AC Theory

Andy Collinson

Sections:
Introduction
Phase
Phase Difference
Time and Frequency
RMS, Peak and Average Values
Resistive Circuits
Reactive Circuits
Phase Change In Reactive Circuits
Inductive Circuits
Capacitive Circuits
Series RC Circuits
Series RL Circuits
Series RLC Circuits
Ohms Law for AC Circuits

Introduction
In electronics, ac (alternating current), sources are widely used. Ac Signals come in many forms, e.g. sine, pulse, square, triangular. Signals consisting of multiple frequencies like music and speech are all ac waveforms. For testing, Sinusoidal waveforms of a single frequency are used for testing audio systems and other applications, and are generated at a power station for transmission of electrical energy. A sinusoidal waveform can be mathematically predicted and is shown below. One complete sinusoidal cycle is completed over 360 degrees. The vertical axis is the amplitude and the horizontal axis is the phase of the waveform. The amplitude starts at zero volts at zero degrees, rises to a positive peak at 90 degrees and then decreases to zero volts at 180 degrees, The amplitude is then reversed and reaches a maximum negative peak at 270 degrees before reaching zero volts again at 360 degrees. These cycles continue as long as the wave is generated.

Phase
The phase of a periodic waveform is an angle that represents the number of periods for that waveform. As shown above one complete cycle takes 360 degrees. The time it takes to complete the cycle depends upon the frequency, but at any given time, the phase angle is represented by a point on the graph. Phase uses the symbol φ and is measured in degrees or radians. Angular velocity, symbol ω takes into account the waveforms frequency a and is equivalent to:

ω = 2 π f

Phase Difference
If two waveforms are displayed and start and finish at the same points in time, then these waves are in phase and have no phase difference. Now look at the graph below showing two waves called V(vc) and V(n001). The waves are not in phase and have a phase shift in time or phase difference. The phase difference is calculated where each wave crosses the horizontal axis at 0 Volts amplitude, and amount of phase shift is shown with the white line.

A phase shift in AC circuits always occurs with non-linear components such as inductors and capacitors. If the phase shift is a positive value then this is called a phase lead, if the value is negative, then it is a phase lag. To convert a phase shift in time to units of degrees then the following formula is used:

Phase φ = 360 * f * Δt
Where f is the frequency of the wave and Δt is the time difference in seconds. The output phase φ will be in units of degrees, So for exmple if two waves 1KHz sine wave have a time differnce of 0.1 ms then the phase difference is 360*1000*0.0001 = 36 °

Time and Frequency
The number of cycles per second is called the frequency, and the time it takes to complete is called the waveform period. The reciprocal of frequency gives the periodic time, and the reciprocal of periodic time gives frequency. For example, a sine wave of 1000 cycles per second (or 1 kHz) has a periodic time of 1/1000 or 0.001 seconds or 1 ms. Figure 1 below, shows two cycles of a 1kHz waveform:

sine wave representatio over time
Figure 1

The shape of the waveform is sinusoidal, and this is the most common waveform used in power supplies and signal generators. As can be seen above, 1 complete cycle takes 1 ms, the frequency is thus 1/periodic time or 1 kHz, and this wave has a positive peak of 10 Volts and a negative peak of -10 volts. The peak to peak value is the sum of positive and negative peaks, or 20 Volts peak to peak.

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

Reactive Circuits
When an AC current or voltage passes through a non-linear (not resistive) component then the current and voltage will be out of phase. This is known as a reactive circuit and the voltage and currents, and phase angle change with frequency.
The difference between current and voltage is called the phase angle and is measured in degrees or radians. Ohms law for AC involving pure resistance is calculated the same way as ohm's law for DC, but when reactive components are present the current and voltage will be out of phase and the resistance to AC is now called the impedance.

Phase Change in Reactive Circuits
The current in a circuit with a capaitor leads the voltage by 90 °. With a circuit containing just an inductor, then the voltage leads the current by 90° With any combination of resistor and capacitor or inductor there will be a phase lead or lag. This can be remembered in one word:

C I V I L
Using the word 'civil' for a capacitor C, then the current 'i' leads the voltage 'v' For an inductor 'l' as 'v' is before 'i' then the voltage leads the current in an inductor.

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 Capacitor Circuits
All capacitors have two plates and a dielectric material. When a voltage is applied across a capacitor, the initial current is very large and as the capacitor charges up, and decreases to zero when fully charges. By contrast the voltage across the capacitor grows slowly and voltage and currents are out of phase. A circuit containing a resistor and capacitor has a varying impedance depending om the frequency of the AC supply voltage. The current will still lead the voltage but the phase angle is now less than 90°. The phase angle, symbol θ is shown on the diagram below.

The impedance, Z of the combined capacitor and resistor network is:

Z = √ R2 + Xc2)

As I = V  then I=  V
Z R2 + Xc2)

The phase angle between voltage and current is:

Tan-1 θ = Xc
R
Where Xc is the capacitive reactance, 1/2πfC

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 + XL2)
As I = V  then I=  V
Z R2 + XL2)

The phase angle between voltage and current is:

Tan-1 θ = XL
R

Where XL 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-1 θ = XL - XC  or   XC - XL
R R

Ohms Law for AC Circuits
Knowing the impedance Z, for a given AC circuit allows you to calculate voltage or current. To calculate power then you must also know the phase angle between the voltage and the current. The pie chart below, figure 3 shows the relationship between voltage, current, impedance and power:

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.

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