Thursday, December 17, 2009

Do I have to use complex numbers to work out impedance networks?

Is there another way?Do I have to use complex numbers to work out impedance networks?
No of course you don't. I used to do the calculations many many years ago before I became aware of complex numbers.





As to the method you just keep track of the phase. It turns out that you are doing all the same calculations that you would if the problem were expressed using complex algebra.Do I have to use complex numbers to work out impedance networks?
yes. use complex number if you want to know whether the current is leading or lagging the voltage and by how many degrees. but most of the application we want to know the magnitude only so we don't have to use complex number.
complex numbers although easier and faster in the impedence calculations is another way of representing the system in ordinary differential equations, eg if a current flow thru an inductor the voltage represented is IxjXL which if u do it using ode , then it is VL=Ldi/dt and if I=sine wave then di/dt=cosine wave etc ....so u can see the simplifications involved .......
Yes, you do.





The impedance of electrical networks stems from the fact that they are representations of linear response functions. Linear response functions can be written as matrices, matrix equations can be solved with characteristic polynomials and the zeros of characteristic polynomials are... complex numbers.





So whatever formalism you used, in the end it would solve for complex numbers. That's in the nature of mathematics and nothing can change that.





Of course, if you want to dumb the problem down for a user who is not willing to learn complex numbers, you could re-formulate everything with real numbers and endlessly complex trigonometric formulas which seem to drop out of nowhere. But it would be nothing but a brute force denial of what you are really doing...
Yes, you do.


A Smith chart would help too.





There a free online tools, but you won't use them right if you don't understand what they're helping you do.
I assume by impedance networks, that you mean combinations of resistors, capacitors and inductors. These can then be combined in various ways to suit various applications. Mathematically, the Impedance contribution of each of these in isolation is resistor = R ohms, capacitor = 1/(2*Pi*j*f*C) where f is the frequency in Hz, C is the capacitance in Farads and j (or i if you prefer) is square root of -1. inductor = 2*Pi*j*f*L where L is inductance in Henrys. It is the inductors and capacitors that provide the so-called ';Imaginary'; part and the Resistors provide the ';Real'; part to make the total ';Complex'; Impedance. Circuit Engineers sometimes replace j*2*Pi*f (can also be writtien (j*w, w is frequency in radians per sec) with s, the Laplace variable. This makes the sums look simpler and various rules have been devised to make approximations for the equations of s easy to plot (Bode Plots). But at the end of the day if you want the fine detail either you or a computer program will still have to work with the complex numbers.
if you know how to ....do it





But I now model in a simulation on the computer (it's quicker and reduces errors in manipulating equations)








and they model the opamps and transistors I use

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