AC Circuit Analysis

AC Circuit Analysis is created by rotating a coil within a magnetic field, and alternating voltages and currents form the basis of AC Theory.

More commonly called Direct Current or DCis a form of electric current or voltage that flows in only one direction around an electrical circuit, making it a "one-way" source.

In general, both DC currents and voltages are produced by power supplies, batteries, dynamo and solar cells, to mention a few.A DC voltage or current has a constant magnitude (amplitude) and a certain direction associated with it.For example, +12V represents 12 volts in a positive direction or 5 volts in a negative direction of -5V.

We also know that DC power supplies do not change their values over time, it is a constant value that flows in the direction of constant state.In other words, dc always maintains the same value, and a fixed one-way DC resource never changes or is negative unless its connections are physically reversed.An example of a simple DC or direct current circuit is shown below.

DC Circuit and WaveForm

ac circuit analysis

On the other hand, the AC WaveFormis defined as a function that changes in both size and direction, more or less evenly over time, making it a "two-way" waveform.An AC function can represent a signal source in the form of an AC waveform that follows a power source or a mathematical sinusoid defined in the following figure in general: A(t) = A max *sin(2πεt).

To give the term AC or the full definition of Alternating Current, it refers to a waveform that changes over time in general, and the most common of these is called a Sinusoid, known as the Sinusoidal WaveForm.Sinusoidal waveforms are more commonly referred to as Sinus Waves by their short definitions.Sinus waves are one of the most important types of AC waveforms used in electrical engineering.

The shape obtained by drawing the instantaneous coordinate values of the voltage or current against time is called ac waveform.An AC waveform continuously changes its polarity every half cycle, ranging from a positive maximum value and a negative maximum value respectively over time, a common example of this is the domestic grid voltage supply that we use in our homes.

This means that the AC WaveForm is a "time-dependent signal" and that the most common type of time-dependent signal is that of the Periodic WaveForm.The periodic or AC waveform is the resulting product of a rotating electric generator.In general, the shape of any periodic waveform can be created using a basic frequency and overlapping it with harmonic signals of varying frequencies and amplitude.

Alternating voltages and currents cannot be stored in batteries or cells such as direct current (DC), it is much easier and cheaper to produce these quantities using alternators or waveform generators when necessary.The type and shape of an AC waveform depends on the generator or device that produces them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves.The main characteristics of an AC WaveForm are defined as follows:

AC WaveForm Properties

  • The period, (T), is the time in seconds that takes for the waveform to repeat itself from start to finish.This can also be called periodic time of waveform for sinus waves or Pulse Width for square waves.
  • Frequency, (ε) is the number of times the waveform repeats itself in a second.The frequency is the opposite of the time period ( ε = 1/T), the unit of which is Hertz (Hz).
  • Amplitude (A) is the size or density of the signal waveform measured in volts or amps.

In our tutorial on waveforms, we looked at different waveforms and said, "Waveforms are basically a visual representation of a voltage or change of current drawn on a time base."In general, for AC waveforms, this horizontal baseline represents the zero condition of voltage or current.Any part of the AC-type waveform on the horizontal zero axis represents a voltage or current flowing in one direction.

Similarly, any part of the waveform that falls below the horizontal zero axis represents a voltage or current that flows in the opposite direction to the first.Usually for sinusoidal AC waveforms, the shape of the waveform above the zero axis is the same as the shape below it.However, this is not always the case for most non-power AC signals, including sound waveforms.

The most common periodic signal waveforms used in Electrical and Electronics Engineering are Sinusoidal WaveForms.However, an alternative AC waveform may not always take a proper shape based on the function of the trigonometric sinus or cosine.AC waveformscan also take the form of Complex Waves , Square Waves, or Triangular Waves, which are shown below.

Periodic WaveForm Types

ac circuit analysis

The time it takes for an AC WaveForm to complete a full pattern from positive to negative half and again to zero baseline is called Loop, and a full cycle contains both a positive half-loop and a negative half loop.The time it takes for the waveform to complete a full loop is called the Periodic Time of the waveform and is indicated by the symbol "T".

The number of full cycles produced within one second (cycles per second) is called the Frequency , symbol ε of the alternate waveform.The frequency is measured in Hertz ( Hz), named after the German physicist Heinrich Hertz.

Then we can see that there is a relationship between cycles (oscillations), periodic time and frequency (number of cycles per second), so if there is a number of cycles per second, each cycle should take 1/ε seconds to complete.

Relationship Between Frequency and Periodic Time

ac circuit analysis

AC WaveForm Questionnaire 1

1. What is the periodic time of a waveform of 50 Hz?

2. What is the frequency of an AC waveform with a periodic time of 10 mS?

1).

ac circuit analysis

2).

ac circuit analysis

Frequency was referred to as "number of revolutions per second", abbreviated as "cps", but is more commonly indicated in units called "Hertz" today.The frequency for a domestic grid supply will be 50Hz or 60Hz depending on the country and is fixed by the rotational speed of the generator.But a hertz is a very small unit, so prefixes are used at higher frequencies such as kHz , MHz and even GHz, indicating the order of magnitude of the waveform.

Definition of Frequency Prefixes

PrefixDefinitionAbbreviationPeriodic Time
KiloThousandKhz1 ms
MegaMillionMhz1 us
GigaBillionGhz1ns
TeraTrillionTHz1ps

Amplitude of AC WaveForm

In addition to knowing the periodic time or frequency of the alternate quantity, another important parameter of the AC waveform is Amplitude,known as Maximum or Peak value, represented by the terms V max for voltage or I max for current.

Peak value is the maximum value of the voltage or current reached by the waveform during each half-cycle measured from the zero baseline. Unlike a DC voltage or current with a fixed state that can be measured or calculated using the Ohm Act, an alternative quantity constantly changes its value over time.

For pure sinusoidal waveforms, this peak value will always be the same for both half cycles ( +Vm = -Vm), but for non-sinusoidal or complex waveforms, the maximum peak value can be very different for each half cycle.Sometimes, alternative waveforms are given a top-to-top, V p-p value, which is the maximum peak value, +V max, and minimum peak value during a full cycle, the distance at the voltage between -V max, or total.

Average Value of AC WaveForm

The average value of continuous DC voltage will always be equal to the maximum peak value, since the DC voltage is constant.This average value will change only when the task cycle of the DC voltage changes.In pure sinus wave, if the average value is calculated over the full cycle, the average value will be equal to zero, as the positive and negative halves will cancel each other out.Therefore, the average value of an AC waveform is calculated or measured only over half a cycle, and this is shown below.

Average Value of a Non-Sinusoidal WaveForm

ac circuit analysis

To find the average value of the waveform, we need to calculate the area below the waveform using the mid-ordinate rule, the trapezope rule, or the Simpson rule, which is commonly found in mathematics.The approximate area below any irregular waveform can be easily found using only the middle coordinate rule.

The zero-axis baseline is divided into any number of equal parts, and in our simple example above, this value is nine (V 1 to V 9).The more ordinate lines are drawn, the more accurate the final average or average value.The average value will be the collection of all snapshot values added together, and then divided by the total number.It is given as this.

Average Value of AC WaveForm

ac circuit analysis

Where: n equals the actual median coordinate number used.

For a pure sine wave, this average or average value will always be equal to 0.637 * V max, and this relationship also applies to the average values of the current.

RMS Value of AC WaveForm

The average value of an AC waveform we calculated above: 0.637*V max is NOT the same as the value we will use for dc welding.This is because, unlike a DC resource with a constant value, an AC waveform changes continuously over time and has no constant value.Therefore, for an alternating current system that provides the same amount of electrical power as a DC equivalent circuit, the equivalent value is called an "active value".

The effective value of the sinus wave produces the same I 2*R heating effect at a load, as we expect to see if the same load is fed by a constant DC feed.The active value of a sine wave is more commonly known as the Root Mean Square or simply the RMS value, as the voltage or square of the current is calculated as the square root of the mean (average).

That is, V rms or I rms is given as a square root of the average of the sum of all square medium coordinate values of the sine wave.The RMS value for any AC waveform can be found from the following modified average value formula, as shown.

RMS Value of AC WaveForm

ac circuit analysis

Where: n equals the number of middle coordinates.

The RMS value for a pure sinusoidal waveform will always be equal to 0.707*VMax,and this relationship also applies to the RMS values of the current. The RMS value of a sinusoidal waveform is always greater than the average value, except for the rectangular waveform. In this case, the heating effect remains constant, so that the average and RMS values are the same.

A final comment on RMS values.Most multimeters, digital or analogue, measure only the RMS values of voltage and current, not the average, unless otherwise specified.Therefore, when using a multimeter in a direct current system, the reading will be equal to I = V/R, and the reading for an alternating current system will be equal to Irms = Vrms/R.

You can also access our content about TRUE RMS Multimeters here.

Also, when calculating RMS or peak voltages, except for average power calculations, use only V RMS or peak current, peak voltage, Vp to find ip values to find I RMS values.Do not mix them together because the Average, RMS, or Peak values of the sinus wave are completely different and your results are absolutely incorrect.

Form Factor and Crest Factor

Although it is rarely used these days, both the Form Factor and the crest factor can be used to provide information about the true shape of the AC waveform. The form factor is the ratio between the average value and the RMS value and is given as follows.

ac circuit analysis

For a pure sinusoidal waveform, the Form Factor will always equal 1.11.The Peak Factor is the ratio between the RMS value of the waveform and the Peak value, and is given as such:

ac circuit analysis

For a pure sinusoidal waveform, the Peak Factor will always equal 1,414

AC WaveForm Questionnaire 2

A 6-amp sinusoidal alternating current passes through a resistance of 40Ω.Calculate the average voltage and peak voltage of the source.

The RMS Voltage value is calculated as follows:

ac circuit analysis

The Average Voltage value is calculated as follows:

ac circuit analysis

The Peak Voltage value is calculated as follows:

peak voltage

The use and calculation of average, RMS, Form factor and Peak Factor can also be used with any type of periodic waveform, including Triangle, Square, Saw Gear or any other irregular or complex voltage/current waveform shape.The transformation between various sinusoidal values can sometimes be confusing, so the following table provides an appropriate way to convert the value of one sinus wave to another.

Sinusoidal WaveForm Transformation Table

ConvertedMultiplication CriteriaMultiplication Criteria SourceDesired
Reaching the climax2(√ 2 ) 2ndHill to Hill
Hill to Hill0.51/2Hill
Hill0.7071/(√ 2 )RMS
Hill0.6372/πAverage
Average1.570π/2Hill
Average1.111π/(2√ 2 )RMS
RMS1.414√ 2Hill
RMS0.901(2√ 2 )/πAverage

In the next lesson on Sinusoidal WaveForms, we will look at the principle of creating a sinusoidal AC waveform (a sinusoid) together with angular speed representation.