Circuit Theory/Phasor Analysis Phasor Analysis[edit] The mathematical representations of individual circuit elements can be converted into phasor notation, and then the circuit can be solved using phasors. Resistance, Impedance and Admittance[edit] In phasor notation, resistance, capacitance, and inductance can all be lumped together into a single term called "impedance". The phasor used for impedance is . is Voltage and is current. And the Ohm's law for phasors becomes: It is important to note at this point that Ohm's Law still holds true even when we switch from the time domain to the phasor domain. Impedance is still measured in units of Ohms, and admittance (like Conductance, its DC-counterpart) is still measured in units of Siemens. Let's take a closer look at this equation: If we break this up into polar notation, we get the following result: Resistors[edit] Resistors do not affect the phase of the voltage or current, only the magnitude. Capacitors[edit] A capacitor with a capacitance of C has a phasor value: Where
Complex plane Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z. In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.[1] Notational conventions[edit] In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this: for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In the Cartesian plane the point (x, y) can also be represented in polar coordinates as where Stereographic projections[edit] Cutting the plane[edit]
Phasor An example of series RLC circuit and respective phasor diagram for a specific Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform, which additionally can be used to (simultaneously) derive the transient response of an RLC circuit.[10][8] However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.[10] Definition[edit] Euler's formula indicates that sinusoids can be represented mathematically as the sum of two complex-valued functions: [a] or as the real part of one of the functions: The term phasor can refer to either [citation needed] or just the complex constant, . An even more compact shorthand is angle notation: See also vector notation. A phasor can be considered a vector rotating about the origin in a complex plane. . represents the angle that the vector forms with the real axis at t = 0. Phasor arithmetic[edit] In electronics, and .
Imaginary unit i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number, except zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι (see alternative notations) is sometimes used. For the history of the imaginary unit, see Complex number: History. Definition[edit] With i defined this way, it follows directly from algebra that i and −i are both square roots of −1. Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint. Similarly, as with any non-zero real number: i and −i[edit] and are solutions to the matrix equation Proper use[edit] (incorrect). (ambiguous). Similarly: Powers[edit]
AC power The blinking of non-incandescent city lights is shown in this motion-blurred long exposure. The AC nature of the mains power is revealed by the dashed appearance of the traces of moving lights. Real, reactive, and apparent power[edit] In a simple alternating current (AC) circuit consisting of a source and a linear load, both the current and voltage are sinusoidal. If the load is purely resistive, the two quantities reverse their polarity at the same time. If the loads are purely reactive, then the voltage and current are 90 degrees out of phase. Practical loads have resistance, inductance, and capacitance, so both real and reactive power will flow to real loads. Engineers care about apparent power, because even though the current associated with reactive power does no work at the load, it heats the wires, wasting energy. Conventionally, capacitors are considered to generate reactive power and inductors to consume it. The complex power is the vector sum of real and reactive power. .
Euler's formula This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic. Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, Euler's formula is ubiquitous in mathematics, physics, and engineering. History[edit] It was Johann Bernoulli who noted that[3] And since the above equation tells us something about complex logarithms. Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Meanwhile, Roger Cotes, in 1714, discovered that ("ln" is the natural logarithm with base e).[4] Applications in complex number theory[edit] where the real part the imaginary part atan2(y, x) . and that both valid for any complex numbers a and b. . . and
What is reactive power Reactive power is an odd topic in AC (Alternating Current) power systems, and it's usually explained with vector mathematics or phase-shift sinewave graphs. However, a non-math verbal explanation is possible. Note that Reactive power only becomes important when an "electrical load" or a home appliance contains coils or capacitors. If the electrical load behaves purely as a resistor, (such as a heater or incandescent bulb for example,) then the device consumes "real power" only. Reactive power is simply this: when a coil or capacitor is connected to an AC power supply, the coil or capacitor stores electrical energy during one-fourth of an AC cycle. In other words, if your electrical appliance contains inductance or capacitance, then electrical energy will periodically return to the power plant, and it will flow back and forth across the power lines. This undesired "energy sloshing" effect can be eliminated. Why is reactive power so confusing? What is imaginary power? What is real power?
Radius of convergence Definition[edit] For a power series ƒ defined as: where cn is the nth complex coefficient, and z is a complex variable. The radius of convergence r is a nonnegative real number or ∞ such that the series converges if and diverges if In other words, the series converges if z is close enough to the center and diverges if it is too far away. Finding the radius of convergence[edit] Two cases arise. then you take certain limits and find the precise radius of convergence. Theoretical radius[edit] The radius of convergence can be found by applying the root test to the terms of the series. "lim sup" denotes the limit superior. and diverges if the distance exceeds that number; this statement is the Cauchy–Hadamard theorem. The limit involved in the ratio test is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite. This is shown as follows. That is equivalent to Practical estimation of radius[edit] Domb–Sykes plot of the function as a function of index . by
Reactive power The first place you should look is search for 'reactive power'. There are contributors to this site who will insist that reactive power doesn't "flow", or that there's no such thing as reactive "power". But for the purposes of this contribution (and per commonly acepted convention--see we will consider that reactive power does indeed exist and that it flows. And remember: the originator of this thread asked for an explanation in layman's terms, which for me, anyway, does not initially include vector diagrams and trigonometry and mathematical formulae. All of these things are necessary for a complete and proper understanding of the phenomenon, but not for a rudimentary understanding on which to base all of the formulae and vector diagrams, etc., which just serve as mathematical proofs of the principles. In an AC power system, there are resistive loads and reactive loads. Okay, a hot, flat beer is worse.
Avogadro constant Previous definitions of chemical quantity involved Avogadro's number, a historical term closely related to the Avogadro constant but defined differently: Avogadro's number was initially defined by Jean Baptiste Perrin as the number of atoms in one gram-molecule of hydrogen. It was later redefined as the number of atoms in 12 grams of the isotope carbon-12 and still later generalized to relate amounts of a substance to their molecular weight.[4] For instance, to a first approximation, 1 gram of hydrogen, which has a mass number of 1 (atomic number 1), has 6.022×1023 hydrogen atoms. Similarly, 12 grams of carbon 12, with the mass number of 12 (atomic number 6), has the same number of carbon atoms, 6.022×1023. Avogadro's number is a dimensionless quantity and has the numerical value of the Avogadro constant given in base units. The Avogadro constant is fundamental to understanding both the makeup of molecules and their interactions and combinations. History[edit] [dubious ] Measurement[edit]
reactive power Question I also faced this Question!! Rank Answer Posted By Re: wat is the importance of reactive power in power generation.....if it is zero means wat will happen to system........ AC power can be classified in to active power and reactive power. importance of reactive powers are to Maintain and control the voltage balance on the system, Avoid damage to the Transmission System, Generation plant and Other connected parties• The provision of Reactive Power by all Generation Units for voltage support is vital to maintain a secure and stable Transmission System. dear gobi.........if reactive power zero means p.f will increase...ok...at that time voltage will rise r not....this will not affect the transmission?.... if reactive power of generator is zero. required reactive power is drawn from the grid or other generators. [Century Cement Baikunth] Inadequate reactive support ! wat is the limit for that reactive power? Asked @ Answers What are the difference between C-Curve and D-Curve for MCB's. Essar