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AC FundamentalsAlternating CurrentSingle Phase AC Circuits

RMS Value, Average Value, Peak Value, Peak Factor And Form Factor in AC

What is RMS Value, Peak Value, Average Value, Instantiations Value, Form Factor, Peak Factor & Other related Terms to AC Circuits and Sine wave?

AC and DC Currents

It is known that the polarity of DC voltage and direction of DC current are always same i.e. it is a unidirectional value which does not change the polarity as well as direction as shown in fig 1.

On the other hand, (AC) Alternating Current or Voltage  is one which regularly changes its direction as well as its value. In other words, alternating current (AC) is a type of current which flows first in one direction and secondly, it flows in the opposite direction. In each cycle, it changes the value from zero to the maximum and again hit the zero value.

The value of alternating current or voltage can be express in AC (Sinusoidal) Sine wave as shown in fig (1) below.

demonstration of ac and dc
Fig – 1 Difference Between AC & DC

In AC, its not possible to represent the magnitudes as its amplitude of AC sine wave continuously changes with time.

This way, we have multiple options to expressed the magnitude and different values related to an AC sine wave such as:

  • RMS Value
  • Average Value
  • Instantaneous Value
  • Maximum or Peak Value
  • Peak to Peak Value
  • Peak Factor
  • Form Factor
  • Other related terms

We will discuses all of them in details as follow.

What is RMS (Root Mean Square) Value ?

The RMS (Root Mean Square) value (also known as effective or virtual value) of  of an alternating current (AC) is the value of direct current (DC) when flowing through a circuit or resistor for the specific time period and produces same amount of heat which produced by the alternating current (AC) when flowing through the same circuit or resistor for a specific time.

RMS Value

The value of an AC which will produce the same amount of heat while passing through in a heating element (such as resistor) as DC produces through  the element is called R.M.S Value.

In short,

The RMS Value of an Alternating Current is that when it compare to the Direct Current, then both AC and DC current produce the same amount of heat when flowing through the same circuit for a specif time period.

For a sinusoidal wave,RMS Value Formulas for Current & Voltage in sine wave

or

IRMS = 0.707 x IM , ERMS = 0.707 EM

Actually, the RMS value of a sine wave is the measurement of heating effect of sine wave. For example, When a resistor is connected to across an AC voltage source, it produce specific amount of heat (Fig 2 – a). When the same resistor is connected across the DC voltage source as shown in (fig 2 – b). By adjusting the value of DC voltage to get the same amount of heat generated before in AC voltage source in fig a. It means the RMS value of a sine wave is equal to the DC Voltage source producing the same amount of heat generated by AC Voltage source.

R.M.S Voltage Values
Fig 2 – RMS Values of Current & Voltage

In more clear words, The domestic voltage level in US is 110V, while 220V AC in UK. This voltage level shows the effective value of ( 110V or 220V R.M.S) and it shows that the home wall socket is capable to provide the same amount of average positive power as 110V or 220V DC Voltage.

Keep in mind that the ampere meters and volt meters connected in AC circuits always showing the RMS values (of current and voltage).

For AC sine wave,  RMS values of current and voltage are:

IRMS = 0.707 x IM , VRMS = 0.707 VM

To find the RMS value of a sine wave, We may use the following two methods.

  1. Mid Ordinate Method
  2. Integration Method.

Lets see how to find the R.M.S values of a sine wave.

Methods for Finding RMS Value of Sine Wave.

Analytical Method

Method 1

We know that the value of sinusoidal alternating current (AC) =

Im Sin ω θ = Im Sin θ

While the mean of square of instantaneous values of current in in half or complete cycle is:

Analytical Method to Find RMS Value

The Square root of this value is:

Methods for Finding RMS Value of Sine Wave

Hence, the RMS value of the current is (while putting I = Im Sin θ):

Analytical Method for RMS

Now,

Analytical Method for RMS Voltage

Analytical Method to Find RMS Voltage

Therefore, We may find that for a symmetrical sinusoidal current:

IRMS = Max Value of Current x 0.707

Method 2.

Let  i = Sin ω θ  = Im Sin θ

Mean value of i2

Analytical Method to find RMS Voltage & Current

Method 3

Let  i = Sin ω θ  = Im Sin θ

Analytical Method for RMS value of V & I

Graphical or Mid-Ordinate Method

This method is known as mid ordinate or graphical method to find the value of RMS voltages by using mid-ordinates or finding the instantaneous value of AC waveform. For clear understanding, a solved example is given below the explanation.

There are many instantaneous voltages in an AC sine wave and its depends on the time interval. As shown in the fig 3 below where the number of mid ordinates are 12, (The more the mid ordinates, the more accurate will be the result). It shows at instance of t = 1, t = 2, t = 3 …. tn, the instantaneous voltages levels are V1, V2, V3 …. Vn respectively.

Graphical or Mid-Ordinate Method for RMS Value
Fig 3 – Mid-Ordinate or Graphical Method for RMS

First, We will find the instantaneous values of voltages for each each time period like t = 1, t =2 … t = n etc. To find the RMS value, We would have to find the square values of each voltage levels in the AC waveform which shows the square part of the RMS Value.

V12 + V22 + V32 + ….. Vn2

Now the squired values of voltages are divided by the number of mid-ordinates which shows the mean value of the RMS voltage.Mid-Ordinate Method for RMS

For Example,

Number of ordinates used in above fig 3 = 12

Assume the peak value voltage (Max Voltage i.e. amplitude = VPK or VMax) is 12V for the alternating waveform. The waveform is divided in 12 mid-ordinates as shown below:

Voltage2V4V6V8V10V12V10V8V6V4V2V0V
Angle15o30o45o60o75o90o105o120o135o150o165o180o

The RMS Voltage is calculated as follow:Mid-Ordinate Graphical Method for RMS VoltageGraphical Method for RMS

This way, The Value of RMS Voltage is 6.97V by using the graphical or mid-ordinate method to find the RMS value of voltage.

RMS Voltage and Current Equations

RMS Values of Current and Voltage related to Peak Value or Max Value (Both are same).

RMS Value of Voltage and Current related to Peak Voltage

VRMS = 0.707 x VPK   ,  IRMS = 0.707 x IPK 

RMS Value of Voltage and Current related to Max value Voltage

VRMS = 0.707 x VM   ,  IRMS = 0.707 x IM 

RMS Values of Current and Voltage related to Peak to Peak Value.

RMS Value of Voltage and Current related to Peak to Peak Voltage

VRMS = 0.3536 x VP-P   ,  IRMS = 0.3536 x IP-P 

RMS Values of Current and Voltage related to Average Value.

RMS Value of Voltage and Current related to Average value of Voltage

VRMS = 1.11 x VAV   ,  IRMS = 1.11 x IAV

Here all in one picture.

RMS Voltage and Current Equations & Formulas

RMS Voltage Value Formulas for Different Wave forms

In the below table, the RMS Voltage Value formulas are shown for different kind of sinusoidal wave forms.

Waveform TypeFormula for RMS Value (VRMS)
Sine WaveVPK / √2
Half wave rectified sine waveVPK / √2
Full wave rectified sine waveVPK / √2
Square waveVPK
Triangle waveformVPK / √3
Sawtooth waveformVPK / √3

RMS Voltage Calculator 

In the RMS Voltage Value Calculator, You can calculate the value of RMS voltage from different related values like Average Value, Peak Value and Peak to Peak Value.

What is Average Value

If we convert the alternating current (AC) sine wave into direct current (DC) sine wave through rectifiers, then the converted value to the DC is known as the average value of that alternating current sine wave.

Average Value of Voltage
Fig 4 – Average Value of Voltage

If the maximum value of alternating current is “IMAX“, then the value of converted DC current through rectifier would be “0.637 IM” which is known as average value of the AC Sine wave (IAV).

Average Value of Current = IAV = 0.637 IM

Average Value of Voltage = EAV = 0.637 EM

The Average Value (also known as Mean Value) of an Alternating Current (AC) is expressed by that Direct Current (DC) which transfers across any circuit the same amount of charge as is transferred by that Alternating Current (AC) during the same time.

Keep in mind that the average or mean value of a full sinusoidal wave is “Zero” the value of current in first half (Positive) is equal to the the next half cycle (Negative) in the opposite direction. In other words, There are same amount of current in the positive and negative half cycles which flows in the opposite direction, so the average value for a complete sine wave would be “0”. That’s the reason that’s why we don’t use average value for plating and battery charging. If an AC wave is converted into DC through a rectifier, It can be used for electrochemical  works.

What is Mean Value in Sine Wave
Fig 5- Average Value of Current

In short, The average value of a sine wave taken over a complete cycle is always zero, because the positive values (above the zero crossing) offset or neutralize the negative values (below the zero crossing.)

Related Post: Star Connection (Y): Three Phase Power, Voltage & Current Values

Methods for Finding Average Value of Sine Wave.

Mid-Ordinate or Graphical Method

In this method, the half cycle of a sin wave is divided in equal number of time periods where the duration of each time period is “t/n”.

Suppose the average values of instantaneous currents in each time interval is I2, 12, I3 … In. To find the average value for each time interval, both the left and right vertical lines are added and divided by two. The same apply to all time intervals to find the average value for each instance.

Now, all the average values are added and divided by the number mid-ordinates (i.e. time periods) which shows the overall average of half cycle of a sine wave. the formula for average value is shown below:

Average Current Formula Average Current Value Formula

Example:

Mid-Ordinate or Graphical Method for Average Value of Current
Fig 6 – Mid-Ordinate or Graphical Method for Average Value of Current

Number of ordinates used in fig 6 = 12

Assume the peak value of current(Max Current i.e. amplitude = IPK or IMax) is 12A for the alternating waveform. The waveform is divided in 12 mid-ordinates as shown below:

Current2A4A6A8A10A12A10A8A6A4A2A0A
Angle15o30o45o60o75o90o105o120o135o150o165o180o

This way, The Average Value of current is 6A by using the graphical or mid-ordinate method to find the average value of current.

Analytical Method

We know that the standard equation of alternating current is

i = Sin ω θ  = Im Sin θ

  • Maximum value of current on sine wave = Im
  • Average value of current on sine wave = IAV
  • Instantaneous value of current on sine wave = i
  • The angle specified fir “i” after zero position of current = θ
  • Angle of half cycle = π radians
  • Angle of full cucle = 2π radians

(a) Average value of complete cycle:

Let i = Sin ω θ  = Im Sin θ

Analytical Method for Average Value Full Cycle

Thus, the average value of a sinusoidal wave over a complete cycle is zero.

(b) Average value of current over a half cycle

Analytical Method for Average Value Half Cycle

Average Value of Current (Half Cycle)

IAV = 0.637 VM

Similarly, the average value of voltage over a half cycle

VAV = 0.637 VM

Average Voltage and Current EquationsAverage Voltage and Current Equations and formulas

Its mean the value of average voltage or current is equal to multiplication of max or peak value of voltage and current respectively with the constant 0.637.

For example, if the max value or peak value of the sinusoidal wave is 360VPK or 360VMax, by using the above equation, the average value of the voltage would be:

 0.637 x 360VPK = 229.32 Av.

Average Value & RMS Value Formulas for Different Wave forms

In the below table, the Average Value and  RMS Voltage Value formulas are shown for different kind of sinusoidal wave forms.

Waveform TypeFormula for RMS VoltageFormula for Avg Voltage
Sine WaveVRMS = VPK/√2VAV = 0
Full rectified waveVRMS = VPK/√2VAV = 0.637 VPK
Half rectified waveVRMS = VPK/2VAV = 0.318 VPK
Sine wave with DC offsetVRMS = √(VDC2+VPK2/2)VAV = Vdc
Half sine with duration T and frequency fVRMS = VPK x √(f x T/2)VAV = 2f x T x VPK/π
Positive square wave with duration T and frequency fVRMS = VPK x √(f x T)VAV = f x T x VPK
Saw tooth wave with duration T and frequency fVRMS = VPK x √(f x T/3)VAV = f x T x VPK/2
Trapezoidal wave with frequency f, top segment T, and base segment B.VRMS =  VPK x √(fx ((B-T)+3xT)/3)VAV = f x VPK x ((T+B)/2)

What is Instantaneous Value

The value attained by an alternating quantity at any instant is known as instantaneous value. It is denoted by “i” and e.

in other words, the value of an alternating current or voltage at any particular moment us called an instantaneous value.

In fig 7 below, different instantaneous values of voltages or currents are shown at specific point and time period. The value of instantaneous current or  voltage are “+” in the positive cycle and  “-” in negative cycle in a sinusoidal wave. The curves are showing the values of different instantaneous voltages while the same curve can be drawn for current as well. In the fig 7, the value of instantaneous voltages are 2.5V at 1μs, 5.1V at 2μs, 8.9V at 3μs. While it is -2.3V at 4μs, -6.1V at 5μs and -9.2V at 6μs.

Instantaneous Value of Voltage
Fig 7 – Instantaneous Value of Voltage

What is Peak Voltage or Maximum Voltage Value ?

Peak value is also known as Maximum Value, Crest Value or Amplitude. It is the maximum value of alternating current or voltage from the “0” position no matter positive or negative half cycle in a sinusoidal wave as shown in fig 8. Its expressed as IM and EM or VP and IM.

Equations of Peak Voltage Value are:

VP = √2 x VRMS = 1.414 VRMS

VP = VP-P/2 = 0.5 VP-P

VPπ/2 x VAV = 1.571 x VAV

In other words, It is the value of voltage or current at the positive or the negative maximum (peaks) with respect to zero. In simple words, it is the instantaneous value with maximum intensity.

Maximum or Peak Value of Voltages
Fig 8 – Peak or Maximum Values of Voltages

Peak to Peak Value

The sum of positive and negative peak values is known as peak to peak value. Its expressed as IPP or VPP.

Equations and formulas for Peak to Peak Voltage are as follow:

VP-P =  2√2 x VRMS = 2.828 x VRMS

VP-P =2 x VP

VP-Pπ  x VAV = 3.141 x VAV

In other words, the peak to peak value of a sine wave, is the voltage or current from positive peak to the negative peak and its value is double as compared to peak value or maximum value as shown in fig 8 above.

What is Peak Factor

Peak Factor is also known as Crest Factor or Amplitude Factor.

It is the ratio between maximum value and RMS value of an alternating wave.

Peak Factor in AC Circuits

For a sinusoidal alternating voltage:

Peak Voltage Formula

For a sinusoidal alternating current:

Peak Current Value

What is Form Factor

The ratio between RMS value and Average value of an alternating quantity (Current or Voltage) is known as Form Factor.Form Factor in AC Circuit Sine wave

Form Factor Value Formula

In the below table, the form factor formulas and values are shown for different kind of sinusoidal wave forms.

Waveform TypeFormula for Form FactorValue
Sine Waveπ/2√21.11072073
Half wave rectified sine waveπ/21.5707963
Full wave rectified sine waveπ/2√21.11072073
Square wave11
Triangle waveform2/√31.15470054
Saw-tooth waveform2/√31.15470054

Other Terms Related To AC Circuits

Waveform

The path traced by a quantity (such as voltage or current) plotted as a function of some variable (such as time, degree, radians, temperature etc.) is called waveform.

Cycle

  1. One complete set of positive and negative values of alternating quality (such as voltage and current) is known as cycle.
  2. The portion of a waveform contained in one period of time is called cycle.
  3. A distance between two same points related to value and direction is known as cycle.
  4. A cycle is a complete alternation.

Period

The time taken by a alternating quantity (such as current or voltage) to complete one cycle is called its time period “T”.

It is inversely proportional to the Frequency “f” and denoted by “T” where the unit of time period is second.

Mathematically;

T = 1/f

Frequency

Frequency is the number if cycles passed through per second. It is denoted by “f” and has the unit cycle per second i.e. Hz (Herts).

The number of completed cycles in 1 second is called frequency.

It is the number of cycles of alternating quantity per second in hertz.

Frequency is the number of cycles that a sine wave completed in one second or the number of cycles that occurs in one second.

f = 1/T

Amplitude

The maximum value, positive or negative, of an alternating quantity such as voltage or current is known as its amplitude. Its denoted by VP, IP or EMAX and IMAX.

Alternation

One half cycle of a sine wave (Negative or Positive) is known as alternation which span is 180° degree.

RMS Value, Peak Value, Average Value, Instantiations Value, Form Factor, Peak Factor & Other related Terms to AC Circuits and Sine wave
Fig 9 – Different Terms used in AC Circuits and Sine Wave

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