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AC Electricity : Phasor Mathematics

Fourth Part of AC Electricity for Industrial Instrumentation

A powerful mathematical technique for analyzing AC circuits is that of phasors: representing AC quantities as complex numbers, a “complex” number defined as one having both a “real” and “imaginary” component. The purpose of this section is to explore how complex numbers relate to sinusoidal waveforms, and show some of the mathematical symmetry and beauty of this approach.


Crank diagrams and phase shifts

Something every beginning trigonometry student learns (or should learn) is how a sine wave is derived from the polar plot of a circle:

 

Phasor_Mathematics_Fig_020.JPG

 This translation from circular motion to a lengthwise plot has special significance to electrical circuits, because the circular diagram represents how alternating current (AC) is generated by a rotating machines, while the lengthwise plot shows how AC is generally displayed on a measuring instrument. The principle of an AC generator is that a magnet is rotated on a shaft past stationary coils of wire. When these wire coils experience the changing magnetic field produced by the rotating magnet, a sinusoidal voltage will be induced in the coils.

Phasor_Mathematics_Fig_021.JPG

While sine and cosine wave graphs are quite descriptive, there is another type of graph that is even more descriptive for AC circuits: the so-called crank diagram. A “crank diagram” represents the sinusoidal wave not as a plot of instantaneous amplitude versus time, but rather as a plot of peak amplitude versus generator shaft angle. This is basically the polar-circular plot seen earlier, which beginning trigonometry students often see near the beginning of their studies:

 

Phasor_Mathematics_Fig_022.JPG

 By representing a sinusoidal voltage as a rotating vector instead of a graph over time, it is easier to see how multiple waveforms will interact with each other. Quite often in alternating-current (AC) circuits, we must deal with voltage waveforms that add with one another by virtue of their sources being connected in series. This sinusoidal addition becomes confusing if the two waveforms are not perfectly in step, which is often the case. However, out-of-step sinusoids are easy to represent and easy to sum when drawn as vectors in a crank diagram. Consider the following example, showing two sinusoidal waveforms, 60 degrees (π 3 radians) out of step with each other:

Phasor_Mathematics_Fig_023.JPG

Graphically computing the sum of these two waves would be quite difficult in the standard graph (right-hand side), but it is as easy as stacking vectors tip-to-tail in the crank diagram:

Phasor_Mathematics_Fig_024.JPG

The length of the dashed-line vector A + B (radius of the dashed-line circle) represents the amplitude of the resultant sum waveform, while the phase shift is represented by the angles between this new vector and the original vectors A and B.

Complex numbers and phase shifts

This is all well and good, but we need to have a symbolic means of representing this same information if we are to do any real math with AC voltages and currents. There is one way to do this, if we take the leap of labeling the axes of the “crank diagram” as the axes of a complex plane (real and imaginary numbers):

Phasor_Mathematics_Fig_025.JPG

If we do this, we may symbolically represent each vector as a complex number. For example, vector B in the above diagram could be represented as the complex number x + jy (using j as the symbol for an imaginary quantity instead of i so as to not confuse it with current):

 

Phasor_Mathematics_Fig_026.JPG

Vector A lies completely on the “real” axis, and so it could be represented as a complex number x + jy where y has a value of zero.

Alternatively, we could express these complex quantities in polar form as amplitudes (A and B) and angles (θ1 and θ2), using the cosine and sine functions to translate this amplitude and angle into rectangular terms:

Phasor_Mathematics_Fig_027.JPG

This is where things get really elegant. A Swiss mathematician named Leonhard Euler (1707- 1783) proved that sine and cosine functions are related to imaginary powers of e. The following equation is known as Euler’s Relation, and it is extremely useful for our purposes here:

Phasor_Mathematics_Fig_028.JPG

With this critical piece of information, we have a truly elegant way to express all the information contained in the crank-diagram vector, in the form of an exponential term:

Phasor_Mathematics_Fig_029.JPG

In other words, these two AC voltages, which are really sinusoidal functions over time, may be symbolized as constant amplitudes A and B (representing the peak voltages of the two waveforms) multiplied by a complex exponential (e), where θ represents the phase of each waveform. What makes this representation very nice is that the complex exponential obeys all the mathematical laws we associate with real exponentials, including the differentiation and integration rules of calculus.

Anyone familiar with calculus knows that exponential functions are extremely easy to differentiate and integrate, which makes calculus operations on AC waveforms much easier to determine than if we had to represent AC voltages as trigonometric functions.

Credit for this mathematical application goes to Charles Proteus Steinmetz (1865-1923), the brilliant electrical engineer. At the time, Steinmetz simply referred to this representation of AC waveforms as vectors. Now, we assign them the unique name of phasors so as to not confuse them with other types of vectors. The term “phasor” is quite appropriate, because the angle of a phasor (θ) represents the phase shift between that waveform and a reference waveform.

The notation has become so popular in electrical theory that even students who have never been introduced to Euler’s Relation use them. In this case the notation is altered to make it easier to understand. Instead of writing Be, the mathematically innocent electronics student would write BLθ.

Phasor expressions of impedance

However, the real purpose of phasors is to make difficult math easier, so this is what we will explore now. Consider the problem of defining electrical opposition to current in an AC circuit. In DC (direct-current) circuits, resistance (R) is defined by Ohm’s Law as being the ratio between voltage (V ) and current (I):


Phasor_Mathematics_Fig_030.JPG

There are some electrical components, though, which do not obey Ohm’s Law. Capacitors and inductors are two outstanding examples. The fundamental reason why these two components do not follow Ohm’s Law is because they do not dissipate energy like resistances do. Rather than dissipate energy (in the form of heat and/or light), capacitors and inductors store and release energy from and to the circuit in which they are connected. The contrast between resistors and these components is remarkably similar to the contrast between friction and inertia in mechanical systems. Whether pushing a flat-bottom box across a floor or pushing a heavy wheeled cart across a floor, work is required to get the object moving. However, the flat-bottom box will immediately stop when you stop pushing it, while the wheeled cart will continue to coast because it has kinetic energy stored in it.

The relationships between voltage and current for capacitors (C) and inductors (L) are as follows:

Phasor_Mathematics_Fig_031.JPG

Expressed verbally, capacitors pass electric current proportional to how quickly the voltage across them changes over time. Conversely, inductors produce a voltage drop proportional to how quickly current through them changes over time. The symmetry here is beautiful: capacitors, which store energy in an electric field, oppose changes in voltage. Inductors, which store energy in a magnetic field, oppose changes in current.

When either type of component is placed in an AC circuit, and subjected to sinusoidal voltages and currents, it will appear to have a “resistance.” Given the amplitude of the circuit voltage and the frequency of oscillation (how rapidly the waveforms alternate over time), each type of component will only pass so much current. It would be nice, then, to be able to express the opposition each of these components offers to alternating current in the same way we express the resistance of a resistor in ohms (Ω). To do this, we will have to figure out a way to take the above equations and manipulate them to express each component’s behavior as a ratio of V / I . I will begin this process by using regular trigonometric functions to represent AC waveforms, then after seeing how ugly this gets I will switch to using phasors and you will see how much easier the math becomes.

Let’s start with the capacitor. Suppose we impress an AC voltage across a capacitor as such:

 

Phasor_Mathematics_Fig_032.JPG

It is common practice to represent the angle of an AC signal as the product ωt rather than as a static angle θ, with ω representing the angular velocity of the circuit in radians per second. If a circuit has a ω equal to 2π, it means the generator shaft is making one full rotation every second. Multiplying ω by time t will give the generator’s shaft position at that point in time. For example, in the United States our AC power grid operates at 60 cycles per second, or 60 revolutions of our ideal generator every second. This translates into an angular velocity ω of 120π radians per second, or approximately 377 radians per second.

We know that the capacitor’s relationship between voltage and current is as follows:


Phasor_Mathematics_Fig_033.JPG

Therefore, we may substitute the expression for voltage in this circuit into the equation and use calculus to differentiate it with respect to time:

Phasor_Mathematics_Fig_034.JPG

The ratio of V / I (the opposition to electric current, analogous to resistance R) will then be:


Phasor_Mathematics_Fig_035.JPG

This might look simple enough, until you realize that the ratio V/I will become undefined for certain values of t, notably π/2 and 3π/2 . If we look at a time-domain plot of voltage and current for a capacitor, it becomes clear why this is. There are points in time where voltage is maximum and current is zero:

Phasor_Mathematics_Fig_036.JPG

At these instantaneous points in time, it truly does appear as if the “resistance” of the capacitor is undefined (infinite), with multiple incidents of maximum voltage and zero current. However, this does not capture the essence of what we are trying to do: relate the peak amplitude of the voltage with the peak amplitude of the current, to see what the ratio of these two peaks are. The ratio calculated here utterly fails because those peaks never happen at the same time.

One way around this problem is to express the voltage as a complex quantity rather than as a scalar quantity. In other words, we use the sine and cosine functions to represent what this wave is doing, just like we used the “crank diagram” to represent the voltage as a rotating vector. By doing this, we can represent the waveforms as static amplitudes (vector lengths) rather than as instantaneous quantities that alternately peak and dip over time. The problem with this approach is that the math gets a lot tougher:


Phasor_Mathematics_Fig_037.JPG

The final result is so ugly no one would want to use it. We may have succeeded in obtaining a ratio of V to I that doesn’t blow up at certain values of t, but it provides no practical insight into what the capacitor will really do when placed in the circuit.

Phasors to the rescue! Instead of representing the source voltage as a sum of trig functions (V = cos ωt + j sin ωt), we will use Euler’s Relation to represent it as a complex exponential and differentiate from there:

 

Phasor_Mathematics_Fig_038.JPG
 
Note how the exponential term completely drops out of the equation, leaving us with a clean ratio strictly in terms of capacitance (C), angular velocity (ω), and of course j. This is the power of phasors: it transforms an ugly math problem into something trivial by comparison.

Next, will will apply this same phasor analysis to inductors. Recall that an inductor’s relationship between voltage and current is as follows:


Phasor_Mathematics_Fig_039.JPG

If an AC current is forced through an inductor (the AC current described by the expression I = ejωt), we may substitute this expression for current into the inductor’s characteristic equation to solve for voltage as a function of time:

 

Phasor_Mathematics_Fig_040.JPG
 
In summary, we may express the impedance (voltage-to-current ratio) of capacitors and inductors by the following equations:
Phasor_Mathematics_Fig_041.JPG

Most students familiar with electronics from an algebraic perspective (rather than calculus) find the expressions XL = 2πfL and XC = 1/2πfC easier to grasp. Just remember that angular velocity (ω) is really “shorthand” notation for 2πf, so these familiar expressions may be alternatively written as XL = ωL and XC = 1/ωC . Furthermore, recall that reactance (X) is a scalar quantity, having magnitude but no direction. Impedance (Z), on the other hand, possesses both magnitude and direction (phase), which is why the imaginary operator j must appear in the impedance expressions to make them complete. The impedance offered by inductors and capacitors alike are nothing more than their reactance values (X) scaled along the imaginary (j) axis (phase-shifted 90o).

Euler’s Relation and crank diagrams

Another detail of phasor math that is both beautiful and practical is the famous expression of Euler’s Relation, the one all math teachers love because it directly relates several fundamental constants in one elegant equation (remember that i and j mean the same thing, just different notational conventions for different disciplines):

Phasor_Mathematics_Fig_042.JPG

If you understand that this equation is nothing more than the fuller version of Euler’s Relation with θ set to the value of π, you may draw a few more practical insights from it:

Phasor_Mathematics_Fig_043.JPG

After seeing this, the natural question to ask is what happens when we set θ equal to other, common angles such as 0, π/2 , or 3π/2 ? The following examples explore these angles:

Phasor_Mathematics_Fig_044.JPG
We may show all the equivalencies on the complex plane, as unit vectors:
Phasor_Mathematics_Fig_045.JPG

If we substitute ωt for θ, describing a continually increasing angle rather than a fixed angle, we see our original “crank diagram” come to life, with the vector arrow spinning about the origin of the graph in a counter-clockwise rotation:

 

Phasor_Mathematics_Fig_046.JPG

 

Going back to the result we got for the capacitor’s opposition to current (V/I ), we see that we can express the i term (or j term, as it is more commonly written in electronics work) as a complex exponential and gain a little more insight:

 

Phasor_Mathematics_Fig_047.JPG

What this means is that the capacitor’s opposition to current takes the form of a phasor pointing down on the complex plane. In other words, it is a phasor with a fixed angle ( 3π/2 , or π/2 radians) rather than rotating around the origin like all the voltage and current phasors do. In electric circuit theory, there is a special name we give to such a quantity, being a ratio of voltage to current, but possessing a complex value. We call this quantity impedance rather than resistance, and we symbolize it using the letter Z.

When we do this, we arrive at a new form of Ohm’s Law for AC circuits:

Phasor_Mathematics_Fig_048.JPG

With all quantities expressed in the form of phasors, we may apply nearly all the rules of DC circuits (Ohm’s Law, Kirchhoff’s Laws, etc.) to AC circuits. What was old is new again!

The s variable

A concept vital to many forms of engineering is something called a Laplace transform. This is a mathematical technique used to convert differential equations (complicated to solve) into algebraic equations (simpler to solve). The subject of Laplace transforms is vast, and requires a solid foundation in calculus to even begin to explore, but one of the elements of Laplace transforms has application right here to our discussion of phasor mathematics and component impedance.

Recall that we may express the impedance (voltage-to-current ratio) of capacitors and inductors by the following equations:

Phasor_Mathematics_Fig_049.JPG

The product keeps appearing again and again in phasor expressions such as these, because is at the heart of Euler’s Relation, where e= cos θ + j sin θ (or, where e= cos ω + j sin ω).

In Laplace transforms, the “imaginary” concept of is extended by adding a “real” portion to it symbolized by the lower-case Greek letter Sigma (σ). Thus, a complex quantity called s is born:

Phasor_Mathematics_Fig_050.JPG

 

We have already seen how ejωt may be used to describe the instantaneous amplitude of a sinusoidal waveform with an angular velocity of ω. What would happen if we used s as the Euler exponent instead of ? Convention dictates we place a negative sign in the exponential, so the expression will look like this:
Phasor_Mathematics_Fig_051.JPG

We may re-write this expression to show the meaning of s, a complex number formed of a real part (σ) and an imaginary part ():

Phasor_Mathematics_Fig_052.JPG

Distributing the negative sign and the variable for time (t) through the parentheses:

Phasor_Mathematics_Fig_053.JPG
Recalling from the algebraic rules of exponents that a sum (or difference) of exponents is equivalent to a product of exponential terms:
Phasor_Mathematics_Fig_056.JPG

This expansion is useful for understanding because each of these exponential terms (both eσt and ejωt) have practical meaning on their own. We have already seen how the expression ejωt defines the instantaneous value of a sinusoidal function of angular velocity (frequency) ω at any point in time t. The term eσt, however, is even simpler than this: it defines an exponential growth (or decay) function. The familiar expression et _ from RC and RL time-constant circuits describing the decay of voltage or current is an example of an exponential decay function. eσt is just a general expression of this concept, where “sigma” (σ) is the decay constant, equivalent to the reciprocal of the system’s “time constant” ( 1τ ). If the value of sigma is positive (σ > 0), the expression eσt describes a process of decay, where the value approaches zero over time. Conversely, if the value of sigma is negative (σ < 0), the expression eσt describes a process of unbounded growth, where the value approaches infinity over time. If sigma happens to be zero (σ = 0), the value of the expression eσt will be a constant 1 (neither growing nor decaying over time).


Therefore, when we multiply an exponential growth/decay function (eσt) by a sinusoidal function (ejωt), what we get is an expression describing a sinusoidal waveform that either decays, grows, or holds at a steady amplitude over time:

 

Phasor_Mathematics_Fig_054.JPG

We can see from the expression and from the graph that ejωt is just a special case of est, when sigma has a value of zero. Focusing on just the exponent, it is safe to say that is just a special case of s when there is no exponential growth or decay over time.

For this reason, engineers often substitute s for in phasor expressions of impedance. So, instead of defining inductor and capacitor impedance in terms of j and ω, they often just define impedance in terms of s:

Phasor_Mathematics_Fig_055.JPG

Go Back to Lessons in Instrumentation Table of Contents


Comments (2)Add Comment
0
phasor diagram
written by samson, December 10, 2012
Hi

I m a looking for a software that can convert 3phase sine wave to vector components .
0
online tool for complex phasors
written by Patrick Chung, April 18, 2013
Now there is online tool for complex phasors of AC signals adding and drawing in both complex and time domain fashion. Please visit:
http://www.cirvirlab.com/simul...-NycqzmhE
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