why we need complex number

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Bhuvanesh123

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i am so much confused with complex exponential.so i need to break that down into basic and rethink about it

1) why do we need complex number why not we stay with real number alone and what are the impossible become after possible after discovery of complex number.Thank you in advance
 

The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary
Work on the problem of general polynomials ultimately led to 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(reference from wikipedia)
Apart from mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics...We cannt think of solving AC electric cicruits(particularly RLC ciruits) with out the help of complex number..In power system we cant think of solving powerflow between different buses without it...Anything with AC supply needs complex number for easy analysis and understanding in Elctrical engineering
 
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For solving the RLC circuits using the differential circuits is hard.But complex number can solve it easily as well with a simpler approach... so it is required...

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For solving the RLC circuits And other calculation related to electronics using the differential circuits is hard.But complex number can solve it easily as well with a simpler approach... so it is required...
 
The difference between complex numbers and ordinary numbers is that the j operator gives the DIRECTION of the number. As an analogy, suppose you are traveling at 5 MPH, all you can say is that after one hour you have traveled 5 miles. If you say you have traveled at 4 MPH North and 3 MPH East, then at the end of an hour you have still traveled 5 miles but now you can calculate the direction.
In circuits containing Inductance and capacitance, the voltages and the currents are not peaking at the same time, some times when the voltage is increasing the current may be decreasing. So using, say the voltage as a reference, the direction of the current and its amplitude can be written in complex numbers (A + j B). The other way would be to give its absolute amplitude and its phase ( Z @SIN theta).
Frank
 

Hi,

when using a simple RC circuit you have phase shift "phi" as angle. Causing a real part "cos(phi)" and an imaginary part "sin(phi)".

You can see "sin(phi)" = the imaginary part = "j" of the complex number.


****

Maybe this is a bit "semi-scientific" explanation, but at least it helps me to get a picture in my head. ...and to see the complex numbers a bit relaxed.


****

Klaus

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Hi,

when using a simple RC circuit you have phase shift "phi" as angle. Causing a real part "cos(phi)" and an imaginary part "sin(phi)".

You can see "sin(phi)" = the imaginary part = "j" of the complex number.


****

Maybe this is a bit "semi-scientific" explanation, but at least it helps me to get a picture in my head. ...and to see the complex numbers a bit relaxed.


****

Klaus
 

Bhuvanesh123 said:
why do we need complex number why not we stay with real number
Click to expand...

Complex numbers allow performing a lot of math operations in a very simple way. On engineering it is quite usual employ the use of such artifacts as for instance transform the current domain into another just for execute there some calculations, and after, return back to original domain ( real numbers, at this case ).
 

Bhuvanesh123 said:
The voltage in a circuit is 45 + j10 volts

what is 45 represent 10 represents ?
Click to expand...

The complex expression is a mathematical representation of a sinusoidal voltage with phase shift. Generally in circuits that contain capacitors and inductors currents and voltages in different points can be shifted in phase. The use of complex numbers simplifies calculations.
The voltage/current formula for capacitors and inductors is a differential equation. If we want to calculate such circuit we need to solve differential equations which are very difficult.
The method of complex numbers makes this very easy, but is limited to the case where all components are linear and work with sinusoidal waveforms at fixed frequency without initial conditions. The complex numbers method is derived from the Fourier series method for solving differential equations where only 1 frequency is taken into account.
 

The voltage in a circuit is 45 + j10 volts


The impedance in one part of a series circuit is 45 + j10

The current in a circuit is 45 + j10 amps

what is the 10 represent in each case?
 

The from 45 + j10 is used to make easier algebraic calculations. To see the the amplitude and the phase shift it must be transformed to the Euler form
45 + j10 = 46.10 ∠12.53​°, thus the signal has amplitude 46.1 and phase shift 12.53°.
The complex impedance represents the relation between voltage and current, similar to the Ohms law, additionally it allows to calculate also the phase shift between the voltage and current.
 
Bhuvanesh123 said:
The voltage in a circuit is 45 + j10 volts


The impedance in one part of a series circuit is 45 + j10

The current in a circuit is 45 + j10 amps

what is the 10 represent in each case?
Click to expand...

1)This is an AC supply with 46.097(12.53degree)...or u can say that 10 part is something that deals with reactive power in the electric circuit(I hope u r clear with active and reactive power)

2)When an electric circuit contains an energy storing element like inductor or capacitor,we will get a complex term in the impedance..In your case (i.e. 10) is an inductor..

3)Similar to the first case it is that part of the current that deals with reactive power...
 

it is quite usual employ the use of such artifacts as for instance transform the current domain into another just for execute there some calculations, and after, return back to original domain ( real numbers, at this case ).
Click to expand...

could you give example for this
 

When it is needed for example to calculate the lag angle resulting from dividing a voltage of sinusoidal waveform by a complex load (ie, composed of resistive and reactive components), after calculating the total impedance of this load, it is sufficient to perform a simple division operation of the amplitude of the sinusoid amplitude by the amplitude of the load, and a subtraction of both phases of the load and voltage.
 

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