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
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
The reciprocal of the period is the (cyclic) frequency (in Hz)
The angular frequency ω and the cyclic frequency f are obviously related by
Let us now consider a more general expression for a sinusoid,
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” .
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:
Therefore, Eqns. (1.6b-d) can be expressed as
Alternatively, we may use the graphical approach , 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
We could use its amplitude and phase to create a related complex number
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.
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 . Complex variable will always have a “hat” above it.
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
then the sinusoid corresponding to it, simply is
Alternatively, the time-domain form of phasor quantities may be obtained by multiplying the phasor form by ejωt and taking the real part of the result.
3. Derivative in Phasor Domain
Let the time-domain sinusoid be expressed as
Its corresponding phasor is
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):
Thus, the phasor representing the derivative of v(t) is
Let’s have a closer look at this phasor.
We arrived at a very important observation.
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.
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
where each of its components is a sinusoidal function
The corresponding phasors are
Thus, the phasor form of the E vector in Eq. (4.1) is
Similarly, if the magnetic field intensity vector H in time domain is given by
with each of its components being a sinusoidal function
Then the corresponding phasors are
Thus, the phasor form of the H vector in Eq. (4.5) is
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].
In Eqns. (4.9) E denotes electric field intensity, while H denotes magnetic field intensity. J stands for volume current density, while ρv denotes volume charge density.
The corresponding phasor-domain equations are
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, :
In Eqns. (4.11) l and c denote inductance and capacitance per unit length, respectively.
The corresponding phasor equations are
The uncoupled equations for a lossy line are, :
In Eqns. (4.13) r and g denote resistance and conductance per unit length, respectively.
The corresponding phasor equations are
where is the propagation constant of the line, defined by
- Adamczyk, B., Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
- Alexander, C. and Sadiku, M., Fundamentals of Electric Circuits, McGraw Hill, 4th Ed., 2009.
- Paul, C., Introduction to Electromagnetic Compatibility, Wiley, 2006.