Amateur Electronics question

[Magnus Danielson]

From: "Jon Darby" <jdarby at lplizard.com>
Subject: Amateur Electronics question
Date: Mon, 10 Jul 2000 10:16:08 -0400

> Howdy!

Jon,

>    I officially became fed up of not knowing EXACTLY how all the little
> electronic projects I've built work and deceided to educate myself on
> electronic theory.

Good.

> I have a seriously juvenile resistance question, so I
> figured since everyone on the list must have had a baby-steps phase in their
> education I would ask and not worry too much about being humiliated. Here is
> the formula my book gives for finding total resistance with two resistors
> wired in parallel:
> 
> Rtotal=1/R1 * 1/R2
> 
> Assuming this is the correct formula, shouldn't two resistors with a value
> of 1 each equal a total resistance of 1? The answer the book gave was .5,
> where am I missing the logic? Grrrr! Day two and I'm already stumped. Thanks
> a pantload for any help!

Well, the formula isn't correct. It's really

 1     1    1
---- = -- + --
Rpar   R1   R2

This is how it works:

A resistor R has the property of conducting a certain amount of current
depending on how much voltage you apply. You could also see this that given
a certain amount of current you will have to have a certain voltage. The
property of the resistor forms a constant in the ratio between voltage over
the resistor and the current passing through the resistor. Now, depending on
which we put over the other we get:

     U        I
c  = -    c = -
 1   I     2  U

The constants c1 and c2 thus reflect the same relation between current and
voltage. We find that:

     1         1
c  = -    c  = -
 1   c     2   c
      2         1

So, these two constants say the same story but with two diffrent values.
It has been defined so that c1 is called resistance and is measured in Ohms
while c2 is called conductance and is measured in Siemens. Both these units
is named after important fellows in the early eara of electrics. Further,
they also carry diffrent symbols with them so that c1 is R and c2 is G.
Thus, a quick summary:

Def    Property Name   Unit Name

    U
R = -  Resistance      Ohm
    I

    I
G = -  Conductance     Siemens
    U

Now, why two diffrent units if they say the same story? Well, they do, but
they are usefull in diffrent environments and depending on which environment
you can quickly interchange between them without loosing any real information
(well, numerical precision, but thats another story).

So, if we have two resistors and connects them in series we have

     R1        R2
I1         I2        Itot
---\/\/\/----\/\/\/---

  ------->  ------->
     U1        U2

 ------------------->
         Utot

What is the total resistance of this curcuit?

Well, we observe that the currents I1 and I2 are the same, since we do not tap
or insert current between the resistors, thus:

Itot = I1 = I2

Then, we observe that Utot must be the summation of U1 and U2, thus:

Utot = U1 + U2

We also know that the total resistance Rtot depends on Utot and Itot according
to

       Utot
Rtot = ----
       Itot

Now, if we express Utot, U1 and U2 in their currents and resistance we get

Rtot * Itot = R1 * I1 + R2 * I2

and then uses the equalence of Itot, I1 and I2 we get

Rtot * Itot = R1 * Itot + R2 * Itot

we can now divide away Itot from all over the place and gets

Rtot = R1 + R2

Now, thats familiar to us! Really, this formula is as we have seen a derived
shorthand based on the definition of resistance, the Kirchcoff Current Law
and the law of voltages. Also, doing this in the resistance form was simple
for some oddball reason.

So, if we now tries to connect them in parallell we have

   I1   R1
  *---\/\/\/---*
  |     U1     |
  |  ------->  |  Itot
--*            *----
  |            |
  |I2   R2     |
  *---\/\/\/---*
        U2
     ------->
       Utot
 ---------------->

Let's try to identify the properties of this curcuit.

We notice that the voltages Utot, U1 and U2 are all the same, thus:

Utot = U1 = U2

For the current we notice that the current I1 may be diffrent to that of I2
but that according to Kirchcoffs Current Law must Itot be the sum of I1 and I2
(or else will I1 + I2 - Itot not be zero, the minus sign comes from the
reference direction), thus:

Itot = I1 + I2

Now, if we are to try the same trick as before we have to express the currents
Itot, I1 and I2 as an expression of their voltages:

Utot   U1   U2
---- = -- + --
Rtot   R1   R2

Them use the equalence on voltages:

Utot   Utot   Utot
---- = ---- + ----
Rtot    R1     R2

Now, we can safely divide away the Utot:

 1     1    1
---- = -- + --
Rtot   R1   R2

Aha, have we seen that formula around somewhere?

But resistance was really messy for this... if we would have used the
conductance properties

     1         1            1
G1 = --   G2 = --   Gtot = ----
     R1        R2          Rtot

we would have got

Gtot * Utot = G1 * U1 + G2 * U2 = G1 * Utot + G2 * Utot

thus

Gtot = G1 + G2

Thus, the conductances will add when you put them in parallell! Putting
a resistor in parallell with another will thus allow more current to flow
between two nodes for the same voltage difference between those nodes.

Naturally, trying to use conductance in the series case will become as messy
as

 1     1    1
---- = -- + --
Gtot   G1   G2

In summation, this will bring for series connection:

                 1    1
Rser = R1 + R2 = -- + --
                 G1   G2

          1         1
Gser = ------- = -------
       R1 + R2   1    1
                 -- + --
                 G1   G2

          1         1
Rpar = ------- = -------
       1    1    G1 + G2
       -- + --
       R1   R2

       1    1
Gpar = -- + -- = G1 + G2
       R1   R2

While resistors rarely is sold with a measurement in Siemens it is usefull to
know of the concept and think in terms of conductance instead of resistance
since it will aid much in understanding how a curcuit works.

I hope that I have helped in removing some of the magic from these formulas
and provided some tools that can aid you in doing your own curcuit analysis.
The fundamental formulas that one has to know is really _very_ few, the rest
is really just methods for attacking diffrent problems.

Cheers,
Magnus