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107-458: 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

214-401: A b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}} , which is valid for non-negative real numbers a and b , and which was also used in complex number calculations with one of a , b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity 1

321-583: A = 1 a {\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}} , even bedeviled Leonhard Euler . This difficulty eventually led to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra , he introduces these numbers almost at once and then uses them in

428-526: A 0 , ...,  a n , the equation a n z n + ⋯ + a 1 z + a 0 = 0 {\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0} has at least one complex solution z , provided that at least one of the higher coefficients a 1 , ...,  a n is nonzero. This property does not hold for the field of rational numbers Q {\displaystyle \mathbb {Q} } (the polynomial x − 2 does not have

535-454: A complex number is an element of a number system that extends the real numbers with a specific element denoted i , called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i {\displaystyle a+bi} , where a and b are real numbers. Because no real number satisfies

642-407: A mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations , even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has

749-486: A standard basis . This standard basis makes the complex numbers a Cartesian plane , called the complex plane . This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line , which is pictured as the horizontal axis of the complex plane, while real multiples of i {\displaystyle i} are

856-490: A wavelength across the component dimensions. A new design model is needed for such cases called the distributed-element model . Networks designed to this model are called distributed-element circuits . A distributed-element circuit that includes some lumped components is called a semi-lumped design. An example of a semi-lumped circuit is the combline filter . Sources can be classified as independent sources and dependent sources. An ideal independent source maintains

963-748: A complex number z is denoted Re( z ) , R e ( z ) {\displaystyle {\mathcal {Re}}(z)} , or R ( z ) {\displaystyle {\mathfrak {R}}(z)} ; the imaginary part is Im( z ) , I m ( z ) {\displaystyle {\mathcal {Im}}(z)} , or I ( z ) {\displaystyle {\mathfrak {I}}(z)} : for example, Re ⁡ ( 2 + 3 i ) = 2 {\textstyle \operatorname {Re} (2+3i)=2} , Im ⁡ ( 2 + 3 i ) = 3 {\displaystyle \operatorname {Im} (2+3i)=3} . A complex number z can be identified with

1070-837: A complex number and as a real number are equal. Using the conjugate, the reciprocal of a nonzero complex number z = x + y i {\displaystyle z=x+yi} can be computed to be 1 z = z ¯ z z ¯ = z ¯ | z | 2 = x − y i x 2 + y 2 = x x 2 + y 2 − y x 2 + y 2 i . {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.} More generally,

1177-591: A complex number as a point in the complex plane ( above ) was first described by Danish – Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra . Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided

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1284-450: A complex number. (This is in contrast to the roots of a positive real number x , which has a unique positive real n -th root, which is therefore commonly referred to as the n -th root of x .) One refers to this situation by saying that the n th root is a n -valued function of z . The fundamental theorem of algebra , of Carl Friedrich Gauss and Jean le Rond d'Alembert , states that for any complex numbers (called coefficients )

1391-407: A consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of ( 2 + i ) ( 3 + i ) = 5 + 5 i . {\displaystyle (2+i)(3+i)=5+5i.} Because

1498-404: A false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, − 1 {\displaystyle {\sqrt {-1}}} positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. In

1605-436: A fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis. The complex numbers form a rich structure that is simultaneously an algebraically closed field , a commutative algebra over the reals, and a Euclidean vector space of dimension two. A complex number

1712-421: A full wavelength λ {\displaystyle \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 π  radians representing one complete cycle. If

1819-532: 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

1926-3028: A linear system stimulated by a sinusoid. The sum of multiple phasors produces another phasor. That is because the sum of sinusoids with the same frequency is also a sinusoid with that frequency: A 1 cos ⁡ ( ω t + θ 1 ) + A 2 cos ⁡ ( ω t + θ 2 ) = Re ⁡ ( A 1 e i θ 1 e i ω t ) + Re ⁡ ( A 2 e i θ 2 e i ω t ) = Re ⁡ ( A 1 e i θ 1 e i ω t + A 2 e i θ 2 e i ω t ) = Re ⁡ ( ( A 1 e i θ 1 + A 2 e i θ 2 ) e i ω t ) = Re ⁡ ( ( A 3 e i θ 3 ) e i ω t ) = A 3 cos ⁡ ( ω t + θ 3 ) , {\displaystyle {\begin{aligned}&A_{1}\cos(\omega t+\theta _{1})+A_{2}\cos(\omega t+\theta _{2})\\[3pt]={}&\operatorname {Re} \left(A_{1}e^{i\theta _{1}}e^{i\omega t}\right)+\operatorname {Re} \left(A_{2}e^{i\theta _{2}}e^{i\omega t}\right)\\[3pt]={}&\operatorname {Re} \left(A_{1}e^{i\theta _{1}}e^{i\omega t}+A_{2}e^{i\theta _{2}}e^{i\omega t}\right)\\[3pt]={}&\operatorname {Re} \left(\left(A_{1}e^{i\theta _{1}}+A_{2}e^{i\theta _{2}}\right)e^{i\omega t}\right)\\[3pt]={}&\operatorname {Re} \left(\left(A_{3}e^{i\theta _{3}}\right)e^{i\omega t}\right)\\[3pt]={}&A_{3}\cos(\omega t+\theta _{3}),\end{aligned}}} where: A 3 2 = ( A 1 cos ⁡ θ 1 + A 2 cos ⁡ θ 2 ) 2 + ( A 1 sin ⁡ θ 1 + A 2 sin ⁡ θ 2 ) 2 , {\displaystyle 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 θ 3 ∈ [ − π 2 , 3 π 2 ] {\textstyle \theta _{3}\in \left[-{\frac {\pi }{2}},{\frac {3\pi }{2}}\right]} , then θ 3 {\displaystyle \theta _{3}} is: or, via

2033-433: A natural way throughout. In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by

2140-422: A powerful tool to understand analog modulations such as amplitude modulation (and its variants) and frequency modulation . x ( t ) = Re ⁡ ( A e i θ ⋅ e i 2 π f 0 t ) , {\displaystyle x(t)=\operatorname {Re} \left(Ae^{i\theta }\cdot e^{i2\pi f_{0}t}\right),} where

2247-393: A quelquefois aucune quantité qui corresponde à celle qu'on imagine. ] A further source of confusion was that the equation − 1 2 = − 1 − 1 = − 1 {\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1} seemed to be capriciously inconsistent with the algebraic identity

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2354-453: A rational root, because √2 is not a rational number) nor the real numbers R {\displaystyle \mathbb {R} } (the polynomial x + 4 does not have a real root, because the square of x is positive for any real number x ). Because of this fact, C {\displaystyle \mathbb {C} } is called an algebraically closed field . It is a cornerstone of various applications of complex numbers, as

2461-487: A rigorous proof of the fundamental theorem of algebra . Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology: If one formerly contemplated this subject from

2568-476: 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. Multiplication of the phasor A e i θ e i ω t {\displaystyle Ae^{i\theta }e^{i\omega t}} by a complex constant, B e i ϕ {\displaystyle Be^{i\phi }} , produces another phasor. That means its only effect

2675-477: 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. 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 components . This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents. In

2782-1042: A set of simultaneous equations that can be solved either algebraically or numerically. The laws can generally be extended to networks containing reactances . They cannot be used in networks that contain nonlinear or time-varying components. [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] To design any electrical circuit, either analog or digital , electrical engineers need to be able to predict

2889-560: A solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions − 1 + 3 i {\displaystyle -1+3i} and − 1 − 3 i {\displaystyle -1-3i} . Addition, subtraction and multiplication of complex numbers can be naturally defined by using

2996-409: A special case the fundamental formula This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number. With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property ,

3103-429: A topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians ( Niccolò Fontana Tartaglia and Gerolamo Cardano ). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it

3210-401: Is "frozen" at some point in time, ( t ) 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

3317-448: Is a network consisting of a closed loop, giving a return path for the current. Thus all circuits are networks, but not all networks are circuits (although networks without a closed loop are often imprecisely referred to as "circuits"). Linear electrical networks, a special type consisting only of sources (voltage or current), linear lumped elements (resistors, capacitors, inductors), and linear distributed elements (transmission lines), have

Phasor - Misplaced Pages Continue

3424-458: Is also denoted by some authors by z ∗ {\displaystyle z^{*}} . Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: z ¯ ¯ = z . {\displaystyle {\overline {\overline {z}}}=z.} A complex number is real if and only if it equals its own conjugate. The unary operation of taking

3531-2987: Is also readily seen that: d d t Re ⁡ ( V c ⋅ e i ω t ) = Re ⁡ ( d d t ( V c ⋅ e i ω t ) ) = Re ⁡ ( i ω V c ⋅ e i ω t ) d d t Im ⁡ ( V c ⋅ e i ω t ) = Im ⁡ ( d d t ( V c ⋅ e i ω t ) ) = Im ⁡ ( i ω V c ⋅ e i ω t ) . {\displaystyle {\begin{aligned}{\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{aligned}}} Substituting these into Eq.1 and Eq.2 , multiplying Eq.2 by i , {\displaystyle i,} and adding both equations gives: i ω V c ⋅ e i ω t + 1 R C V c ⋅ e i ω t = 1 R C V s ⋅ e i ω t ( i ω V c + 1 R C V c ) ⋅ e i ω t = ( 1 R C V s ) ⋅ e i ω t i ω V c + 1 R C V c = 1 R C V s . {\displaystyle {\begin{aligned}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{aligned}}} Solving for

3638-425: Is an expression of the form a + bi , where a and b are real numbers , and i is an abstract symbol, the so-called imaginary unit , whose meaning will be explained further below. For example, 2 + 3 i is a complex number. For a complex number a + bi , the real number a is called its real part , and the real number b (not the complex number bi ) is its imaginary part . The real part of

3745-403: Is assumed to be located ("lumped") at one place. This design philosophy is called the lumped-element model and networks so designed are called lumped-element circuits . This is the conventional approach to circuit design. At high enough frequencies, or for long enough circuits (such as power transmission lines ), the lumped assumption no longer holds because there is a significant fraction of

3852-461: Is computed as follows: For example, ( 3 + 2 i ) ( 4 − i ) = 3 ⋅ 4 − ( 2 ⋅ ( − 1 ) ) + ( 3 ⋅ ( − 1 ) + 2 ⋅ 4 ) i = 14 + 5 i . {\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.} In particular, this includes as

3959-483: Is defined only up to adding integer multiples of 2 π {\displaystyle 2\pi } , since a rotation by 2 π {\displaystyle 2\pi } (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval ( − π , π ] {\displaystyle (-\pi ,\pi ]} , which

4066-590: Is denoted by C {\displaystyle \mathbb {C} } ( blackboard bold ) or C (upright bold). In some disciplines such as electromagnetism and electrical engineering , j is used instead of i , as i frequently represents electric current , and complex numbers are written as a + bj or a + jb . Two complex numbers a = x + y i {\displaystyle a=x+yi} and b = u + v i {\displaystyle b=u+vi} are added by separately adding their real and imaginary parts. That

4173-449: Is detailed further below. There are various proofs of this theorem, by either analytic methods such as Liouville's theorem , or topological ones such as the winding number , or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. The solution in radicals (without trigonometric functions ) of a general cubic equation , when all three of its roots are real numbers, contains

4280-410: Is independent of time. In particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as

4387-526: Is linear if its signals obey the principle of superposition ; otherwise it is non-linear. Passive networks are generally taken to be linear, but there are exceptions. For instance, an inductor with an iron core can be driven into saturation if driven with a large enough current. In this region, the behaviour of the inductor is very non-linear. Discrete passive components (resistors, capacitors and inductors) are called lumped elements because all of their, respectively, resistance, capacitance and inductance

Phasor - Misplaced Pages Continue

4494-399: Is referred to as the principal value . The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function. For any complex number z , with absolute value r = | z | {\displaystyle r=|z|} and argument φ {\displaystyle \varphi } , the equation holds. This identity

4601-590: Is referred to as the polar form of z . It is sometimes abbreviated as z = r c i s ⁡ φ {\textstyle z=r\operatorname {\mathrm {cis} } \varphi } . In electronics , one represents a phasor with amplitude r and phase φ in angle notation : z = r ∠ φ . {\displaystyle z=r\angle \varphi .} If two complex numbers are given in polar form, i.e., z 1 = r 1 (cos  φ 1 + i  sin  φ 1 ) and z 2 = r 2 (cos  φ 2 + i  sin  φ 2 ) ,

4708-404: Is reinserted prior to the real part of the result. The function A e i ( ω t + θ ) {\displaystyle Ae^{i(\omega t+\theta )}} is an analytic representation of A cos ⁡ ( ω t + θ ) . {\displaystyle A\cos(\omega t+\theta ).} Figure 2 depicts it as

4815-440: Is sometimes called " rationalization " of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator. The argument of z (sometimes called the "phase" φ ) is the angle of the radius Oz with the positive real axis, and is written as arg z , expressed in radians in this article. The angle

4922-661: Is that A 3 and θ 3 do not depend on ω or t , which is what makes phasor notation possible. The time and frequency dependence can be suppressed and re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In angle notation , the operation shown above is written: A 1 ∠ θ 1 + A 2 ∠ θ 2 = A 3 ∠ θ 3 . {\displaystyle A_{1}\angle \theta _{1}+A_{2}\angle \theta _{2}=A_{3}\angle \theta _{3}.} Another way to view addition

5029-730: Is that two vectors with coordinates [ A 1 cos( ωt + θ 1 ), A 1 sin( ωt + θ 1 )] and [ A 2 cos( ωt + θ 2 ), A 2 sin( ωt + θ 2 )] are added vectorially to produce a resultant vector with coordinates [ A 3 cos( ωt + θ 3 ), A 3 sin( ωt + θ 3 )] (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

5136-430: Is that: cos ⁡ ( ω t ) + cos ⁡ ( ω t + 2 π 3 ) + cos ⁡ ( ω t − 2 π 3 ) = 0. {\displaystyle \cos(\omega t)+\cos \left(\omega t+{\frac {2\pi }{3}}\right)+\cos \left(\omega t-{\frac {2\pi }{3}}\right)=0.} In

5243-524: Is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that | z | = 1 {\displaystyle |z|=1} . If z = x = x + 0 i {\displaystyle z=x=x+0i} is a real number, then | z | = | x | {\displaystyle |z|=|x|} : its absolute value as

5350-417: Is the ratio of two phasors, which is not a phasor, because it does not correspond to a sinusoidally varying function. With phasors, the techniques for solving DC circuits can be applied to solve linear AC circuits. 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 forth. We can also define

5457-616: Is the unknown quantity to be determined. In the phasor shorthand notation, the differential equation reduces to: i ω V c + 1 R C V c = 1 R C V s . {\displaystyle i\omega V_{\text{c}}+{\frac {1}{RC}}V_{\text{c}}={\frac {1}{RC}}V_{\text{s}}.} Since this must hold for all t {\displaystyle t} , specifically: t − π 2 ω , {\textstyle t-{\frac {\pi }{2\omega }},} it follows that: It

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5564-444: Is the usual (positive) n th root of the positive real number r .) Because sine and cosine are periodic, other integer values of k do not give other values. For any z ≠ 0 {\displaystyle z\neq 0} , there are, in particular n distinct complex n -th roots. For example, there are 4 fourth roots of 1, namely In general there is no natural way of distinguishing one particular complex n th root of

5671-638: Is the vector ( 0 , 1 ) {\displaystyle (0,\,1)} or the number e i π / 2 = i . {\displaystyle 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 ∠ θ 1 {\displaystyle v_{1}=A_{1}\angle \theta _{1}} and v 2 = A 2 ∠ θ 2 {\displaystyle v_{2}=A_{2}\angle \theta _{2}} ,

5778-1045: Is to change the amplitude and phase of the underlying sinusoid: Re ⁡ ( ( A e i θ ⋅ B e i ϕ ) ⋅ e i ω t ) = Re ⁡ ( ( A B e i ( θ + ϕ ) ) ⋅ e i ω t ) = A B cos ⁡ ( ω t + ( θ + ϕ ) ) . {\displaystyle {\begin{aligned}&\operatorname {Re} \left(\left(Ae^{i\theta }\cdot Be^{i\phi }\right)\cdot e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\left(ABe^{i(\theta +\phi )}\right)\cdot e^{i\omega t}\right)\\={}&AB\cos(\omega t+(\theta +\phi )).\end{aligned}}} In electronics, B e i ϕ {\displaystyle Be^{i\phi }} would represent an impedance , which

5885-599: Is to say: a + b = ( x + y i ) + ( u + v i ) = ( x + u ) + ( y + v ) i . {\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.} Similarly, subtraction can be performed as a − b = ( x + y i ) − ( u + v i ) = ( x − u ) + ( y − v ) i . {\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.} The addition can be geometrically visualized as follows:

5992-759: Is unaffected. When we solve a linear differential equation with phasor arithmetic, we are merely factoring e i ω t {\displaystyle 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 : d v C ( t ) d t + 1 R C v C ( t ) = 1 R C v S ( t ) . {\displaystyle {\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

6099-1072: Is written A ∠ θ . {\displaystyle A\angle \theta .} 1 ∠ θ {\displaystyle 1\angle \theta } can represent either the vector ( cos ⁡ θ , sin ⁡ θ ) {\displaystyle (\cos \theta ,\,\sin \theta )} or the complex number cos ⁡ θ + i sin ⁡ θ = e i θ {\displaystyle \cos \theta +i\sin \theta =e^{i\theta }} , according to Euler's formula with i 2 = − 1 {\displaystyle 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 ∠ 90 {\displaystyle 1\angle 90} would be assumed to be 1 ∠ 90 ∘ , {\displaystyle 1\angle 90^{\circ },} which

6206-536: The PLECS interface to Simulink uses piecewise-linear approximation of the equations governing the elements of a circuit. The circuit is treated as a completely linear network of ideal diodes . Every time a diode switches from on to off or vice versa, the configuration of the linear network changes. Adding more detail to the approximation of equations increases the accuracy of the simulation, but also increases its running time. Complex number In mathematics ,

6313-899: The arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π . The n -th power of a complex number can be computed using de Moivre's formula , which is obtained by repeatedly applying the above formula for the product: z n = z ⋅ ⋯ ⋅ z ⏟ n  factors = ( r ( cos ⁡ φ + i sin ⁡ φ ) ) n = r n ( cos ⁡ n φ + i sin ⁡ n φ ) . {\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).} For example,

6420-409: The commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field , the same way as the rational or real numbers do. The complex conjugate of the complex number z = x + yi is defined as z ¯ = x − y i . {\displaystyle {\overline {z}}=x-yi.} It

6527-871: The complex power S = P + jQ and the apparent power which is the magnitude of S . The power law for an AC circuit expressed in phasors is then S = VI (where I is the complex conjugate of I , and the magnitudes of the voltage and current phasors V and of I are the RMS values of the voltage and current, respectively). Given this we can apply the techniques of analysis of resistive circuits with phasors to analyze single frequency linear AC circuits containing resistors, capacitors, and inductors . 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 (using Fourier series ) with magnitude and phase then analyzing each frequency separately, as allowed by

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6634-436: The fundamental theorem of algebra , which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field , where any polynomial equation has a root . Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by

6741-879: 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 ∘ − Δ θ ) = A 1 2 + A 2 2 + 2 A 1 A 2 cos ⁡ ( Δ θ ) , {\displaystyle A_{3}^{2}=A_{1}^{2}+A_{2}^{2}-2A_{1}A_{2}\cos(180^{\circ }-\Delta \theta )=A_{1}^{2}+A_{2}^{2}+2A_{1}A_{2}\cos(\Delta \theta ),} where Δ θ = θ 1 − θ 2 . {\displaystyle \Delta \theta =\theta _{1}-\theta _{2}.} A key point

6848-422: The ordered pair of real numbers ( ℜ ( z ) , ℑ ( z ) ) {\displaystyle (\Re (z),\Im (z))} , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram , . The horizontal axis is generally used to display the real part, with increasing values to

6955-451: The superposition theorem . This solution method applies only to inputs that are sinusoidal and for solutions that are in steady state, i.e., after all transients have died out. The concept is frequently involved in representing an electrical impedance . In this case, the phase angle is the phase difference between the voltage applied to the impedance and the current driven through it. In analysis of three phase AC power systems, usually

7062-548: 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. 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 {\displaystyle A} and angle θ {\displaystyle \theta }

7169-771: The 1st century AD , where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term 81 − 144 {\displaystyle {\sqrt {81-144}}} in his calculations, which today would simplify to − 63 = 3 i 7 {\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}} . Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive 144 − 81 = 3 7 . {\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.} The impetus to study complex numbers as

7276-546: The Italian mathematician Rafael Bombelli . A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton , who extended this abstraction to the theory of quaternions . The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in

7383-450: The above equation, i was called an imaginary number by René Descartes . For the complex number a + b i {\displaystyle a+bi} , a is called the real part , and b is called the imaginary part . The set of complex numbers is denoted by either of the symbols C {\displaystyle \mathbb {C} } or C . Despite the historical nomenclature, "imaginary" complex numbers have

7490-489: The circuit conform to the voltage/current equations governing that element. Once the steady state solution is found, the operating points of each element in the circuit are known. For a small signal analysis, every non-linear element can be linearized around its operation point to obtain the small-signal estimate of the voltages and currents. This is an application of Ohm's Law. The resulting linear circuit matrix can be solved with Gaussian elimination . Software such as

7597-587: The complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. For any complex number z = x + yi , the product is a non-negative real number. This allows to define the absolute value (or modulus or magnitude ) of z to be the square root | z | = x 2 + y 2 . {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.} By Pythagoras' theorem , | z | {\displaystyle |z|}

7704-483: 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 θ , {\displaystyle Ae^{i\theta },} and the common factor e i ω t {\displaystyle e^{i\omega t}}

7811-460: 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. The technique of synchrophasors uses digital instruments to measure the phasors representing transmission system voltages at widespread points in a transmission network. Differences among the phasors indicate power flow and system stability. The rotating frame picture using phasor can be

7918-856: The division of an arbitrary complex number w = u + v i {\displaystyle w=u+vi} by a non-zero complex number z = x + y i {\displaystyle z=x+yi} equals w z = w z ¯ | z | 2 = ( u + v i ) ( x − i y ) x 2 + y 2 = u x + v y x 2 + y 2 + v x − u y x 2 + y 2 i . {\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.} This process

8025-423: 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,

8132-413: The factor multiplying V s {\displaystyle V_{\text{s}}} represents differences of the amplitude and phase of v C ( t ) {\displaystyle v_{\text{C}}(t)} relative to V P {\displaystyle V_{\text{P}}} and θ . {\displaystyle \theta .} In polar coordinate form,

8239-915: The first few powers of the imaginary unit i are i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , … {\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots } . The n n th roots of a complex number z are given by z 1 / n = r n ( cos ⁡ ( φ + 2 k π n ) + i sin ⁡ ( φ + 2 k π n ) ) {\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)} for 0 ≤ k ≤ n − 1 . (Here r n {\displaystyle {\sqrt[{n}]{r}}}

8346-1281: The first term of the last expression is: 1 − i ω R C 1 + ( ω R C ) 2 = 1 1 + ( ω R C ) 2 ⋅ e − i ϕ ( ω ) , {\displaystyle {\frac {1-i\omega RC}{1+(\omega RC)^{2}}}={\frac {1}{\sqrt {1+(\omega RC)^{2}}}}\cdot e^{-i\phi (\omega )},} where ϕ ( ω ) = arctan ⁡ ( ω R C ) {\displaystyle \phi (\omega )=\arctan(\omega RC)} . Therefore: v C ( t ) = Re ⁡ ( V c ⋅ e i ω t ) = 1 1 + ( ω R C ) 2 ⋅ V P cos ⁡ ( ω t + θ − ϕ ( ω ) ) . {\displaystyle 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 )).} A quantity called complex impedance

8453-799: The following de Moivre's formula : ( cos ⁡ θ + i sin ⁡ θ ) n = cos ⁡ n θ + i sin ⁡ n θ . {\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .} In 1748, Euler went further and obtained Euler's formula of complex analysis : e i θ = cos ⁡ θ + i sin ⁡ θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta } by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. The idea of

8560-403: The following is true: A real-valued sinusoid with constant amplitude, frequency, and phase has the form: where only parameter t {\displaystyle t} is time-variant. The inclusion of an imaginary component : gives it, in accordance with Euler's formula , the factoring property described in the lead paragraph: whose real part is the original sinusoid. The benefit of

8667-400: 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° ( 2 π ⁄ 3  radians), or one third of a wavelength λ ⁄ 3 . 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

8774-500: 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

8881-404: 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, t = 0 . When the vector is horizontal the tip of the vector represents the angles at 0°, 180°, and at 360°. Likewise, when

8988-493: The network indefinitely. A passive network does not contain an active source. An active network contains one or more sources of electromotive force . Practical examples of such sources include a battery or a generator . Active elements can inject power to the circuit, provide power gain, and control the current flow within the circuit. Passive networks do not contain any sources of electromotive force. They consist of passive elements like resistors and capacitors. A network

9095-521: The phasor capacitor voltage gives: V c = 1 1 + i ω R C ⋅ V s = 1 − i ω R C 1 + ( ω R C ) 2 ⋅ V P e i θ . {\displaystyle V_{\text{c}}={\frac {1}{1+i\omega RC}}\cdot V_{\text{s}}={\frac {1-i\omega RC}{1+(\omega RC)^{2}}}\cdot V_{\text{P}}e^{i\theta }.} As we have seen,

9202-454: 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

9309-1632: The previous Phase Difference tutorial. The time derivative or integral of a phasor produces another phasor. For example: Re ⁡ ( d d t ( A e i θ ⋅ e i ω t ) ) = Re ⁡ ( A e i θ ⋅ i ω e i ω t ) = Re ⁡ ( A e i θ ⋅ e i π / 2 ω e i ω t ) = Re ⁡ ( ω A e i ( θ + π / 2 ) ⋅ e i ω t ) = ω A ⋅ cos ⁡ ( ω t + θ + π 2 ) . {\displaystyle {\begin{aligned}&\operatorname {Re} \left({\frac {\mathrm {d} }{\mathrm {d} t}}{\mathord {\left(Ae^{i\theta }\cdot e^{i\omega t}\right)}}\right)\\={}&\operatorname {Re} \left(Ae^{i\theta }\cdot i\omega e^{i\omega t}\right)\\={}&\operatorname {Re} \left(Ae^{i\theta }\cdot e^{i\pi /2}\omega e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\omega Ae^{i(\theta +\pi /2)}\cdot e^{i\omega t}\right)\\={}&\omega A\cdot \cos \left(\omega t+\theta +{\frac {\pi }{2}}\right).\end{aligned}}} Therefore, in phasor representation,

9416-1013: The product and division can be computed as z 1 z 2 = r 1 r 2 ( cos ⁡ ( φ 1 + φ 2 ) + i sin ⁡ ( φ 1 + φ 2 ) ) . {\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).} z 1 z 2 = r 1 r 2 ( cos ⁡ ( φ 1 − φ 2 ) + i sin ⁡ ( φ 1 − φ 2 ) ) , if  z 2 ≠ 0. {\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.} (These are

9523-455: The property that signals are linearly superimposable . They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms , to determine DC response , AC response , and transient response . A resistive network is a network containing only resistors and ideal current and voltage sources. Analysis of resistive networks is less complicated than analysis of networks containing capacitors and inductors. If

9630-568: The real and imaginary part of 5 + 5 i are equal, the argument of that number is 45 degrees, or π /4 (in radian ). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan (1/3) and arctan(1/2), respectively. Thus, the formula π 4 = arctan ⁡ ( 1 2 ) + arctan ⁡ ( 1 3 ) {\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)} holds. As

9737-472: The right, and the imaginary part marks the vertical axis, with increasing values upwards. A real number a can be regarded as a complex number a + 0 i , whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi , whose real part is zero. As with polynomials, it is common to write a + 0 i = a , 0 + bi = bi , and a + (− b ) i = a − bi ; for example, 3 + (−4) i = 3 − 4 i . The set of all complex numbers

9844-461: The rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative , commutative , and distributive laws . Every nonzero complex number has a multiplicative inverse . This makes the complex numbers a field with the real numbers as a subfield. The complex numbers also form a real vector space of dimension two , with { 1 , i } {\displaystyle \{1,i\}} as

9951-520: The rules for complex arithmetic, trying to resolve these issues. The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature: ... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine. [ ... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y

10058-507: The same voltage or current regardless of the other elements present in the circuit. Its value is either constant (DC) or sinusoidal (AC). The strength of voltage or current is not changed by any variation in the connected network. Dependent sources depend upon a particular element of the circuit for delivering the power or voltage or current depending upon the type of source it is. A number of electrical laws apply to all linear resistive networks. These include: Applying these laws results in

10165-530: The sources are constant ( DC ) sources, the result is a DC network. The effective resistance and current distribution properties of arbitrary resistor networks can be modeled in terms of their graph measures and geometrical properties. A network that contains active electronic components is known as an electronic circuit . Such networks are generally nonlinear and require more complex design and analysis tools. An active network contains at least one voltage source or current source that can supply energy to

10272-569: The square roots of negative numbers , a situation that cannot be rectified by factoring aided by the rational root test , if the cubic is irreducible ; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna , though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture." This

10379-447: The sum of two complex numbers a and b , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O , and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A , B , respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent . The product of two complex numbers

10486-454: The term in brackets is viewed as a rotating vector in the complex plane. Electrical network An electrical network is an interconnection of electrical components (e.g., batteries , resistors , inductors , capacitors , switches , transistors ) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources , current sources , resistances , inductances , capacitances ). An electrical circuit

10593-638: The time derivative of a sinusoid becomes just multiplication by the constant i ω = e i π / 2 ⋅ ω {\textstyle i\omega =e^{i\pi /2}\cdot \omega } . Similarly, integrating a phasor corresponds to multiplication by 1 i ω = e − i π / 2 ω . {\textstyle {\frac {1}{i\omega }}={\frac {e^{-i\pi /2}}{\omega }}.} The time-dependent factor, e i ω t , {\displaystyle e^{i\omega t},}

10700-428: The time, cost and risk of error involved in building circuit prototypes. More complex circuits can be analyzed numerically with software such as SPICE or GNUCAP , or symbolically using software such as SapWin . When faced with a new circuit, the software first tries to find a steady state solution , that is, one where all nodes conform to Kirchhoff's current law and the voltages across and through each element of

10807-414: The tip of the vector is vertical it represents the positive peak value, ( + A max ) at 90° or π ⁄ 2 and the negative peak value, ( − A max ) at 270° or 3 π ⁄ 2 . 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

10914-432: The vertical axis. A complex number can also be defined by its geometric polar coordinates : the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle . Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by

11021-1082: The voltage source in this circuit is sinusoidal: v S ( t ) = V P ⋅ cos ⁡ ( ω t + θ ) , {\displaystyle v_{\text{S}}(t)=V_{\text{P}}\cdot \cos(\omega t+\theta ),} we may substitute v S ( t ) = Re ⁡ ( V s ⋅ e i ω t ) . {\displaystyle v_{\text{S}}(t)=\operatorname {Re} \left(V_{\text{s}}\cdot e^{i\omega t}\right).} v C ( t ) = Re ⁡ ( V c ⋅ e i ω t ) , {\displaystyle v_{\text{C}}(t)=\operatorname {Re} \left(V_{\text{c}}\cdot e^{i\omega t}\right),} where phasor V s = V P e i θ , {\displaystyle V_{\text{s}}=V_{\text{P}}e^{i\theta },} and phasor V c {\displaystyle V_{\text{c}}}

11128-462: The voltages and currents at all places within the circuit. Simple linear circuits can be analyzed by hand using complex number theory . In more complex cases the circuit may be analyzed with specialized computer programs or estimation techniques such as the piecewise-linear model. Circuit simulation software, such as HSPICE (an analog circuit simulator), and languages such as VHDL-AMS and verilog-AMS allow engineers to design circuits without

11235-413: The waveform above that the waveform starts at time t = 0 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, Φ of the waveform. Consider the diagram below from

11342-427: Was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro , in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless. Work on the problem of general polynomials ultimately led to

11449-415: Was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed

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