V pk. The peak to peak value would be double or 33.84V accounting for the positive and negative half cycles.

Resistive Circuits
AC circuits involving purely resistance, behave the same way as DC circuits. Both voltage and current have 0 phase shift as shown in the diagram below:

The AC voltage source is shown by V feeding a resitive load, R. The current and voltage shown in the waveform have no phase difference at all. Current i is given by:

i = V
R

Inductive Circuits
An AC generator with a parallel connected inductor is shown below. The inductor presents a high impedance to AC current, and as the voltage continues to grow, the current builds up slowly. The current (blue) lags the voltage waveform (green) by 90°. The phase angle, symbol θ is shown on the diagram below.

The inductors resistance to AC is called inductive reactance, symbol XL and is calculated with the formula:

XL = 2πfL

As I = V  then I=  V
XL 2πfL

Capacitive Circuits
An AC generator with a parallel connected capacitor is shown below. The capacitor initially presents a low impedance to AC current, decreasing as the voltage rises. The current (blue) leads the voltage waveform (green) by 90°. The phase angle, symbol θ is shown on the diagram below.

The capacitors resistance to AC is called capacitive reactance, symbol XC and is calculated with the formula:

XC = 1
2πfC

As I = V  then I=  2πfCV
XC

Series Resistor Inductor Circuits
All inductors have resistance as they are made from a coil of wire. Often additional series resistance will be added to shape the response of the circuit. A fixed resistor will offer the same resistance to an AC circuit at all frequencies, whereas the inductors resistance will change with frequency. The combined resistance is now an impedance, and again the current will lag the voltage through the inductor. The phase angle, symbol θ is shown on the diagram below.

The impedance, Z of the combined resitor and inductor is now:

Z = √ R2 + X2)

As I = V  then I=  V
Z R2 + X2)

The phase angle between voltage and current is:

Tan θ = X
R

Where X is the inductive reactance, 2πfL

Series RLC Circuits
The combination of series resistor, inductor and capacitance is often used in filter circuits to tailor frequency response.

The combined impedance is:

Z = √ R2 + (XL - XC)2)

I = V
Z

XL = 2πfL

XC = 1
2πfC

Current Lags if XL > X C

Current Leads if XC > X L

Phase angle θ given by:

Tan θ = XL - XC  or   XC - XL
R R


Ohms Law for AC Circuits
Knowing the phase angle between voltage and current in an AC circuit allows you to calculate ohms law. This is shown in the pie chart of figure 3 below:


Ohms Law for AC Circuits.


Figure 3

In Figure 3, V is Volts, I is Amps, Z is impedance in ohms and P is true power in watts, and θ is phase angle in degrees.