Phasor

and respective phasor diagram for a specific . The arrows in the upper diagram are phasors, drawn in a phasor diagram (complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current. In physics and engineering, a phasor (a portmanteau of phase vector ) is a complex number representing a sinusoidal function whose amplitude and initial phase are time-invariant and whose angular frequency is fixed. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, and (in older texts) sinor or even complexor. A common application is in the steady-state analysis of an electrical network powered by time varying current where all signals are assumed to be sinusoidal with a common frequency. Phasor representation allows the analyst to represent the amplitude and phase of the signal using a single complex number. The only difference in their analytic representations is the complex amplitude (phasor).…

and respective phasor diagram for a specific . The arrows in the upper diagram are phasors, drawn in a phasor diagram (complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current.]] In physics and engineering, a phasor (a portmanteau of phase vector ) is a complex number representing a sinusoidal function whose amplitude and initial phase are time-invariant and whose angular frequency is fixed. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, and (in older texts) sinor or even complexor. A common application is in the steady-state analysis of an electrical network powered by time varying current where all signals are assumed to be sinusoidal with a common frequency. Phasor representation allows the analyst to represent the amplitude and phase of the signal using a single complex number. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be represented as a linear combination of phasors (known as phasor arithmetic or phasor algebra ) and the time/frequency dependent factor that they all have in common. The origin of the term phasor rightfully suggests that a (diagrammatic) calculus somewhat similar to that possible for vectors is possible for phasors as well. An important additional feature of the phasor transform is that differentiation and integration of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple algebraic operations on the phasors; the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain. The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century. He got his inspiration from Oliver Heaviside. Heaviside's operational calculus was modified so that the variable p becomes jω. The complex number j has simple meaning: phase shift. Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform (limited to a single frequency), which, in contrast to phasor representation, can be used to (simultaneously) derive the transient response of an RLC circuit. However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required. rotates around the origin. Its magnitude is A, and it completes one cycle every 2 /ω. θ is the angle it forms with the positive real axis at (and at for all integer values of ).]] Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering. A vector whose polar coordinates are magnitude A and angle \theta is written A \angle \theta. 1 \angle \theta can represent either the vector (\cos \theta,\, \sin \theta) or the complex number \cos \theta + i \sin \theta = e^{i\theta} , according to Euler's formula with i^2 = -1 , both of which have magnitudes of 1. The angle may be stated in degrees with an implied conversion from degrees to radians. For example 1 \angle 90 would be assumed to be 1 \angle 90^\circ, which is the vector (0,\, 1) or the number e^{i\pi/2} = i. Multiplication and division of complex numbers become straight forward through the phasor notation. Given the vectors v_1 = A_1 \angle \theta_1 and v_2 = A_2 \angle \theta_2 , the following is true: v_1 \cdot v_2 = A_1 \cdot A_2 \angle (\theta_1 + \theta_2) , \frac{v_1}{v_2} = \frac{A_1}{A_2} \angle (\theta_1 - \theta_2) . v(t) = A\cos(\omega t + \theta) A phasor is a complex number representation that captures the amplitude and phase of this sinusoid. The phasor corresponding to the above sinusoid is defined as: \mathbf{V} = Ae^{i\theta} Using Euler's formula ( ), this can be expanded to: \mathbf{V} = A\cos(\theta) + i\cdot A\sin(\theta) Alternatively, the phasor can be expressed in polar form as: \mathbf{V} = A \angle \theta which reads as "amplitude at phase angle ", whose real part is the original sinusoid. The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids. Furthermore, all the mathematics can be done with just the phasors A e^{i \theta}, and the common factor e^{i\omega t} is reinserted prior to the real part of the result. The function Ae^{i(\omega t + \theta)} is an analytic representation of A\cos(\omega t + \theta). Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a phasor, as we do in the next section. \begin{align} &\operatorname{Re}\left( \left(A e^{i\theta} \cdot B e^{i\phi}\right) \cdot e^{i\omega t} \right) \\ \begin{align} &A_1\cos(\omega t + \theta_1) + A_2\cos(\omega t + \theta_2) \\[3pt] ={} &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, and, if we take \theta_3 \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right] , then \theta_3 is: \sgn(A_1 \sin(\theta_1) + A_2 \sin(\theta_2)) \cdot \frac{\pi}{2}, if A_1 \cos\theta_1 + A_2 \cos\theta_2 = 0, with \sgn the signum function; \arctan\left(\frac{A_1 \sin\theta_1 + A_2 \sin\theta_2}{A_1 \cos\theta_1 + A_2 \cos\theta_2}\right), if A_1 \cos\theta_1 + A_2 \cos\theta_2 > 0 ; \pi + \arctan\left(\frac{A_1 \sin\theta_1 + A_2 \sin\theta_2}{A_1 \cos\theta_1 + A_2 \cos\theta_2}\right), if A_1 \cos\theta_1 + A_2 \cos\theta_2 . or, via the law of cosines on the complex plane (or the trigonometric identity for angle differences): A_3^2 = A_1^2 + A_2^2 - 2 A_1 A_2 \cos(180^\circ - \Delta\theta) A_1 \angle \theta_1 + A_2 \angle \theta_2 = A_3 \angle \theta_3. Another way to view addition is that two vectors with coordinates and are added vectorially to produce a resultant vector with coordinates (see animation). In physics, this sort of addition occurs when sinusoids interfere with each other, constructively or destructively. The static vector concept provides useful insight into questions like this: "What phase difference would be required between three identical sinusoids 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, so the angle between each phasor to the next is 120° (  radians), or one third of a wavelength . So the phase difference between each wave must also be 120°, as is the case in three-phase power. In other words, what this shows is that: \cos(\omega t) + \cos\left(\omega t + \frac{2\pi}{3}\right) + \cos\left(\omega t - \frac{2\pi}{3}\right) = 0. In the example of three waves, the phase difference between the first and the last wave was 240°, while for two waves destructive interference happens at 180°. 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 \lambda . This is why in single slit diffraction, the minima occur when light from the far edge travels a full wavelength further than the light from the near edge. As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360° or 2  radians representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, . When the vector is horizontal the tip of the vector represents the angles at 0°, 180°, and at 360°. Likewise, when the tip of the vector is vertical it represents the positive peak value, ( ) at 90° or and the negative peak value, ( ) at 270° or . Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represents a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, ( ) and in our example above, this is at an angle of 30°. Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time with a corresponding phase angle in either degrees or radians. But if a second waveform starts to the left or to the right of this zero point, or if we want to represent in phasor notation the relationship between the two waveforms, then we will need to take into account this phase difference, of the waveform. Consider the diagram below from the previous Phase Difference tutorial. \begin{align} &\operatorname{Re}\left( \frac{\mathrm{d}}{\mathrm{d}t} \mathord\left(A e^{i\theta} \cdot e^{i\omega t}\right) \right) \\ Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant i \omega = e^{i\pi/2} \cdot \omega . Similarly, integrating a phasor corresponds to multiplication by \frac{1}{i\omega} = \frac{e^{-i\pi/2}}{\omega}. The time-dependent factor, e^{i\omega t}, is unaffected. When we solve a linear differential equation with phasor arithmetic, we are merely factoring e^{i\omega t} out of all terms of the equation, and reinserting it into the answer. For example, consider the following differential equation for the voltage across the capacitor in an RC circuit: \frac{\mathrm{d}\, v_\text{C}(t)}{\mathrm{d}t} + \frac{1}{RC}v_\text{C}(t) = \frac{1}{RC} v_\text{S}(t). When the voltage source in this circuit is sinusoidal: v_\text{S}(t) = V_\text{P} \cdot \cos(\omega t + \theta), we may substitute v_\text{S}(t) = \operatorname{Re}\left( V_\text{s} \cdot e^{i \omega t} \right). v_\text{C}(t) = \operatorname{Re}\left(V_\text{c} \cdot e^{i \omega t} \right), where phasor V_\text{s} = V_\text{P} e^{i\theta}, and phasor V_\text{c} is the unknown quantity to be determined. In the phasor shorthand notation, the differential equation reduces to: i \omega V_\text{c} + \frac{1}{RC} V_\text{c} = \frac{1}{RC}V_\text{s}. {{math proof|title=Derivation|proof= {{NumBlk2|:| \frac{\mathrm{d}}{\mathrm{d}t} \operatorname{Re}\left(V_\text{c} \cdot e^{i\omega t}\right) + \frac{1}{RC}\operatorname{Re}(V_\text{c} \cdot e^{i\omega t}) = \frac{1}{RC}\operatorname{Re}\left(V_\text{s} \cdot e^{i\omega t}\right) |Eq.1}} Since this must hold for all t , specifically: t - \frac{\pi}{2\omega}, it follows that: {{NumBlk2|:| \frac{\mathrm{d}}{\mathrm{d}t} \operatorname{Im}\left(V_\text{c} \cdot e^{i\omega t}\right) + \frac{1}{RC} \operatorname{Im}\left(V_\text{c} \cdot e^{i\omega t}\right) = \frac{1}{RC}\operatorname{Im}\left(V_\text{s} \cdot e^{i\omega t}\right). |Eq.2}} It is also readily seen that: \begin{align} \frac{\mathrm{d}}{\mathrm{d}t} \operatorname{Re}\left( V_\text{c} \cdot e^{i\omega t} \right) &= \operatorname{Re}\left(\frac{\mathrm{d}}{\mathrm{d}t} \mathord\left( V_\text{c} \cdot e^{i\omega t} \right)\right) = \operatorname{Re}\left(i\omega V_\text{c} \cdot e^{i\omega t} \right) \\ \frac{\mathrm{d}}{\mathrm{d}t} \operatorname{Im}\left( V_\text{c} \cdot e^{i\omega t} \right) &= \operatorname{Im}\left(\frac{\mathrm{d}}{\mathrm{d}t} \mathord\left( V_\text{c} \cdot e^{i\omega t} \right) \right) = \operatorname{Im}\left(i\omega V_\text{c} \cdot e^{i\omega t} \right). \end{align} Substituting these into and , multiplying by i, and adding both equations gives: \begin{align} i\omega V_\text{c} \cdot e^{i\omega t} + \frac{1}{RC}V_\text{c} \cdot e^{i\omega t} &= \frac{1}{RC}V_\text{s} \cdot e^{i\omega t} \\ \left(i\omega V_\text{c} + \frac{1}{RC}V_\text{c} \right) \!\cdot e^{i\omega t} &= \left(\frac{1}{RC}V_\text{s}\right) \cdot e^{i \omega t} \\ i\omega V_\text{c} + \frac{1}{RC}V_\text{c} &= \frac{1}{RC}V_\text{s}. \end{align} }} Solving for the phasor capacitor voltage gives: V_\text{c} = \frac{1}{1 + i \omega RC} \cdot V_\text{s} = \frac{1 - i\omega R C}{1 + (\omega RC)^2} \cdot V_\text{P} e^{i\theta}. As we have seen, the factor multiplying V_\text{s} represents differences of the amplitude and phase of v_\text{C}(t) relative to V_\text{P} and \theta. In polar coordinate form, the first term of the last expression is: \frac{1 - i\omega R C}{1 + (\omega RC)^2}=\frac{1}{\sqrt{1 + (\omega RC)^2}}\cdot e^{-i \phi(\omega)}, where \phi(\omega) = \arctan(\omega RC) . Therefore: v_\text{C}(t) =\operatorname{Re}\left(V_\text{c} \cdot e^{i \omega t} \right)= \frac{1}{\sqrt{1 + (\omega RC)^2}}\cdot V_\text{P} \cos(\omega t + \theta - \phi(\omega)). x(t) = \operatorname{Re}\left( A e^{i \theta} \cdot e^{i 2\pi f_0 t} \right), where the term in brackets is viewed as a rotating vector in the complex plane. The phasor has length A , rotates anti-clockwise at a rate of f_0 revolutions per second, and at time t = 0 makes an angle of \theta with respect to the positive real axis. The waveform x(t) can then be viewed as a projection of this vector onto the real axis. A modulated waveform is represented by this phasor (the carrier) and two additional phasors (the modulation phasors). If the modulating signal is a single tone of the form Am \cos{2\pi f_m t} , where m is the modulation depth and f_m is the frequency of the modulating signal, then for amplitude modulation the two modulation phasors are given by, {1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi (f_0+f_m) t}, {1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi (f_0-f_m) t}. The two modulation phasors are phased such that their vector sum is always in phase with the carrier phasor. An alternative representation is two phasors counter rotating around the end of the carrier phasor at a rate f_m relative to the carrier phasor. That is, {1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi f_m t}, {1 \over 2} Am e^{i \theta} \cdot e^{-i 2\pi f_m t}. Frequency modulation is a similar representation except that the modulating phasors are not in phase with the carrier. In this case the vector sum of the modulating phasors is shifted 90° from the carrier phase. Strictly, frequency modulation representation requires additional small modulation phasors at 2f_m, 3f_m etc., but for most practical purposes these are ignored because their effect is very small. == Footnotes==

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