Phasor (sine waves) - Citizendia

In physics and engineering, a phasor is a representation of a sine wave whose amplitude (A), phase (θ), and frequency (ω) are time-invariant. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and It is a subset of a more general concept called analytic representation. In Mathematics and Signal processing, the analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal Phasors reduce the dependencies on these parameters to three independent factors, thereby simplifying certain kinds of calculations. In particular, the frequency factor, which also includes the time-dependence of the sine wave, is often common to all the components of a linear combination of sine waves. Using phasors, it can be factored out, leaving just the static amplitude and phase information to be combined algebraically (rather than trigonometrically). Similarly, linear differential equations can be reduced to algebraic ones. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential The term phasor therefore often refers to just those two factors. In older texts, a phasor is also referred to as a sinor.

1 Definition
2 Phasor arithmetic
3 Circuit laws
4 Power engineering
5 References
6 External links

Definition

Euler's formula indicates that sine waves can be represented mathematically as the sum of two complex-valued functions:

$A\cdot \cos(\omega t + \theta) = A/2\cdot e^{j(\omega t + \theta)} + A/2\cdot e^{-j(\omega t + \theta)},$ ^[1]

or as the real part of one of the functions:

$\begin{align} A\cdot \cos(\omega t + \theta) &= \operatorname{Re} \left\{ A\cdot e^{j(\omega t + \theta)}\right\} \\ &= \operatorname{Re} \left\{ A e^{j\theta} \cdot e^{j\omega t}\right\}. \end{align}$

As indicated above, phasor can refer to either $A e^{j\theta} e^{j\omega t}\,$ or just the complex constant, $A e^{j\theta}\,$ . This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In the latter case, it is understood to be a shorthand notation, encoding the amplitude and phase of an underlying sinusoid.

An even more compact shorthand is angle notation: $A \angle \theta.\,$

Phasor arithmetic

Multiplication by a constant (scalar)

Multiplication of the phasor A•e^{j θ} by a complex constant, B•e^{j φ}, produces another phasor. Angle notation or phasor notation is a notation used in electronics using the \ang \!\ sign That means its only effect is to change the amplitude and phase of the underlying sinusoid:

$\begin{align} \operatorname{Re}\{(A e^{j\theta} \cdot B e^{j\phi})\cdot e^{j\omega t} \} &= \operatorname{Re}\{(AB e^{j(\theta+\phi)})\cdot e^{j\omega t} \} \\ &= AB \cos(\omega t +(\theta+\phi)) \end{align}$

Although B•e^{j φ} has the form of the shorthand notation for a phasor, it is not a phasor. In electronics, it is an impedance, and the phase shift is actually caused by the time-delay associated with a reactive circuit element. The product of two phasors (or squaring a phasor) would represent the product of two sine waves, which is a non-linear operation and does not produce another phasor.

Differentiation and integration

The time derivative or integral of a phasor produces another phasor^[2]. For example:

$\begin{align} \operatorname{Re}\left\{\frac{d}{dt}(A e^{j\theta} \cdot e^{j\omega t})\right\} &= \operatorname{Re}\{A e^{j\theta} \cdot j\omega e^{j\omega t}\} \\ &= \operatorname{Re}\{A e^{j\theta} \cdot e^{j\pi/2} \omega e^{j\omega t}\} \\ &= \operatorname{Re}\{\omega A e^{j(\theta + \pi/2)} \cdot e^{j\omega t}\} \\ &= \omega A\cdot \cos(\omega t + \theta + \pi/2) \end{align}$

Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant, $j \omega = (e^{j\pi/2} \cdot \omega).\,$ Similarly, integrating a phasor corresponds to multiplication by $\frac{1}{j\omega} = \frac{e^{-j\pi/2}}{\omega}.\,$ The time-dependent factor, $e^{j\omega t}\,$ , is unaffected. When we solve a linear differential equation with phasor arithmetic, we are merely factoring $e^{j\omega t}\,$ out of all terms of the equation, and reinserting it into the answer. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential For example, consider the following differential equation for the voltage across the capacitor in an RC circuit:

$\frac{d\ v_C(t)}{dt} + \frac{1}{RC}v_C(t) = \frac{1}{RC}v_S(t)$

When the voltage source in this circuit is sinusoidal:

$v_S(t) = V_P\cdot \cos(\omega t + \theta),\,$

we may substitute:

$\begin{align} v_S(t) &= \operatorname{Re} \{V_s \cdot e^{j\omega t}\} \\ \end{align}$

$v_C(t) = \operatorname{Re} \{V_c \cdot e^{j\omega t}\},$

where phasor $V_s = V_P e^{j\theta},\,$ and phasor $V_c\,$ is the unknown quantity to be determined. A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by

In the phasor shorthand notation, the differential equation reduces to^[3]:

$j \omega V_c + \frac{1}{RC} V_c = \frac{1}{RC}V_s$

Solving for the phasor capacitor voltage gives:

$V_c = \frac{1}{1 + j \omega RC} \cdot (V_s) = \frac{1-j\omega R C}{1+(\omega R C)^2} \cdot (V_P e^{j\theta})\,$

As we have see, the complex constant factor represents differences of the amplitude and phase of $v_C(t)\,$ relative to $V_P\,$ and $\theta.\,$

In polar coordinate form, the factor is:

$\frac{1}{\sqrt{1 + (\omega RC)^2}}\cdot e^{-j \phi(\omega)},\,$ where $\phi(\omega) = \arctan(\omega RC).\,$

Therefore:

$v_C(t) = \frac{1}{\sqrt{1 + (\omega RC)^2}}\cdot V_P \cos(\omega t + \theta- \phi(\omega))$

Addition

The sum of multiple phasors produces another phasor. That is because the sum of sine waves of one frequency is also a sine wave:

$\begin{align} A_1 \cos(\omega t + \theta_1) + A_2 \cos(\omega t + \theta_2) &= \operatorname{Re} \{A_1 e^{j\theta_1}e^{j\omega t}\} + \operatorname{Re} \{A_2 e^{j\theta_2}e^{j\omega t}\} \\ &= \operatorname{Re} \{A_1 e^{j\theta_1}e^{j\omega t} + A_2 e^{j\theta_2}e^{j\omega t}\} \\ &= \operatorname{Re} \{(A_1 e^{j\theta_1} + A_2 e^{j\theta_2})e^{j\omega t}\} \\ &= \operatorname{Re} \{(A_3 e^{j\theta_3})e^{j\omega t}\} \\ &= A_3 \cos(\omega t + \theta_3), \end{align}$

where:

$A_3^2 = (A_1 \cos(\theta_1)+A_2 \cos(\theta_2))^2 + (A_1 \sin(\theta_1)+A_2 \sin(\theta_2))^2$

$\tan(\theta_3) = \frac{A_1 \sin(\theta_1)+A_2 \sin(\theta_2)}{A_1 \cos(\theta_1)+A_2 \cos(\theta_2)}.$

In physics, this sort of addition occurs when sine waves "interfere" with each other, constructively or destructively. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern Another way to view the calculations above is that two vectors with coordinates $[A_1 \cos(\theta_1), A_1 \sin(\theta_1)]\,$ and $[A_2 \cos(\theta_2), A_2 \sin(\theta_2)]\,$ are added (see vector addition) to produce a resultant vector with coordinates $[A_3 \cos(\theta_3), A_3 \sin(\theta_3)].\,$

Phasor diagram of three waves in perfect destructive interference

The vector concept provides useful insight into questions like this: "What phase difference would be required between three identical waves for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral triangle, and the angle between each phasor to the next is 120° (2π/3 radians), or one third of a wavelength $λ / 3. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line$ So the phase difference between each wave must also be 120°.

In other words, what this shows is:

$\cos(\omega t) + \cos(\omega t + 2\pi/3) + \cos(\omega t +4\pi/3) = 0.\,$

In the example of three waves, the phase difference between the first and the last wave was 240 degrees, while for two waves destructive interference happens at 180 degrees. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength $λ$ . This is why in single slit diffraction, the minima occurs when light from the far edge travels a full wavelength further than the light from the near edge. Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700

Phasor diagrams

Electrical engineers, electronics engineers, and electronic engineering technicians use phasor diagrams to visualize complex constants and variables (phasors). Like vectors, arrows drawn on graph paper or computer displays represent phasors. Cartesian and polar representations each have advantages.

Notes

^
- j is the Imaginary unit ( $j 2 = - 1$ ). Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation
- The frequency of the wave, in Hz, is given by $ω / 2π$ .
^ This results from: $\frac{d}{dt}(e^{j \omega t}) = j \omega e^{j \omega t}$ which means that the complex exponential is the eigenfunction of the derivative operation. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change
^ Proof:

$\frac{d\ \operatorname{Re} \{V_c \cdot e^{j\omega t}\}}{dt} + \frac{1}{RC}\operatorname{Re} \{V_c \cdot e^{j\omega t}\} = \frac{1}{RC}\operatorname{Re} \{V_s \cdot e^{j\omega t}\}$

(Eq. 1)

Since this must hold for all $t\,$ , specifically: $t-\frac{\pi}{2\omega },\,$ it follows that:

$\frac{d\ \operatorname{Im} \{V_c \cdot e^{j\omega t}\}}{dt} + \frac{1}{RC}\operatorname{Im} \{V_c \cdot e^{j\omega t}\} = \frac{1}{RC}\operatorname{Im} \{V_s \cdot e^{j\omega t}\}$

(Eq. 2)

It is also readily seen that:

$\frac{d\ \operatorname{Re} \{V_c \cdot e^{j\omega t}\}}{dt} = \operatorname{Re} \left\{ \frac{d\left( V_c \cdot e^{j\omega t}\right)}{dt} \right\} = \operatorname{Re} \left\{ j\omega V_c \cdot e^{j\omega t} \right\}$

$\frac{d\ \operatorname{Im} \{V_c \cdot e^{j\omega t}\}}{dt} = \operatorname{Im} \left\{ \frac{d\left( V_c \cdot e^{j\omega t}\right)}{dt} \right\} = \operatorname{Im} \left\{ j\omega V_c \cdot e^{j\omega t} \right\}$

Substituting these into (1) and (2), and multiplying (2) by $j\,$ , and adding both equations gives:

$j\omega V_c \cdot e^{j\omega t} + \frac{1}{RC}V_c \cdot e^{j\omega t} = \frac{1}{RC}V_s \cdot e^{j\omega t}$

$\left(j\omega V_c + \frac{1}{RC}V_c = \frac{1}{RC}V_s\right) \cdot e^{j\omega t}$

$j\omega V_c + \frac{1}{RC}V_c = \frac{1}{RC}V_s \quad\quad(QED)$

Circuit laws

With phasors, the techniques for solving DC circuits can be applied to solve AC circuits. Direct current ( DC) is the unidirectional flow of Electric charge. A list of the basic laws is given below.

Ohm's law for resistors: a resistor has no time delays and therefore doesn't change the phase of a signal therefore V=IR remains valid.
Ohm's law for resistors, inductors, and capacitors: V=IZ where Z is the complex impedance. Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal Alternating current (AC
In an AC circuit we have real power (P) which is a representation of the average power into the circuit and reactive power (Q) which indicates power flowing back and forward. We can also define the complex power S=P+jQ and the apparent power which is the magnitude of S. This article deals with power in AC systems See Mains electricity for information on utility supplied AC power The power law for an AC circuit expressed in phasors is then S=VI^* (where I^* is the complex conjugate of I). In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.
Kirchhoff's circuit laws work with phasors in complex form

Given this we can apply the techniques of analysis of resistive circuits with phasors to analyze single frequency AC circuits containing resistors, capacitors, and inductors. For other laws named after Gustav Kirchhoff, see Kirchhoff's laws. A network in the context of Electronics, is a collection of interconnected components Multiple frequency linear AC circuits and AC circuits with different waveforms can be analyzed to find voltages and currents by transforming all waveforms to sine wave components with magnitude and phase then analyzing each frequency separately.

Power engineering

In analysis of three phase AC power systems, usually a set of phasors is defined as the three complex cube roots of unity, graphically represented as unit magnitudes at angles of 0, 120 and 240 degrees. This article deals with the basic mathematics and principles of three-phase electricity By treating polyphase AC circuit quantities as phasors, balanced circuits can be simplified and unbalanced circuits can be treated as an algebraic combination of symmetrical circuits. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents. In the context of power systems analysis, the phase angle is often given in degrees, and the magnitude in rms value rather than the peak amplitude of the sinusoid. This article describes the unit of angle For other meanings see Degree. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the

References

Douglas C. Giancoli (1989). Physics for Scientists and Engineers. Prentice Hall. ISBN 0-13-666322-2.

External links

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