Hello,

I am going through concept of impedance where it is represented as a complex number.
Same representation is done for AC voltage and current.

I understand that AC signals would need both magnitude and phase representation but I don't
understand why it should be complex representation. For example, why not in a simple 2 dimensional plane
with x,y co-ordinates? In fact, complex representation has sqrt(-1) which has no physical significance ...

Thanks,

 

I understand that AC signals would need both magnitude and phase representation but I don't understand why it should be complex representation. For example, why not in a simple 2 dimensional plane with x,y co-ordinates? In fact, complex representation has sqrt(-1) which has no physical significance ...

1) Using a graphical notation is often inconvenient and uses up a lot of paper when you can say the same thing with a few numbers.
2) The imaginary notation represents a real relationship just as much as 240 volts represents a real quantity. You need to become able to interchange the forms at will. It is merely learning the language of how to describe a vector with numbers, and vise-versa.

 

The complex representation is used because it makes the math a LOT easier, though it may not seem that way right now.

Because of the Euler relationship you have:

\(
Ae^{j\omega t} \; = \; A \cos(\omega t) \; + \; j A \sin(\omega t)
\)

This allows you to use all of the very powerful mathematical relationships involving exponentials, particularly when it comes to derivatives and integrals, which are part of the constitutive relations for capacitors and inductors.

 

Complex numbers can be (and are) represented on X,Y coordinates with the X-axis representing the real part and the Y-axis representing the imaginary part.

Historically, the word imaginary was perhaps an unfortunate selection to use for the name for the right half of a complex number since it can represent real quantities just as much as the left hand part.
Reactance is just as real as resistance.

 

Thanks all. I do understand that some physical phenomenon have two parameters.
But my question was the choice of i/j/sqrt(-1) as the prefix for y axis value.
Does using sqrt(-1) help in any special way that one would not get without ...

 

Thanks all. I do understand that some physical phenomenon have two parameters.
But my question was the choice of i/j/sqrt(-1) as the prefix for y axis value.
Does using sqrt(-1) help in any special way that one would not get without ...

Again, the use of the complex plane allows you to use complex exponentials to describe your signals and systems, which makes the math a lot easier.

 

The easiest way to see the importance is to understand the connection between 2nd order differential equations and 2nd order polynomials. Second order differential equations occur in electronic circuits and mechanical systems all the time. For all 2nd order polynomials with real coefficients there are two real roots, or a single repeated root, or a complex conjugate pair. Each of these three situations describes different dynamics. Without the imaginary unit in these cases, there would be no way to describe the dynamics of the complex conjugate pair, because the algebra of real numbers just won't allow the third case to exist.

 

Without the imaginary unit in these cases, there would be no way to describe the dynamics of the complex conjugate pair, because the algebra of real numbers just won't allow the third case to exist.

I wouldn't go that far. We can certainly solve second order differential equations for systems that are underdamped without requiring complex numbers. But the math is more tedious and convoluted because we have robbed ourselves of the simplicity and elegance afforded by complex numbers.

 

Without the imaginary unit in these cases, there would be no way to describe the dynamics of the complex conjugate pair, because the algebra of real numbers just won't allow the third case to exist.

I wouldn't go that far. We can certainly solve second order differential equations for systems that are underdamped without requiring complex numbers. But the math is more tedious and convoluted because we have robbed ourselves of the simplicity and elegance afforded by complex numbers.

 

.........................
Does using sqrt(-1) help in any special way that one would not get without ...

The use of i or j (sqrt(-1) allows you to mathematically keep the resistance part of the impedance separate from the reactance part so that you can readily generate and solve the complex impedance, voltage, and current equations for circuits that contain resistive and reactive elements.
Without the i to keep them separate the math would become much more complex and messy.
You're welcome to try and solve some circuit response problems for complex circuits with the i if you want an interesting exercise.

 

Life is complex. Get used to it.

 

Thank you all. Though I don't get this fully now, I hope to understand the concept sooner than later ...

Life is complex. Get used to it.

Dear Brownout, on a lighter note - I am at a level where complex appears too complex, while for most of the esteemed members here, even complex looks quite simple ...

 

Dear Brownout, on a lighter note - I am at a level where complex appears too complex, while for most of the esteemed members here, even complex looks quite simple ...

My advise: just learn the math and don't worry so much about what it means. Some things take awhile to sink in. In the end you'll probably notice it's all just a simple time/phase relationship.

 

My advise: just learn the math and don't worry so much about what it means. Some things take awhile to sink in. In the end you'll probably notice it's all just a simple time/phase relationship.

Dear Sir,

I accept your input. In fact, I do take time some time to grab concepts but if I don't get the concept then something keeps lingering in my mind until I get it.

 

Dear Sir,

I accept your input. In fact, I do take time some time to grab concepts but if I don't get the concept then something keeps lingering in my mind until I get it.

And it is also a very positive attitude seeking help from others when you feel you need it. Some of us bang our heads against the wall to no end... too proud to admit that sometimes we need assistance.

 

I wouldn't go that far. We can certainly solve second order differential equations for systems that are underdamped without requiring complex numbers. But the math is more tedious and convoluted because we have robbed ourselves of the simplicity and elegance afforded by complex numbers.

I was only talking about finding the roots of the polynomial, which leads to the solution of the differential equation. I understand that you can still solve the differential equation without resorting to complex numbers. It is as you say more difficult however.

 

The easiest way to see the importance is to understand the connection between 2nd order differential equations and 2nd order polynomials. Second order differential equations occur in electronic circuits and mechanical systems all the time. For all 2nd order polynomials with real coefficients there are two real roots, or a single repeated root, or a complex conjugate pair. Each of these three situations describes different dynamics. Without the imaginary unit in these cases, there would be no way to describe the dynamics of the complex conjugate pair, because the algebra of real numbers just won't allow the third case to exist.

Hola PB

Could you post an example of those three (six?) cases to see if I finally start getting the feeling of imaginary numbers?

Maybe I fail because I try to imagine them in a 2D or 3D system.

 

Well, there are only three cases for the roots of a quadratic polynomial:
Case 1:
\(x^2+5x+6=0\)
has a pair of real roots at x= {-2,-3}

Case 2:
\(x^2+4x+4=0\)
has a single root with a multiplicity of 2 at x = {-2}

Case 3:
\(x^2+2x+5=0\)
has a pair of complex conjugate roots at {-1 ± j2}
Without complex numbers this equation has no solutions in ℝ, the set of real numbers.

The differential equation corresponding to case three has real solutions in the set of real functions of a real variable. As has already been mentioned, this case is harder to solve than the other two.