Understanding Phasors: Complex Numbers in Sinusoidal Analysis
Introduction to Phasors
Phasors are complex numbers that represent the amplitude and phase of sinusoidal signals. For example, a sinusoid VT = 3 cos(Ωt + 30°) has an amplitude of 3 and a phase angle of 30°. The corresponding phasor encodes these two pieces of information as a complex number.
Phasors and RMS Values
Phasors can represent either the maximum amplitude or the RMS (root mean square) value of a sinusoid. The RMS value is the maximum amplitude divided by √2, so for VT = 3 cos(Ωt + 30°), the RMS phasor magnitude is 3/√2 with the same phase angle.
Advantages of Using Phasors
Phasors simplify the analysis of linear circuits excited by sinusoidal sources by converting time-dependent sinusoids into time-independent complex numbers. This method was introduced by Charles Steinmetz in 1893.
Review of Complex Numbers
Phasors are complex numbers, so understanding their forms is essential:
- Rectangular form: Z = X + jY, where X is the real part and Y is the imaginary part.
- Polar form: Z = R∠θ, where R is the magnitude and θ is the phase angle.
- Exponential form: Z = Re^(jθ), based on Euler's identity.
Conversions Between Forms
- Magnitude: R = √(X2 + Y2)
- Phase angle: θ = tan−1(Y/X)
- Real part: X = R cos θ
- Imaginary part: Y = R sin θ
Euler's Identity and Phasors
Euler's identity states e^(jθ) = cos θ + j sin θ. Using this, a sinusoid VT = VM cos(Ωt + θ) can be expressed as the real part of VM e^(j(Ωt + θ)). Suppressing the time-dependent term e^(jΩt) yields the phasor VM e^(jθ), a time-independent complex number representing the sinusoid's amplitude and phase.
Phasor Representation and Notation
Phasors are often written in polar form as VM∠θ or in bold uppercase letters (e.g., V). They represent the sinusoid without the time component, making calculations more straightforward.
Plotting Phasors
Phasors can be visualized as vectors in the complex plane:
- The real axis represents the real part.
- The imaginary axis represents the imaginary part.
- The vector's length is the magnitude (VM or IM).
- The angle with the real axis is the phase angle (θ).
For example, two phasors V = VM∠θ1 and I = IM∠-θ2 can be plotted with angles measured counterclockwise (positive) and clockwise (negative), respectively.
Important Considerations in Phasor Analysis
Phasor analysis is valid only when all signals have the same frequency. If signals have different frequencies, phasor analysis cannot be applied.
Visualization
An animation demonstrates a phasor vector rotating at angular frequency Ω, with the sinusoidal waveform corresponding to points on the circle traced by the vector.
Conclusion
Phasors provide a powerful tool for simplifying sinusoidal signal analysis in AC circuits by converting time-dependent functions into time-independent complex numbers. The next lecture will cover practical problem-solving using phasors.
For further reading on related topics, check out these resources:
we are done with sinusoids and now we will have discussion on phasers phaser is a complex number that represents the
amplitude and phase of a sinusoid so phasor is a complex number representing two things of a sinusoid first one is
the amplitude and the second one is the phase for example if we have a sinusoid VT and it is equal to 3 cos Omega T plus
30 degrees here the amplitude is 3 and the phases 30 degrees then corresponding to this sinusoid we will have a complex
number represented like this known as the phasor and it will have two informations about VD first one is the
amplitude that is 3 and the second one is the phase angle that is 30 degrees now here you can notice one thing 3 is
the maximum value of VT so this phasor corresponding to VT is for the maximum value we can also have the phasor
corresponding to the RMS value of VT and the RMS value will be 3 over root 2 and the phase angle will remain as it is and
we can see that the RMS value is 3 over root 2 because we know in case of sinusoids the RMS value is equal to the
maximum value divided by root 2 and when we convert the sinusoids to phasers then it is more convenient to work with
phasers as compared to sine or cosine functions and the phasers provide a simple means of analyzing the linear
circuits which are excited by the sinusoidal sources and the notion of solving the AC circuits using the
fizzles was first introduced by Charles statements in 1893 now before diving into more details of
phasers we will quickly revise the basics of complex numbers and we are required to know the
six of complex numbers because phasor is nothing but a complex number and away we'll start with the rectangular form of
complex numbers and let us say that our complex number is represented by Zed and we know it has two parts the first part
is known as the real part of the complex number and the second part we call imaginary part of the complex number and
let us say that the real part of the complex number is equal to X and the imaginary part of the complex number is
equal to Y and therefore we can say that our complex number Z is equal to X plus J Y and we know J is equal to under root
minus 1 so this form we call as rectangular form of the complex number and now we will talk about the polar and
exponential forms of the complex number the complex number is Ed in polar form is written as R which is the magnitude
with angle theta which is the phase angle and the exponential form of the complex number is equal to the magnitude
multiplied to E power J theta and as I told you our is the magnitude of the complex number and theta is the phase of
the complex number and now we will find out the relation between polar and rectangular forms that is we will try to
have R and theta when x and y are given and we will try to have x and y when R and theta are given and for this I have
taken the rectangular coordinate system the x axis is for the real values of the complex number and the y axis is for the
imaginary values of the complex number and let us say at our complex number Z is equal to X
plus J Y in the rectangular form and ER this means from origin to the value X on the real axis is our real part of the
complex number and from origin to the value Y on the imaginary axis is the imaginary part of the complex number and
the corresponding to x and y we will have a point on the complex plane and now we can visualize our complex number
as a vector like this and this vector will have the magnitude equal to R and this line will make an angle equal to
theta which is the phase of the complex number now when you focus on this triangle you
will find R is the hypotenuse of the triangle and we know it is equal to under root square of the base which is X
plus square of the perpendicular which is y so if we have x and y we can have R from this particular result and we know
that this slope is equal to 10 theta and 10 theta will be Y over X Y over X so from here we will have the phase angle
theta equal to 10 inverse Y over X so if we know Y and X we can have theta as well now what if we know R and theta and
we are interested in finding out Y and X then you can see that cos theta in this triangle will be equal to X divided by R
so from here we will have the real part of the complex number equal to R multiplied to cos theta and sine theta
will be equal to Y over R so we can say that the imaginary power is equal to R sine theta and therefore
our complex number said will be equal to R cos theta plus J sine theta now we will continue with our discussion on
phasers the phasor representation is based on Euler's identity and the we know the Euler's identity is e power J
theta is equal to cos theta plus J sine theta and when you compare the right hand side with this you will find the
real part of E power J theta is equal to cos theta and the imaginary part of E power J theta is equal to sine theta and
let us say that the sinusoidal function we are having is VD and it is equal to VM cos Omega T plus theta now compare VM
cos Omega T plus theta with cos theta you will find in place of theta we have Omega T plus theta and here one is
multiplied so one is multiplied to cos theta but here VM is multiplied so we will have VM real part of e power J
Omega T plus theta equal to V T so we can write V T is equal the real part of VM e power J Omega T plus theta now we
can further write VT equal to the real part of VM e power J Omega T multiplied to E power J theta now we will suppress
this term e power J Omega T which is the time component so here we are suppressing the time component and after
suppressing the time component we will have vm e power j theta and when you compare this with this you will find VM
is the magnitude and theta is the phase and this is the exponential form of the complex number which we call as the
phasor and we can represent phasor like this also the upper case and bold V and we usually write our phasor in the polar
form which is VM with the phase angle so now it is clear that V phasor is the phasor of sinusoid VT and it is a
complex number representing the amplitude and phase of sinusoid VT and we are getting this complex number by
suppressing the time component and therefore we can say that a phasor is the mathematical equivalent of a
sinusoid when the time dependences dropped and therefore VT is a time dependent quantity but V phaser is a
time independent quantity and if you have the sinusoid VM sine Omega T plus theta in place of course if you have
sine then also the phasor is going to be same now I will take two phasers and I will try to plot them in the same
complex plane this is our real axis and this one is our imaginary axis and the first phasor is v phasor and it is equal
to VM angle theta1 and the second phaser his i phaser and it is equal to I am angle negative of theta2 we will first
plot v phaser we will have a vector like this with magnitude equal to VM and it will make an angle equal to theta one
in anti-clockwise direction with respect to the real axis we are done with this phasor now we will plot this phasor we
will have a vector making angle theta two but in the clockwise direction with respect to the real axis why because
here we have negative of theta two and we know angle measured in clockwise direction is negative and angle measured
in anti-clockwise direction is positive and this will have the magnitude equal to I M so this vector is our phasor V
phasor and this vector is our phasor I phasor and there is one point which is very important the phasor analysis is
applicable only when the frequency is same for example if this phasor is having the parent signal VT and VT is
having the frequency Omega and this phasor is having the parent signal heidi and ID is also having the frequency
Omega then we can do the phasor analysis of I T and VT but if i t is having the frequency 2 times omega then we cannot
do the phasor analysis of VT and hi T so this is one very important point related to the phasor analysis and now I will
show you one animation and in the animation you can see that the vector is rotating at the angular frequency Omega
and their corresponding to the points on this circle we are having the sinusoidal waveform now you can easily imagine that
if you increase or decrease the angular frequency then the sinusoid will change so this is all for this lecture in the
coming lecture we will solve some questions related to phasers
[Applause] [Music]
Heads up!
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