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# Complex Numbers

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#### Electronman

##### New Member
Hi,

What's the influence of complex numbers in RLC circuits?
I have seen several technical books do not pay any attention to those numbers yet are able to analyze RLC circuits?!
For instance I have seen several books use the formula 1/jwC and the other books just use 1/wC for a cap, and the same for an inductor?!
So why and when those numbers are coming in?

1/wC is not correct notation, it should be written as a complex number, 1/jwC. But if you are just interested in the impedance of a capacitor or inductor you can omit the j (but recognize that it is a reactance with a 90 degree phase shift between current and voltage). If you have any resistance in the circuit than you must use the complex form of the equations to do any calculations.

So Is J for RL or RC circuits?
Can I ignore J but keep in mind the 90 degree of phase shift? I.e I mean Is J there just to indicate 90 degrees of phase shift?

So Is J for RL or RC circuits?
both.

Can I ignore J but keep in mind the 90 degree of phase shift? I.e I mean Is J there just to indicate 90 degrees of phase shift?
The phase shift of an L is opposite to that of a C. Depending on what you want to calculate, you may be able to ignore it; but not usually.

If you are performing calculations to determine the impedance of a circuit with R's, L's, and C's, you need to use complex notion to get the correct answer. For example if you have 100Ω of resistance in series with 100Ω of inductive reactance, the circuit impedance is not 200Ω, it is the square root of 200Ω as determined by the complex number calculation. The phase shift of an arbitrary RLC circuit can be anywhere from -90 degrees to +90 degrees depending upon the frequency and the component values.

both.

The phase shift of an L is opposite to that of a C. Depending on what you want to calculate, you may be able to ignore it; but not usually.

I think I need a more complete respond when we are able to ignore it and when we can not. Why J was created here in RC and RL (and maybe LC?) circuits

Ignore it when you have a very simple circuit consisting of only Rs, Cs or Ls.

You can ignore it when R >> XC and/or R >> XL but you will lose accuracy of the result.

If you want to get frequency response of anything but a simple circuit, it's best to keep the j in-place. Why not just keep it complex? It's much more useful.

j is for impedance.

The impedance of a RΩ Resistor is R

The impedance of a L Henries inductor is jωL

The impedance of a C Farads capacitor is 1/(jωC)

When you see ωL and 1/(ωC), you are talking about Reactance, not Impedance.

j is for impedance.

The impedance of a RΩ Resistor is R

The impedance of a L Henries inductor is jωL

The impedance of a C Farads capacitor is 1/(jωC)

When you see ωL and 1/(ωC), you are talking about Reactance, not Impedance.

I think I can calculate the Impedance by Pythagorean theorem??? If so then why do I need "J"??

I think I can calculate the Impedance by Pythagorean theorem??? If so then why do I need "J"??
For complex circuits with several L's or C's the Pythagorean theorem won't work, you have to use complex numbers.

You are trying to avoid the use of complex numbers but, for any but the simplest RLC circuits, it is necessary to generate the right answer. Sorry.

I think I can calculate the Impedance by Pythagorean theorem??? If so then why do I need "J"??

You need j to separate what is real from imaginary.

Try to solve a 5 stage filter with only Pyt. theorem.

If you are performing calculations to determine the impedance of a circuit with R's, L's, and C's, you need to use complex notion to get the correct answer. For example if you have 100Ω of resistance in series with 100Ω of inductive reactance, the circuit impedance is not 200Ω, it is the square root of 200Ω as determined by the complex number calculation. The phase shift of an arbitrary RLC circuit can be anywhere from -90 degrees to +90 degrees depending upon the frequency and the component values.

No. The resulting impedance of 100 ohms resistive plus 100 ohms reactive is 100 times sqrt 2. Or, 141.4 ohms at a 45 deg angle, V leading I. You probably know that, but made a slight faux pas.

Hi there,

My advice is to not try to avoid the use of the 'j' operator, but try instead to
embrace it and incorporate it into your work.

Another way of thinking about this is that j is equal to the square root of minus 1
j=sqrt(-1)

so it simplifies if it ends up in an equation where it is squared, because the square of
j is taken to equal minus 1 (-1) for most problems:
j=sqrt(-1)
j^2=-1

The use of j allows you to solve circuits that would otherwise be more difficult
so get used to it now and you will see how easy it gets after a while.
You should probably get a scientific calculator that can work with complex
numbers too so you can see just how simple this ends up being.
TI makes some good ones with graphics too, and there are probably free ones

Get used to 'j' now and save yourself lots of time and effort.

Here are a few more identities:

j=sqrt(-1)
j^2=-1
j^3=-j
j^4=1

and the list repeats for exponents of 5 and above.

Note the lower case J is used in electronic work, but it's the same as lower case I in
most math texts, so watch out for either j or i as the operator. A lot of electronic
work uses I for current so the lower case J was adopted for electronic work instead
of lower case I.

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I think I can calculate the Impedance by Pythagorean theorem??? If so then why do I need "J"??

You can't ignore j because the phase shift of a combined impedance (R with any L or C) will not always be 90 degrees. The phase of R is always 0, and the phase of L is always +90, and the phase of C is always -90, but when you combine them it can be anything from 0 to +/- 180 degrees. It's not enough just to keep in mind that L and C have +/-90 phase.

Ignoring j when analyzing RLC circuits is exactly the same thing as ignoring the three angles of a triangle when doing trigonometry and trying to do everything with just the 3 side lengths.

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No. The resulting impedance of 100 ohms resistive plus 100 ohms reactive is 100 times sqrt 2. Or, 141.4 ohms at a 45 deg angle, V leading I. You probably know that, but made a slight faux pas.
You are correct. My face is red.

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