Circuit Theory/Phasors

Input and Output

If the input to a linear circuit is a sinusoid, then the output from the circuit will be a sinusoid. Specifically, if we have a voltage sinusoid as such:

$v(t)=M_v\cos (\omega t+{\varphi }_v)$

Then the current through the linear circuit will also be a sinusoid, although its magnitude and phase may be different quantities:

$i(t)=M_i\cos (\omega t+{\varphi }_i)$

Note that both the voltage and the current are sinusoids with the same radial frequency, but different magnitudes, and different phase angles. Passive circuit elements cannot change the frequency of a sinusoid, only the magnitude and the phase. Why then do we need to write $\omega$ in every equation, when it doesnt change? For that matter, why do we need to write out the cos( ) function, if that never changes either? The answers to these questions is that we don't need to write these things every time. Instead, engineers have produced a short-hand way of writing these functions, called "phasors".

Euler's Equation

An important mathematical rule that needs to be understood for this discussion to continue is Euler's Equation. This rule is very difficult to understand at first, and it is often useful to derive the rule for better understanding, but in this wikibook we will take Euler's Equation as being axiomatic:

$M{e}^{j\omega t+\varphi }=M\cos (\omega t+\varphi )+jM\sin (\omega t+\varphi )$

This equation allows us to view sinusoids as complex exponential functions. Specifically, this allows us to consider the magnitude and phase angles directly, without having to worry about annoying trigonometry. Also, by viewing these quantities in terms of the complex quantity $ℂ=X+jY$, We can graph the point (X, Y) on the complex plane. On the complex plane, the Imaginary axis is the vertical axis, and the horizontal axis is the real axis, and points can be plotted exactly the same as they are plotted on a regular Cartesian plane.

Using this fact, we can get the angle from the origin of the complex plane to out point (X, Y) with the function:

${\theta }_C={\tan }^{-1}(\frac{Y}{X})$

And using the pythagorean theorem, we can find the magnitude of C -- the distance from the origin to the point (X, Y) -- as:

$M_C=|ℂ|=\sqrt{X^2+Y^2}$.

The system of using an angle and a distance to describe a point, instead of an X and Y coordinate system is known as Polar Graphing. Using X and Y coordinates is known as Rectangular Graphing.

Phasors

If the radial frequency for all sources is the same then we can employ a representation called "Phasors". Phasors are similar to vectors, but they represent a sinusoid instead of a location in space. This is an important fact, because at any time a phasor can be converted into a cosine function, and any cosine function can be converted into a phasor. However, the user has to remember to write down the value for the radial frequency, because phasor notation does not include the radial frequency.

If we have the following complex exponential:

$v(t)=M_v{e}^{(j\omega t+\varphi )}$

We know what the value of $\omega$ is because we have already figured it out and written it down someplace safe. The only important pieces of information then are the phase angle $\varphi$, and the magnitude M. We can then separate out these quantities into a new representation called a phasor:

$𝕍=M_v\mathrm{\angle }\varphi$

Phasors will always be written out either with a large bold letter (as above), or will be written out with a vector notation, such as the letter $\overrightarrow{V}$ This wikibook prefers the former notation for the simple reason that phasors are not vectors (or else presumably, we would call them "vectors").

Cosine Convention

It is important to remember which trigonometric function your phasors are mapping to. Since a phasor only includes information on magnitude and phase angle, it is impossible to know whether a given phasor maps to a sin( ) function, or a cos( ) function instead. By convention, this wikibook will say that all phasors map to cosine functions. This way, there is no ambiguity when we are moving from one notation to another.

Granted, the other convention could just as well be used, but it is important to pick a single convention and to stick with it. In this book, we assume that all phasors map to cosine functions, just as an arbitrary convention.

Any Sinusoid Can Be a Phasor

It is also important to remember that phasors can represent any quantity that can be represented by either a sinusoid, or a complex exponential function (as per Eulers Equation, above). For this reason, Current and Voltage can both be written as phasors, using the notation $𝕀$ and $𝕍$, respectively. Other complex values can also be expressed as phasors, including Impedance, and Complex Power, which are discussed in later chapters.

When dealing with phasors, we must transform every element in the circuit into a phasor representation. This means that every source, and every load element (resistor, capacitor, or inductor) can be transformed into a phasor quantity. We will discuss all these transformations later.

Phasor Arithmetic

Phasors can be mapped into a variety of different representations, each of which is good for different calculations. Since phasors map neatly to exponential functions, the multiplication of 2 phasors is incredibly easy. If we have a phasor A and a phasor B, we can multiply them together, as such:

$𝔸\times 𝔹=(M_a\times M_b)\mathrm{\angle }({\varphi }_a+{\varphi }_b)$

Keep in mind that as an exponential function, we are multiplying the coefficients (the magnitudes), and we are adding the exponents (the phase angles). As an exercise, the reader should convert the above phasors A and B into exponential representation, and do the math by yourself to prove this point.

Likewise, division is done extremely easily with phasors in their exponential notation, because the magnitudes can be divided into each other, and the phase angles can be subtracted from one another:

$𝔸/𝔹=(M_a/M_b)\mathrm{\angle }({\varphi }_a-{\varphi }_b)$

It is important to remember that the phasor on the bottom of the division (the denominator) is subtracted in the phase angles, and is the denominator of the magnitude division.

However, phasors are not good for the operations of addition and subtraction. For this, we need to convert the phasors into rectangular notation:

$ℂ=X+jY$

$X=M\cos (\varphi )$, $Y=M\sin (\varphi )$

This notation makes it very easy to add and subtract, so long as we follow some very simple rules:

1. Real parts get added together.
2. Imaginary parts get added together.

So, to add together 2 phasors, in rectangular form, we just add the real parts and imaginary parts, as such:

$ℂ=𝔸+𝔹=(X_A+X_B)+j(Y_A+Y_B)=X_C+j{Y}_C$

We often include subscripts on the different quantities, so we can remember which phasor each value belongs to. If we want to convert a phasor in rectangular form back into it's exponential form, we can use the following relations to make the transformation:

$ℂ=M_c\mathrm{\angle }{\varphi }_c=\sqrt{X^2+Y^2}\mathrm{\angle }{\tan }^{-1}(\frac{Y}{X})$

The first part of the equation (the magnitudes) many readers will recognize as being the pythagorean theorem, and the inverse tangent part some readers will recognize as being used to find the angle from the origin to a point on the cartesian plane. This makes intuitive sense when we consider that complex values can be graphed on the complex plane. On the complex plane, the value of the magnitude is simply the distance from the origin to the point, and the phase angle is the angle the line makes with the positive Real axis.

Once we have a phasor in exponential form, we can convert it back into a sinusoid (remember, we are using cosine functions by convention) by following 2 simple steps:

1. Use Eulers equation to separate the exponential function out into sin and cos functions.
2. Take the Real value as the sinusoid result.

For instance:

$\mathrm{Re}(M{e}^{(j\omega t+\varphi )})=M\cos (\omega t+\varphi )$

Variable Frequency

If the radial frequency of a circuit is variable, we can include a notation to show that the phasors are dependant on the radial frequency:

$𝔸(\omega )$

This does not, however change the representation of the phasor, we still only write it in terms of the magnitude and the phase angle. However, different circuit components respond differently to different frequencies (we will see this in detail later), so when the frequency is variable we need to keep that information handy so we don't get confused. The downside to this is that for every different value for the frequency, conceptually we need to recompute all our phasors. Also, phasors of different frequencies cannot be combined, or used in the same equation. All the phasors need to be in relation to the same radial frequency before they can be used.

Using Phasors

It is important to remember a few details when working with phasors:

1. Phasors may represent Current, Voltage, Power, or Impedance, so it is important to remember which quantity you are describing.
2. Phasors do not include information about the radial frequency, so this information needs to be written down somewhere.
3. Phasors are a short-hand way of writing a complex exponential, so phasors are manipulated in the same way that exponentials are manipulated.
4. Phasors can be used to represent either Sine or Cosine functions, so the exact relationship being used needs to be considered. By convention, this wikibook maps all phasors to cosine functions.

Different Representations, Same Values

Phasors are simply a different way to write values that are equivalent to the "normal" time-domain way we have been writing them. However, phasors make many operations and methods of analysis much easier then they have been. Specifically, phasors allow us to ignore differential equations (good), they allow us to combine resistors, capacitors, and inductors together (good), and they allow us to focus our attention on the quantities that change in a circuit (the phase and the magnitude), without having to write out alot of other repetitive things (very good). Here are a few of the representations that we will be using throughout the rest of our discussion of phasors:

$ℂ=M\mathrm{\angle }\varphi$ "Polar Notation" or "Phasor Notation"

$ℂ=M{e}^{j\omega t\varphi }$ "Exponential Notation"

$ℂ=A+jB$ "Rectangular Notation"

$ℂ=M\cos (\omega t+\varphi )+jM\sin (\omega t+\varphi )$ "sinusoid notation"

These 3 notations are all just different ways of writing the same exact thing.

Note on Notation

There are other ways to denote a phasor. Some common notations are:

• $\mathbb{P}$ (the large bold block-letters we use in this wikibook)
• $\overline{P}$ ("bar" notation, used by Wikipedia)
• $\overrightarrow{P}$ (vector arrow notation)
• $\tilde{P}$

further reading

• Wikipedia:Euler's_Identity uses Euler's equation to prove "the most beautiful theorem in mathematics".