Advertisement

Phase Difference and Phase Shift

Phase Difference and Phase Shift

Phase Difference is used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values

Phasors are an effective way of analysing the behavour of elements within an AC circuit when the circuit frequencies are the same. The result of adding together two phasors depends on their relative phase, whether they are “in-phase” or “out-of-phase” due to some phase difference.

A sinusoidal waveform is an alternating quantity that can be presented graphically in the time domain along a horizontal axis. As a time-varying quantity, sinusoidal waveforms have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and points.

However, not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but may be “shifted” to the right or to the left of 0o by some value when compared to another sine wave.

For example, comparing a voltage waveform to that of a current waveform. This then produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any sine wave that does not pass through zero at t = 0 has a phase shift.

The difference or phase shift as it is also called of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. In other words phase shift is the lateral difference between two or more waveforms along a common axis and sinusoidal waveforms of the same frequency can have a phase difference.

The difference between phases, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ  = 0 to 360o depending upon the angular units used.

Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, +10mS or – 50uS but generally it is more common to express phase difference as an angular measurement.

Then the equation for the instantaneous value of a sinusoidal voltage or current waveform we developed in the previous Sinusoidal Waveform will need to be modified to take account of the phase angle of the waveform and this new general expression becomes.

Phase Difference Equation

phase angle
  • Where:
  •   Am  –  is the amplitude of the waveform.
  •   ωt  –  is the angular frequency of the waveform in radian/sec.
  •   Φ (phi)  –  is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.

If the positive slope of the sinusoidal waveform passes through the horizontal axis “before” t = 0 then the waveform has shifted to the left so Φ >0, and the phase angle will be positive in nature, giving a leading phase angle. In other words it appears earlier in time than 0o producing an anticlockwise rotation of the vector.

Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal x-axis some time “after” t = 0 then the waveform has shifted to the right so Φ <0, and the phase angle will be negative in nature producing a lagging phase angle as it appears later in time than 0o producing a clockwise rotation of the vector. Both cases are shown below.

Phase Relationship of a Sinusoidal Waveform

sinusoidal phase relationship

 

Firstly, lets consider that two alternating quantities such as a voltage, v and a current, i have the same frequency ƒ in Hertz. As the frequency of the two quantities is the same the angular velocity, ω must also be the same. So at any instant in time we can say that the phase of voltage, v will be the same as the phase of the current, i.

Then the angle of rotation within a particular time period will always be the same and the phase difference between the two quantities of v and i will therefore be zero and Φ = 0. As the frequency of the voltage, v and the current, i are the same they must both reach their maximum positive, negative and zero values during one complete cycle at the same time (although their amplitudes may be different). Then the two alternating quantities, v and i are said to be “in-phase”.

Two Sinusoidal Waveforms – “in-phase”

in-phase sinusoids

 

Now lets consider that the voltage, v and the current, i have a phase difference between themselves of  30o, so (Φ  = 30o or π/6 radians). As both alternating quantities rotate at the same speed, i.e. they have the same frequency, this phase difference will remain constant for all instants in time, then the phase difference of  30o between the two quantities is represented by phi, Φ as shown below.

Phase Difference of a Sinusoidal Waveform

phase difference of two sinusoids

 

The voltage waveform above starts at zero along the horizontal reference axis, but at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30o later. Then there exists a difference in the phases between the two waveforms as the current cross the horizontal reference axis reaching its maximum peak and zero values after the voltage waveform.

As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by an amount determined by phi, Φ and in our example this is 30o. So we can say that the two waveforms are now 30o out-of phase. The current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle, Φ. Then in our example above the two waveforms have a Lagging Phase Difference so the expression for both the voltage and current above will be given as.

lagging phase difference

 

  Where current, i “lags” voltage, v by phase angle Φ

Likewise, if the current, i has a positive value and crosses the reference axis reaching its maximum peak and zero values at some time before the voltage, v then the current waveform will be “leading” the voltage by some phase angle. Then the two waveforms are said to have a Leading Phase Difference and the expression for both the voltage and the current will be.

leading phase difference

 

  Where current, i “leads” the voltage v by phase angle Φ

The phase angle of a sine wave can be used to describe the relationship of one sine wave to another by using the terms “Leading” and “Lagging” to indicate the relationship between two sinusoidal waveforms of the same frequency, plotted onto the same reference axis. In our example above the two waveforms are out-of-phase by 30o. So we can correctly say that i lags v or we can say that v leads i by 30o depending upon which one we choose as our reference.

The relationship between the two waveforms and the resulting phase angle can be measured anywhere along the horizontal zero axis through which each waveform passes with the “same slope” direction either positive or negative.

In AC power circuits this ability to describe the relationship between a voltage and a current sine wave within the same circuit is very important and forms the bases of AC circuit analysis.

The Cosine Waveform

So we now know that if a waveform is “shifted” to the right or left of 0o when compared to another sine wave the expression for this waveform becomes Am sin(ωt ± Φ). But if the waveform crosses the horizontal zero axis with a positive going slope 90o or π/2 radians before the reference waveform, the waveform is called a Cosine Waveform and the expression becomes.

Cosine Expression

Cosine Wave

The Cosine Wave, simply called “cos”, is as important as the sine wave in electrical engineering. The cosine wave has the same shape as its sine wave counterpart that is it is a sinusoidal function, but is shifted by +90o or one full quarter of a period ahead of it.

Difference between a Sine wave and a Cosine wave

phase difference

 

Alternatively, we can also say that a sine wave is a cosine wave that has been shifted in the other direction by -90o. Either way when dealing with sine waves or cosine waves with an angle the following rules will always apply.

Sine and Cosine Wave Relationships

sine and cosine relationship

 

When comparing two sinusoidal waveforms it more common to express their relationship as either a sine or cosine with positive going amplitudes and this is achieved using the following mathematical identities.

sine and cosine identities

 

By using these relationships above we can convert any sinusoidal waveform with or without an angular or phase difference from either a sine wave into a cosine wave or vice versa.

In the next tutorial about Phasors we will use a graphical method of representing or comparing the phase difference between two sinusoids by looking at the phasor representation of a single phase AC quantity along with some phasor algebra relating to the mathematical addition of two or more phasors.

75 Comments

Join the conversation

Error! Please fill all fields.

  • NAGESH PANDE

    I wish to parallel R & Y phases for inter-phase power flow using phase shifting techniques
    kindly let me know how it can be achieved for Railway traction 25 kV system

  • Fidelia

    How do I change to a different phase when my current is low

    • T.S.Ganesh

      Usually there is a Box with three knobs for three phases. If one phase doesn’t work the knob of the non-working phase is shifted to one of the remaining two working one’s

  • Jay

    Now that’s what we call a tutorial!!!The best, I gv u that!!!

    Wonderful!!

  • WILFRED CHEWE

    VERY GOOD TUTORIALS AND WOULD LOVE TO LEARN MORE.

  • nazari

    hello
    please show a circuit with phase diffrence between current and voltage with oscilosecope

  • T.S.Ganesh

    Your Tutorials are Lucid.

  • Ali Khaled

    In the beginning of the lesson in the negative slope. How it is going to make a clockwise rotation

  • Anuja bagul

    9689388766

  • Maria

    Theta or Phi? Phasor lagging and leading equations seem wrong. Theta is wt. If it is wt-theta then it must be ending in zero. Leaving mit with ?????
    Please comment.

  • Nduka ebuka

    This article or whatever I may call it is very resourceful.

    • Wayne Storr

      It’s called a “Tutorial”, which is why the website is known as “Electronics Tutorials”. The clue is in the name.

  • Justin

    You completely lost me at “When comparing two sinusoidal waveforms it more common to express their relationship as either a sine or cosine with positive going amplitudes and this is achieved using the following mathematical identities.”

  • lindo

    none

  • Alejandro Nava

    You have an error in the last image, when you said “±sin(ωt) = cos(ωt ± 90°)”, but that’s wrong. As correctly shown in the second last image, it should be “±sin(ωt) = cos(ωt ∓ 90°)”. You should fix it.

  • Emmanuel

    Good information

    • Len

      That is very good information but i still like to speak with a video engineer. Sometime. It’s about this color phase circuit I am trying to design.

  • Anup Sadhu

    what is the phase difference and phasor diagram of v = vmcos (wt-30⁰) and i = imsin (wt+60⁰)

    • Wayne Storr

      v = Vm cos (wt – 30o)

      i = Im sin(wt + 60o)
      i = Im cos(wt + 60o – 90o)
      i = Im cos(wt – 30o)

      Thus the phase difference is zero

  • James

    Sample of phase difference between current and voltage

    • Len horowitz

      Well, that is good information. I need to build, quickly a funny :”hue” control for the first Ampex color video recording system. They had a crazy way of modifying a color receiver to use the color pilot signal; (511Kc) which was 3.58 divided by seven- of all things, to lock up color with good fidelity. They added a phase control which shifts the phase by increments of 90 degrees using 3 coils I wanted a vari-cap or, a way to go from zero, to 90, to 270 and back to zero with a knob, Not a switch. as it turns out, it more tricky than I thought. It is for the early Ampex VR 7500, the 1 inch “A” format recorder (you can look that up). Fascinating piece of engineering for its day in September of 1966.

  • Len

    I need a circuit for our history of Recorded Sound museum.. This concerns a pilot color system where I want to shift 3.58 Mhz 270 degrees. I couldn’t find anything except phase shift oscillators or low frequency circuits (audio band). The original circuit, employed a rotory switch with four possible selections: “0” shift, 90 180 270 degrees. I wanted one knob. For color hue.
    Any clues?

    • Wayne Storr

      There are many different circuits and applications for phase shifting in the high frequency radio range, using discrete R, L and C passive devices such as connecting reactive components spaced one-quarter wavelength apart across the supply/transmaission line will produce the required phase shift. Op-amps, timers, filters and other such active circuits can also be used. Phase shifting is basically about time delaying the base frequency. Your 3.58MHz frequency will have a period of 279.33nS, thus delaying the input signal by multiples of 69.83nS would give the desired 90, 180, and 270o of phase shift. Then you need to research time delay phase shifter circuits with low insertion loss.

  • Anas aamer

    Where the wave move when the phase angle is positive or negative

  • UAJ

    Can someone please tell that how am I supposed to find the angle by which one sinusoidal wave leads or lags the other one? I cant solve the 2nd part of this question:
    “Find the angle by which i1 lags v1 if v1 = 120 cos(120πt − 40◦) V
    and i1 equals (a) 2.5 cos(120πt + 20◦) A; (b) 1.4 sin(120πt − 70◦) A;
    (c) −0.8 cos(120πt − 110◦) A.”

    • Wayne Storr

      For sines or cosines with an angle, the cosine wave is a sine wave shifted by +90o, and a sine wave is a cosine wave shifted by -90o.

      As the frequency is the same and V and I are both given as cosines, the result for question (a) is:
      V1 = 120 cos(120πt − 40o) = 120 cos(377t – 40o) volts
      I1 = 2.5 cos(120πt + 20o) = 2.5 cos(377t + 20o) amps

      The phase difference is: 20 – (-40) = 60o, That is i leads v by 60o
      Note (b) is sine -70. and (c) is -0.8 so phasor is (-110 – 180) = -290

  • Rohit Singh

    What is the effect of phase difference between any two electrical quantities?