© 2026 - Sandro Ronca - All rights reserved - Legal notes - Technical notes

Alternating Current

Sinusoidal electrical quantities. Average and RMS values.
Rotating vectors and phasors. The symbolic method.

Summary: we discuss how alternating sinusoidal electromotive forces are generated and how these quantities can be represented by rotating vectors with a given angular velocity and by phasors. We define their basic features: amplitude, frequency, angular frequency, phase, average value, RMS value. Finally we introduce the symbolic method which, thanks to complex numbers, turns operations on sinusoids into algebraic calculations.

1 - Sinusoids, rotating vectors, phasors

Introduction.
An electromotive force (abbr. emf, symbols: \(\small e, e(t), E\)) is physically a potential difference, measured in volts, between two points in space. The potential difference \(\small \Delta V\) is inseparably tied to the presence of an electric field \(\small \vec{E}\), so much so that the potential difference is often called "the work of the electric field", \(\small \Delta V = \vec{E} \cdot \Delta \vec{l}\), by analogy with the work of a force.
When this electric field acts on mobile charges (e.g. in a conductor) it separates them: positive charges move along the field, negative charges against it, giving rise to an electrical polarity.
The emf is a potential difference typically produced inside a generator.

1.1 - Generating a sinusoidal alternating electromotive force

To generate an emf, according to the general law of electromagnetic induction, the magnetic flux (see below) linked with an electric circuit must change. One way to obtain this is to produce a relative motion between a magnetic field and a conductor or an electric circuit. ⁽¹⁾. Since the generating system must be kept reasonably small, it is convenient to rotate a conductor inside a magnetic field with constant angular velocity (uniform circular motion). Note also that it makes no difference whether the conductor moves with respect to the magnetic field or vice versa. What matters, for generating the induced emf, is the relative motion between magnetic field and conductor.

1.1.1 - The Lorentz force

An electric charge q moving with velocity v in a magnetic field B is subject to a force:
\[ \vec F = q \cdot \vec v \times \vec B \] called the Lorentz force ⁽²⁾.
If velocity and magnetic field are perpendicular to each other we simply have, in magnitude: \[ F = q v B \] and otherwise, in general: \[ F = q v B \; sin(\alpha) \] where α is the angle at which the conductor cuts the magnetic field lines.
This force separates the electric charges.
In our example the positive charges, if they were free to move, would be pushed in the direction shown by the arrow while the conductor moves; the free electrons are instead pushed towards the opposite end. During the first half turn, positive charges therefore pile up at end R of the conductor (the arrow tip) and electrons pile up correspondingly at the other end. The situation is reversed during the second half turn.
The charges separated by the Lorentz force give rise to an electric field \( \small \vec{E}\) such that, at equilibrium: \[ q\vec E = - q \cdot \vec v \times \vec B \] in magnitude: \[ qE = q v B \; sin(\alpha) \] with \(\small \alpha\) depending on time according to the law: \(\small \alpha=\pi-\omega t \) (see the figure below for the names of the angles), since this is uniform circular motion with angular velocity \(\small \omega\); but \(\small sin(\pi-\omega t) = sin(\omega t) \). Hence: \[ qE = q v B \; sin(\omega t) \] which tells us that the electric field inside the conductor is a sinusoidal alternating field: \[ E(t) = v B \; sin(\omega t) \] It results from a continuous oscillation of the electric charges caused by the Lorentz force and by the uniform circular motion of the conductor.
The polarities at the ends of the conductor change continuously with a frequency equal to that of the sinusoid, \( f = \frac{\omega}{2 \pi}\), or, if you prefer, of the uniform circular motion.
At every instant  the electric field \(\vec{E}\) is constant along the whole length \(\vec{l}\) (taken as a displacement vector) of the conductor, so between the ends of the conductor there will be a potential difference or electromotive force (emf):⁽³⁾ \[ e(t) = \vec{E} \cdot \vec{l} = v B \; sin(\omega t) l\] (do not confuse \(e(t)\), the emf or potential difference, with \(E(t)\), the electric field).
So, as a result of the Lorentz force, we get: \[e(t) = Blv \; sin(\omega t) \] which is the potential difference across the ends of the conductor.
This is the so-called induced electromotive force.

1.1.2 - The role of the transverse component of velocity

In our case the magnetic field \(\small B\), also called the inducing field, is kept constant, the length of the conductor \(l\) (also called the active side) does not change either, and the speed \(\small v\) is constant in magnitude since this is uniform circular motion with frequency \(\small f\) and constant angular velocity \(\small \omega = 2 \pi f\); its direction, however, changes continuously as the angle \(\small \omega t\) changes.
When \(\small \omega t = 0 \), \(\small \alpha = \pi - \omega t = \pi\) and when \(\small \omega t = \pi \), \(\small \alpha = \pi - \omega t = 0\). In both cases \(\small sin(\alpha) = 0\). From this we understand that if the conductor moves parallel or antiparallel to the magnetic field the induced emf is zero: \[e(t) = Blv \; sin(0) = Blv \; sin (\pi) = 0 \] while the effect is greatest when the conductor cuts the field lines at right angles, \(\small \omega t = \frac{\pi}{2} + k \pi \quad (k = 0, 1, 2, ...)\); in that case: \[e(t) = Blv \; sin(\frac{\pi}{2} + k \pi) = \pm Blv \] where the double sign \(\pm\) shows the alternating direction of the emf, that is, its polarity. The peripheral velocity can be split into two directions: one parallel (or antiparallel) to the field lines and the other perpendicular to them.

Only the transverse component of the velocity matters for producing the induced emf; the parallel component of the velocity has no effect at all.

1.1.3 - Direction of the induced electromotive force

The direction of electromotive forces is tied to the principle of conservation of energy which, for electromagnetic induction phenomena, is expressed by Lenz's law,  according to which:
phenomena due to electromagnetic induction always take place in such a way as to oppose the causes that produced them.
The minus sign that appears in some formulations of the law of electromagnetic induction has to do with this law and so with the directions (once a positive direction has been chosen by convention) of the quantities involved.
The animation on the board highlights the directions of the emfs, which can also be found with an empirical rule called the right-hand rule (or generator rule) ⁽⁴⁾ .

Go deeper:

Faraday's and Lenz's laws and the directions of induced emfs

1.2 - Sinusoids

As is well known, a sinusoid is the graph of the trigonometric function \(\small sin(\alpha)\) or \(\small cos(\alpha)\)⁽⁵⁾ as a function of the angle \(\small \alpha\), measured in radians.
These curves are especially important in physics for describing oscillations, from harmonic motion to waves.
Uniform circular motion, for example, gives as its projection a harmonic oscillation which, plotted against time, is exactly a sinusoid.
In the mathematical description of harmonic oscillations, \(\small \alpha\) depends on time through the relation \(\small \alpha = \omega t \), where \(\small \omega = 2 \pi f\) is a quantity related to the frequency \(\small f\) of the oscillation, called the angular frequency. On the other hand, the angular frequency \(\small \omega\) is defined exactly as the angular velocity of a uniform circular motion. We will see later how the oscillations of electrical quantities can be matched to the uniform circular motions of special mathematical objects: rotating vectors.
Since \(\small \alpha\) is measured in radians and \(\small t\) in seconds, the dimensions of \(\small \omega\) are radians/second [rad/s].
A sinusoid describing an oscillation has the general analytic expression:
\[y(t) = Y_M sin(\omega t + \varphi) \] where \(\small Y_M\) is the amplitude (the subscript M stands for Maximum. \(\small Y\) without subscripts usually means the RMS value), \(\small \omega = 2 \pi f \) is the angular frequency, \(\small f\) is the frequency, measured in Hz, that is the number of oscillations per unit of time, and \(\small \varphi\) is the phase.
The unit of \(\small Y_M\) depends on the physical quantity being represented: for example, if \(\small y(t)\) is a current, \(\small Y_M\) is measured in amperes; if \(\small y(t)\) is a voltage, \(\small Y_M\) is measured in volts, and so on.
We also define the period \(\small T = \frac{1}{f}\), the time needed to complete one full oscillation. If we plot the sinusoid against the angle, the period is given by \(\small \omega T = 2 \pi\).
The role of these parameters can be understood both by interacting with the figure above on the right and by reading the rest of the lesson.

1.2.1 - Average value of a sinusoidal quantity

We now give the important definitions of the average value and the RMS value of a sinusoid. We will try to understand these concepts both from a mathematical and from a physical point of view.
 Imagine that the graph on the board on the left shows a sinusoidal alternating current \(\small i(t) = I_M sin(\omega t)\). The definition of electric current is \(\small i(t) = \frac{\Delta Q}{\Delta t}\), where \(\small \Delta Q\) is the amount of electric charge that crosses a section of the conductor in the time \(\small \Delta t\). Let us draw under the curve a rectangle with base \(\small \Delta t\) and height \(\small i(t_1)\), the instantaneous value of the current at time \(\small t_1\), for example. The area of this rectangle measures exactly the amount of charge that would flow in the circuit if during this time interval the current stayed constant at the value \(\small i(t_1)\): \(\small Q = i(t_1) \cdot \Delta t\).

By increasing the number of intervals, and so of rectangles under the first half-wave, we see that we can compute, with better and better approximation, the area bounded by the curve and the time axis.

Activity 1


For readers who know calculus, here is the analytic proof:
\[\int_{0}^{\frac{T}{2}} i(t)\, dt = \int_{0}^{\frac{T}{2}} I_M sin(\omega t)\, dt = \frac{I_M}{\omega} \int_{0}^{\pi} sin(\omega t)\, d (\omega t) = \] \[= \frac{I_M}{\omega} [- cos (\omega t)]_0^\pi = \frac{I_M}{\omega} [-cos(\pi)+ cos(0)] = \frac{I_M}{\omega} [-(-1)+1] = 2\frac{I_M}{\omega} \] From the above we conclude that this area represents the amount of electric charge flowing in one direction.
We can repeat the reasoning for the second half period of the sinusoid and compute its area in the same way.
In this case, however, we get a negative area. This makes sense mathematically, since the height of the rectangles under the negative half-wave is a negative segment.
Physically, the negative area represents not so much a negative charge as a charge flowing in the opposite direction to the first. This tells us that if we watch the charge crossing a given section of the conductor over a whole period we will record a total charge flow of zero, because as much charge flows in one direction as flows in the opposite direction. The total charge that has crossed the section of the conductor in one period of oscillation of the current is zero, just like the total area bounded by the sinusoid.
How, then, do we define the average value of a sinusoid?
The physical interpretation can help. If the sinusoid represents a current, its average value is a current constant in time (a direct current) that carries, in a given time interval, the same amount of charge.
If the time interval considered is the period \(\small T\), the average value is clearly zero.
For this reason, for a sinusoid we define an average value over half a period; indeed, when we talk about the average value of a sinusoid we mean precisely the average value over the half period.
So for \(\small I_m\), the average value of the sinusoid over the half period, using the previous result, we have:
\[ I_m = \frac{Area}{\frac{T}{2}}= \frac{\int_{0}^{\frac{T}{2}} i(t)\, dt}{\frac{T}{2}} = \frac{2 I_M}{\omega \frac{T}{2}}= \frac{4 I_M}{\omega T} = \frac{4 I_M}{2 \pi} = \frac{2}{\pi}I_M \] Hence, since \( \small \frac{2}{\pi} = 0.637\): \[ I_m = 0.637 I_M \].

1.2.2 - RMS (effective) value of a sinusoidal quantity

Now imagine multiplying the sinusoid by itself, that is, evaluating and plotting the function \(\small i^2(t) = I^2_M sin^2(\omega t) \).
At first sight the resulting curve looks like a sinusoid too, but this must be proved. Indeed, a well-known trigonometric identity lets us state that \(\small sin^2(\omega t)= \frac{1 - cos(2\omega t)}{2} \) and therefore:
\[ i^2(t) = \frac{I^2_M}{2}[1 - cos(2 \omega t)]= \frac{I^2_M}{2} - \frac{I^2_M}{2} cos(2 \omega t) \] in which we can see a component constant in time: \(\small \frac{I^2_M}{2} \) and a cosine curve with angular frequency \(\small 2 \omega \) and amplitude \(\small \frac{I^2_M}{2} \): \[ - \frac{I^2_M}{2}cos(2 \omega t)= \frac{I^2_M}{2} cos(2 \omega t \pm \pi) \] Note how we have absorbed  the minus sign into the phase ±π.
Let us rewrite the previous expression:
\[ i^2(t) = \frac{I^2_M}{2} + \frac{I^2_M}{2} cos(2 \omega t \pm \pi) \] So we have a cosine curve shifted by the amount \(\small \frac{I^2_M}{2} \) and with twice the frequency of \(\small i(t) \), exactly as the graph of \(\small i^2(t) \) shows.
Now suppose that the current \(\small i(t) \) flows through a resistance R. We can then plot the power \(\small p(t) = R \,i^2(t) \), which obviously has the same shape as \(\small i^2(t) \), R being a constant (if we neglect its change with temperature).

In the same way as before we can compute the area bounded by the curve \(\small R\, i^2(t) \). The dimensions of this area are [Power x time]= [W x s]=[J], so it is an energy: the energy that, through the Joule effect, is dissipated as heat in the resistance R carrying the current \(\small i(t) \).
The average value of \(\small R i^2(t) \) is represented by the height of a rectangle whose base is the period, with a value equal to half the maximum value. It is easy to see that the area under the sinusoid equals that of the rectangle if you imagine filling the gaps of the rectangle with the parts of the sinusoid's area that lie outside the rectangle. Since the height of the rectangle is \(\small R \frac{I^2_M}{2} \) and
\(\small \frac{I^2_M}{2}= \frac{I_M}{\sqrt{2}} \cdot \frac{I_M}{\sqrt{2}} \), we define the effective value or RMS value (Root Mean Square) of i(t): \[I = \frac{I_M}{\sqrt{2}} \]
We can therefore say that the RMS value of an alternating current is the value a direct current would need to have in order to deliver the same energy as the alternating current.

Let us now look at an analytic proof of what we have said.
First we compute the area under the curve \(i^2(t) \):

\[\int_{0}^{T} i^2(t)\, dt = \int_{0}^{T} I^2_M sin^2(\omega t)\, dt = \frac{I^2_M}{\omega} \int_{0}^{2 \pi} sin^2(\omega t)\, d (\omega t) = \frac{I^2_M}{\omega} \pi \] and then the average value of this area, which is the RMS value (usually written simply as \(\small I \), with no subscript):
\[I = \sqrt{\frac{1}{T}\int_{0}^{T} i^2(t)\, dt} = \sqrt{\frac{I^2_M}{\omega T} \pi} = \sqrt{\frac{I^2_M}{2 \pi} \pi} = \sqrt{\frac{I^2_M}{2}} = \frac{I_M}{\sqrt{2}}\]

1.3 - Vector representation of sinusoids 

Sinusoids can be represented by a special kind of vectors: rotating vectors with a given angular velocity \(\omega \).


Activity 2


By projecting the vector \(\small \vec{E_1} \) onto the vertical axis and plotting these values against the corresponding instants, we get exactly the sinusoid \(\small e_1(t) \). Representing sinusoids by rotating vectors is especially useful, as we will see, for calculations on the sinusoids themselves.
For example, to add two sinusoids we would have to compute the sum point by point (instant by instant) and then plot the resulting sinusoid. Clearly this is a very laborious procedure.
If we could turn operations on sinusoids into operations we know how to perform on vectors (addition, subtraction, dot product, multiplication and division by a scalar, etc.), things would become much simpler.
First of all, there must be no possible ambiguity in the correspondence sinusoid \(\iff \) vector. That is, it must not be possible for the same vector to represent several sinusoids, or, conversely, for one sinusoid to correspond to several vectors. The correspondence must be one-to-one.
If we match:

• the angular velocity \(\small \omega= \frac{\Delta \alpha}{\Delta t} \) of the vector with the angular frequency \(\small \omega = 2 \pi f \) of the sinusoid;
• the magnitude of the vector with the amplitude of the sinusoid;
• the phase of the vector with the phase of the sinusoid

there is no possible ambiguity.

1.3.1 - Angular frequency and frequency

The angular frequency \(\small \omega \), measured in rad/s [radians per second], is related to the frequency by the relation \(\small \omega = 2 \pi f \). The frequency is the number of oscillations per unit of time and is measured in Hz [hertz].
The angular frequency of the sinusoid corresponds to the angular velocity of the rotating vector.

Activity 3

1.3.2 - Amplitude

It is the maximum value the sinusoid can take. In our case I_M.
The amplitude of the sinusoid corresponds to the magnitude of the corresponding rotating vector.

Activity 4

1.3.3 - Phase

The phase angle of the sinusoid is what sets its initial value.  
The phase of the sinusoid corresponds to that of the rotating vector. 

Activity 5


If you have completed the actions suggested in activity 5 you will certainly have noticed that the phase sets the instantaneous value of the sinusoid at the start, that is, at the initial instant.
What does all this mean?
If we enter a phase value, for example 30°, we see that at time t=0, that is, at the instant we start observing the phenomenon (and start counting time), the sinusoid starts  with a value corresponding to \(\small sin(30°)\), or better: \(\small sin( \pi/6)\), and the conductor is also at an angle of 30°, \(\small \omega t = \pi/6\). In other words, when I start observing the sinusoid the conductor has already covered an angle of 30°.
The corresponding rotating vector, in turn, makes an angle of 30° with the reference axis: the horizontal axis, or real axis as we will see, which is often not drawn in vector diagrams.
Likewise: a phase of −30° means that initially \(\small e(t)|^{t=0} = sin(- \pi/6)\). The conductor must travel 30° before passing again through the origin (0°) of the rotation. The representative vector is initially at −30°.
In general, then, a sinusoidal quantity will be represented by:
\[ e(t) = E \, sin(\omega t + \varphi)\] where \(\small E\) is the amplitude of the sinusoid (and also the magnitude of the rotating vector) and \( \small \varphi\) is the phase.

1.3.4 - Lead and lag

It is common to say that a sinusoid leads if it has a positive phase and lags if its phase is negative.
Of course we must specify the reference: the sinusoid leads or lags with respect to what?
The reference, as seems obvious, is a sinusoid with a phase of 0° or 0 radians.
This becomes clear if, with the sinusoid displayed, we move the slider φ to the right (lead) or to the left (lag). In the first case the sinusoid shifts to the left (its instantaneous values come before the corresponding ones of the sinusoid with φ=0). In the second the sinusoid shifts to the right (its instantaneous values come after the corresponding values of the sinusoid with φ=0).
    If we show several sinusoids at the same time we are also interested in defining their mutual phase shift, that is, saying by how much one sinusoid (one vector) leads or lags  the other.

Activity 6


The phase shift between the two sinusoids, \(\small e_1(t)= E\,sin(\omega t +\varphi)\) with respect to \( \small e_2(t) = E\,sin(\omega t +\psi)\), is given by the angle: \(\small \beta =\varphi - \psi\) or \( \small \beta =\psi - \varphi\) for the phase shift of \(\small e_2(t)\) with respect to \(\small e_1(t)\).
If \(\small |\beta| = \pi/2\) (90°) the sinusoids are said to be in quadrature (leading or lagging).
For example, if \(\small \varphi\) = 30° and \(\small \psi\) = −60°, \(\small \beta\) = 90° and \( \small e_1(t)\) is in leading quadrature with respect to \(\small e_2(t)\).

In physics, when studying oscillations, sinusoids are often written using the cosine function, \( \small e_1(t) = E_M \, cos(\omega t + \psi)\). In that case it is enough to take the projection of the vector on the horizontal axis instead of the vertical one, and to give the phase \(\small \psi\) the value \( \small \psi = \varphi- \pi/2\) (\( \small E_M = Blv\)).

1.4 - Phasors

We have seen how sinusoids can be represented by rotating vectors. Indeed, we have established that under certain conditions there is a one-to-one correspondence between these vectors and sinusoids.
A key condition for this kind of representation is that the sinusoids and the corresponding rotating vectors have the same frequency. This frees us from the time dependence, since all the vectors rotate at the same speed, so the quantity is fully determined by the vector at the initial instant. We can then set up a one-to-one correspondence between the set of sinusoids with the same frequency and phasors, that is, the representation of the rotating vector at the initial instant. A phasor is therefore characterized only by amplitude (magnitude) and phase.
A phasor \( \small \textbf Y \) can be written in polar form (Steinmetz form): \( \small \textbf{Y} = Y \angle \alpha °\) where \( \small Y \) is the magnitude of the vector and \( \small \alpha \) is the phase in degrees. The sign \( \small \angle \) stands for the angle.
The magnitude of the phasor is usually the RMS value of the sinusoidal quantity: \( \small Y = Y_M / \sqrt{2}\)

Activity 7


In short:
phasors are fixed vectors. They keep the information about the initial state. Since the quantities have the same frequency, they will all evolve over time in the same way, so for studying the system only the initial state matters. However, not all systems can be described with same-frequency phasors. Only linear systems meet the requirements for this kind of representation.
Phasors play an important role in the analysis of alternating current circuits by means of the symbolic method and complex numbers.

1.5 - Sinusoids and complex numbers

Analyzing alternating current circuits through operations between sinusoids is not practical at all. Fortunately we have already pointed out the correspondence between sinusoids and phasors, which lets us translate mathematical operations between sinusoids into mathematical operations between phasors. Turning these operations into algebra requires a further step: the introduction of complex arithmetic.
We have discussed the vector representation of complex numbers (1.2.2), reaching the conclusion that every complex number can be represented on the Argand-Gauss plane by a vector. The phasor, as we said, represents a rotating vector frozen in its initial state, generally with a magnitude equal to the RMS value of the quantity. The phasor is characterized by amplitude and phase, which correspond to the modulus \( \small \sqrt{a^2 + b^2}\) and the argument \( \small \beta\) of the complex number (figure 2).
A sinusoid \( \small y(t)\) with angular frequency \( \small \omega\) can then be represented by the complex number \( \small a+jb\), where \( \small j =\sqrt{-1}\) is the imaginary unit, if: \( \small y(t) = \sqrt{a^2 + b^2}\,sin(\omega t + \beta)\).
So we have the correspondence:
\[ a + jb \;\leftrightarrow\; \sqrt{a^2 + b^2}\,sin(\omega t + \beta)\] or, writing the complex number in exponential form:

\[ \sqrt{a^2 + b^2}\, e^{j \beta} \;\leftrightarrow\; \sqrt{a^2 + b^2}\,sin(\omega t + \beta)\]
The argument \( \small \beta\) is found from the ratio between the imaginary part b and the real part a of the complex number, since \( \small \beta = arctan(b/a)\).
These correspondences are the basis of the symbolic method, or phasor analysis, for solving electric circuits in sinusoidal alternating current.

2 - The basics of the symbolic method

Assuming the reader is familiar with complex numbers and with Euler's formula, we give here the mathematical basis of the symbolic method for analyzing sinusoidal alternating current circuits.
The symbolic method consists precisely in representing sinusoidal quantities by phasors and complex numbers.

Let us write the following complex function⁽⁶⁾:

(2.1) \(\qquad \dot Y(t) = A [ cos(\omega t + \varphi) + j\,sin(\omega t + \varphi)] \)

Its real and imaginary parts are respectively:

\(\qquad Re[\dot Y(t)] = A\,cos(\omega t + \varphi) \qquad Im[\dot Y(t)] = A\,sin(\omega t + \varphi) \)

Both can describe a harmonic oscillation.
Usually the real part of (2.1) is taken to represent the physical phenomenon, while no physical meaning is given to the imaginary part.
So if (2.1) were the complex form of a current, the physical quantity, the current intensity, would be given by:
\[ i(t) = Re[\dot Y(t)] = A\,cos(\omega t + \varphi) \] In an introductory study of AC circuits, however, it is more straightforward to use the sine function, which amounts to  giving physical meaning to the imaginary part of \(\small \dot Y(t)\):
\[ i(t) = A\,sin(\omega t + \varphi) \] mainly because, to get the instantaneous values of the sinusoidal quantity, it is more convenient to project the rotating vectors onto the vertical axis.
The substance does not change anyway, since the cosine and sine functions are obtained from each other through a phase shift of \(\small \pi/2) \) \[ cos(\omega t + \varphi - \frac{\pi}{2}) = sin(\omega t + \varphi) \] Using Euler's formula, (2.1) can be rewritten as:

(2.2) \(\qquad \qquad \qquad \dot Y(t) = A\, e^{j\,(\omega t + \varphi)}\)

and, by well-known properties of the exponential:

(2.3) \(\quad \dot Y(t) = A\,e^{j\,\varphi} e^{j\, \omega t} = A \,[ cos (\varphi) + j\,sin(\varphi)] [cos(\omega t) + j\,sin(\omega t)]\)

with which we have separated the time dependence \( e^{j\,\omega t} \) from the phasor part: \( A\,e^{j\,\varphi} \).
So we write:

(2.4) \(\qquad \qquad \qquad \dot Y(t) = \dot Y\, e^{j\,(\omega t}\)

where:

(2.5) \(\qquad \qquad \qquad \dot Y = A\,e^{j\,\varphi} =A \,[ cos (\varphi) + j\,sin(\varphi)]\)

is the phasor.

From (2.3) it is also clear that:

(2.6)\(\qquad \dot Y(0) = A\,e^{j\,\varphi} e^{j\, 0} = A\,e^{j\,\varphi} \cdot 1 = A\,e^{j\,\varphi} = \dot Y\)

which proves what we said before: the phasor represents the initial state of the rotating vector (apart from a possible conversion of the amplitude into the RMS value).
Since the time dependence \( e^{j\,\omega t} \) is common to all the quantities (the sinusoids all have the same frequency), the relations between them involve only the phasors. This lets us turn every operation into algebra through complex numbers.
The sum and difference of sinusoids translate immediately into the sum or difference of phasors, and therefore of complex numbers.
The product of sinusoids deserves a few more remarks.

The symbolic method uses complex numbers to represent:

1. actual sinusoids;
2. operators that turn sinusoids into others of a different kind and with different features

The first point has already been covered.
If we multiply the phasor (2.5) by a complex number \( \small \dot Z = R +j\,X = Z\,e^{j\,\beta}\)

(2.7)\(\quad \dot Z \cdot \dot Y = Z\,e^{j\,\beta}\,A\, e^{j\, \varphi} = Z\,A \,e^{j\,(\beta +\varphi)} = Z\,A\,[ cos (\beta +\varphi) + j\,sin(\beta + \varphi)]\)

we get a new phasor with magnitude \(\small Z \, A\), rotated by \(\small \beta\) radians with respect to \(\small \dot Y\).
In this sense we regard \(\small \dot Z\) as an operator that changes the magnitude and phase of the phasor.
If \(\small \dot Z\) also has physical dimensions, applying it to \(\small \dot Y\) will also change the kind of physical quantity. For example, if \(\small \dot Y\) were a current and \(\small \dot Z\) were a quantity measured in ohms, \(\small \dot Z \cdot \dot Y\) would be a quantity measured in volts, that is, a voltage.

Continue with:

• Alternating Current 2: impedance, reactance and series RLC circuits, voltage resonance (in Italian)
• Alternating Current 3: admittance, parallel RLC circuits, current resonance (in Italian)

Rev. 12/28/2020


Books by the author and further reading

The author’s books

• Complex numbers: definitions and basic operations (in Italian)
• Euler's formula and identity. Complex powers and logarithms (in Italian)

---

⁽¹⁾ Actually, electric fields and hence potential differences can be created by magnetic fields that change over time, whether or not conductors and/or electric circuits are present.

⁽²⁾ The product between vectors written with the operator \(\times\) (in Europe the symbol \(\wedge\) is also used) is the so-called cross product: it is a vector perpendicular to the plane defined by the operands \(\small \vec{v}\) and \(\small \vec{B}\), with magnitude \(\small v\,B\,sin(\alpha)\), where \(\small \alpha\) is the angle between the vectors \(\small \vec{v}\) and \(\small \vec{B}\).

⁽³⁾ \(\small \vec{E} \cdot \vec{l}\) is the so-called dot product of vectors, taking \(\small \vec{l}\) as a displacement vector: \(\small \vec{E} \cdot \vec{l} = E\,l\,cos(\beta)\), where \(\small \beta\) is the angle, if any, between the field vector and the displacement vector. So the formula, written correctly, would be \(\small e(t) = \vec{E} \cdot \vec{l} = v\,B\,sin(\omega t)\,l\,cos(\beta)\), but here \(\small \beta = 0 \Rightarrow cos(\beta) = 1\), since the field is parallel to the displacement.

⁽⁴⁾ Thumb, index and middle fingers of the right hand are held perpendicular to one another (forming a right-handed set). The thumb gives the direction of the velocity, the index finger that of the magnetic field (North-South), and the middle finger gives the direction of the induced electromotive force.

⁽⁵⁾ In that case it is sometimes called a cosinusoid. Recall that \(\small cos(\alpha) = sin(\alpha + \pi/2)\), so the graph of \(\small cos(\alpha)\) is, to all intents and purposes, a sinusoid.

⁽⁶⁾ We use the dotted letter \(\small \dot Y \) for complex quantities (complex numbers). In the technical and scientific literature bold type is also often used for vectors, phasors and complex quantities in general. The imaginary unit \(i = \sqrt{-1}\) is called \(j\) here, as is customary in electrical engineering and electronics, to avoid confusion with the symbol used for current.