Concept of a Phasor in Sinusoidal Steady State Analysis

This article introduces a concept of a phasor which is extensively used in several EMC topics including electromagnetic waves, antennas, and transmission lines. The following series of articles will use phasors to determine voltages and currents along the transmission line, and subsequently the input impedance to transmission line. 

1. Sinusoidal Voltages 

Consider a single frequency sinusoidal signal

 (1.1) 

where V is the amplitude of the sinusoid and ω is the angular frequency in radians per second, rad/s. The period T and the angular frequency ω are related by

 (1.2)

The reciprocal of the period is the (cyclic) frequency (in Hz)

 (1.3)

The angular frequency ω and the cyclic frequency f are obviously related by

 (1.4)

Let us now consider a more general expression for a sinusoid,

 (1.5)

where (ωt + φ) is called the argument of the cosine function, and φ is its phase.

The sinusoidal functions may, in general, be expressed in any of the four different forms: either as a sine or a cosine function, with either positive or negative “amplitude” [1].

For example,

 (1.6a)

 (1.6b)

 (1.6c)

 (1.6d)

As we will see in the next section, we often need the sinusoid to be expressed as a cosine function with positive amplitude, as shown in Eq. (1.6a). Therefore, we need to be able to transform the other three forms into the positive cosine form. To accomplish that, we could use the following trigonometric identities:

(1.7a)

(1.7b)

(1.7c)

Therefore, Eqns. (1.6b-d) can be expressed as

(1.8a)

(1.8b)

(1.8c)

Alternatively, we may use the graphical approach [2], as follows. Consider the set of axis shown in Figure 1. The horizontal axis represents the cosine, while the vertical axis (pointing down) denotes the sine. Angles are measured positively counterclockwise from the horizontal, as in polar coordinates.

This figure can be used to obtain a positive cosine out of the other three forms, as follows. Negative cosine is equivalent to positive cosine plus or minus 180°. Positive sine is equivalent to positive cosine minus 90°. Negative sine is equivalent to positive cosine plus 90°.

2. Sinusoids and Phasors 

Consider a positive cosine function of the form

 (2.1)

We could use its amplitude and phase to create a related complex number

(2.2)

Obviously, the complex number in expression (2.2) is related to the sinusoid in expression (2.1). We often say that this complex number represents the respective sinusoid.

Is this representation useful?  Extremely! Instead of performing mathematical operations on sinusoids in time domain (which is often difficult to do), we can perform the operations on complex numbers, related to these sinusoids, in complex domain (which is relatively easy to do).

Note that the sinusoid exists in time domain, while the complex number representing it exists in complex domain. Therefore, they are not equal; they correspond to each other.

(2.3)

When a complex number represents a sinusoid we call it a phasor. By representing the sinusoid as a phasor we transform the sinusoid from the time domain to the phasor or frequency domain.

So what is the difference between a phasor and a complex number? Every phasor is a complex number, but not every complex number is a phasor. Only when the complex number represents a sinusoid, it is referred to as a phasor.

In electromagnetic compatibility we often deal with complex voltages and currents. These complex expressions represent sinusoids in time domain, and therefore are phasors. We also encounter complex impedance, but the impedance does not represent a time-domain sinusoid; therefore, it is not a phasor, it is just a complex expression.

To distinguish between the time domain variables and the complex variables, we will adopt the notation from [3]. Complex variable will always have a “hat” above it. 

(2.4)

In the above expression, the magnitude V and the angle θ are real, thus do not have “hats” but the phasor is complex.

Given a phasor, in polar or exponential form, we can easily determine the time-domain sinusoid corresponding to it. For instance, if the phasor is given by

(2.5)

then the sinusoid corresponding to it, simply is 

(2.6)

Alternatively, the time-domain form of phasor quantities may be obtained by multiplying the phasor form by e^jωt and taking the real part of the result. 

(2.7)

3. Derivative in Phasor Domain 

Let the time-domain sinusoid be expressed as

(3.1)

Its corresponding phasor is

(3.2)

If we take the derivative of v(t) in expression (3.1) we will obtain another sinusoid; a negative sine function, to be exact. That negative sine function can be expressed as a positive cosine using the transformations discussed earlier. 

Therefore, we could create a phasor representing it. The question we pose is as follows: what is the relationship between the original phasor representing v(t) and the phasor representing its derivative? 

To answer this question let us take the derivative of v(t):

(3.3)

Thus, the phasor representing the derivative of v(t) is

(3.4)

Let’s have a closer look at this phasor.

(3.5)

However,

(3.6)

and therefore

(3.7)

We arrived at a very important observation.

 (3.8a)

(3.8b)

That is, to obtain the phasor representing the derivative of a (sinusoidal) function, we simply take the phasor representing that function and multiply it by jω. What about the phasor representing the second derivative? We simply multiply the phasor representing the first derivative by another jω term.

 (3.8c)

4. EMC Applications 

4.1  Electromagnetic Fields

Of major interest in EMC are the sinusoidal electromagnetic fields with the time-domain vectors and scalar expressions being the sinusoidal functions of time and space. 

For instance, the electric field intensity vector E in time domain is given by

 (4.1)

where each of its components is a sinusoidal function

 (4.2a)

 (4.2b)

 (4.2c)

The corresponding phasors are

 (4.3a)

 (4.3b)

 (4.3c)

Thus, the phasor form of the E vector in Eq. (4.1) is

 (4.4)

Similarly, if the magnetic field intensity vector H in time domain is given by

 (4.5)

with each of its components being a sinusoidal function

 (4.6a)

 (4.6b)

 (4.6c)

Then the corresponding phasors are

 (4.7a)

 (4.7b)

 (4.7c)

Thus, the phasor form of the H vector in Eq. (4.5) is

 (4.8)

4.2  Maxwell’s Equations

Maxwell’s equations can be expressed in several forms. Here, we present the differential, time-domain version of these equations in simple medium [1,3].

 (4.9a)

 (4.9b)

 (4.9c)

 (4.9d)

In Eqns. (4.9) E denotes electric field intensity, while H denotes magnetic field intensity. J stands for volume current density, while ρ_v denotes volume charge density.

The corresponding phasor-domain equations are

 (4.10a)

 (4.10b)

 (4.10c)

 (4.10d)

4.3  Transmission Line Equations

In time domain, the transmission line equations can be expressed either as a a set of coupled first order equations or a set of uncoupled second order equations. The coupled equations for a lossless line are, [1]:

 (4.11a)

 (4.11b)

In Eqns. (4.11) l and c denote inductance and capacitance per unit length, respectively. 

The corresponding phasor equations are

 (4.12a)

 (4.12b)

The uncoupled equations for a lossy line are, [1]:

 (4.13a)

 (4.13b)

In Eqns. (4.13) r and g denote resistance and conductance per unit length, respectively. 

The corresponding phasor equations are

 (4.14a)

 (4.14b)

where  is the propagation constant of the line, defined by

 (4.15)

References

1. Adamczyk, B., Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017. 
2. Alexander, C. and Sadiku, M., Fundamentals of Electric Circuits, McGraw Hill, 4^th Ed., 2009. 
3. Paul, C., Introduction to Electromagnetic Compatibility, Wiley, 2006.