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 or imaginary numbers - A complete course in algebra The defining property of i The square root of a negative number Powers of i Algebra with complex numbers The real and imaginary components Complex conjugates IN ALGEBRA, we want to be able to say that every polynomial equation has a solution; specifically, this one: x2 + 1 = 0. That implies, x2 = −1. But there is no real number whose square is negative. i2 = −1. That is the defining property of the complex unit i. In other words, i = The complex number i is purely algebraic. Example 1. 3i· 4i = 12i2 = 12(−1) = −12. Example 2. −5i· 6i = −30i2 = 30. We see, then, that the factor i2 changes the sign of a product. Problem 1. To see the answer, pass your mouse over the colored area. The square root of a negative number If a radicand is negative -- , where a > 0, -- then we can simplify it as follows: = i In other words: The square root of −a is equal to i times the square root of a. Problem 2. Powers of i Let us begin with i0, which is 1. And we are back at 1 -- the cycle of powers will repeat. 1, i, −1, or −i
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 .
Complex Numbers A Complex Number A Complex Number is a combination of a Real Number and an Imaginary Number Real Numbers are numbers like: Nearly any number you can think of is a Real Number! Imaginary Numbers when squared give a negative result. Normally this doesn't happen, because: when we square a positive number we get a positive result, and when we square a negative number we also get a positive result (because a negative times a negative gives a positive), for example −2 × −2 = +4 But just imagine such numbers exist, because we will need them. The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of −1 Because when we square i we get −1 i2 = −1 Examples of Imaginary Numbers: And we keep that little "i" there to remind us we need to multiply by √−1 Complex Numbers A Complex Number is a combination of a Real Number and an Imaginary Number: Examples: Can a Number be a Combination of Two Numbers? Can we make up a number from two other numbers? We do it with fractions all the time.
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. .
Complex Number -- from Wolfram MathWorld The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of . is used to denote a complex number, it is sometimes called an "affix." can be written . The set of complex numbers is implemented in the Wolfram Language as Complexes. can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex. Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. Through the Euler formula, a complex number may be written in "phasor" form Here, is known as the complex modulus (or sometimes the complex norm) and is known as the complex argument or phase. , where the dashed circle represents the complex modulus of and the angle represents its complex argument. Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities.
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?
Complex Numbers: Introduction Complex Numbers: Introduction (page 1 of 3) Sections: Introduction, Operations with complexes, The Quadratic Formula Up until now, you've been told that you can't take the square root of a negative number. That's because you had no numbers which were negative after you'd squared them (so you couldn't "go backwards" by taking the square root). Every number was positive after you squared it. So you couldn't very well square-root a negative and expect to come up with anything sensible. Now, however, you can take the square root of a negative number, but it involves using a new number to do it. (But then, when you think about it, aren't all numbers inventions? Anyway, this new number was called "i", standing for "imaginary", because "everybody knew" that i wasn't "real". Then: Now, you may think you can do this: But this doesn't make any sense! Simplify sqrt(–9). (Warning: The step that goes through the third "equals" sign is " ", not " ". Simplify sqrt(–25). Simplify sqrt(–18). Simplify –sqrt(–6).