Reciprocal

Reciprocal is somehow anti-intuitive. y = x is a very simple function, a simple straight line but its reciprocal : y = 1/x is not simple at all as shown below.

(Note: y = 1/x. This can be changed to yx = 1. also think of a rectangular having area 1 (fixed) and one side y , the other side x. Think about the meaning of this.)

Ohm's Law

         V
I =  ------
         R (in Ohm)


Ohm's Law shows a simple relation between I (current), V(voltage) and R(Resistance). When V is fixed, the relation between I and R becomes simply reciprocal. Say, to make V = 1 (volt), it becomes

         1
I =  ------
         R


When R is very large, say 1M ohm ( 1,000,000 ohm) and 1G ohm (1,000,000, 000 ohm), the current differs very little from 1/1,000,0000 A (1uA) to 1/1,000,000,000 (1 nano A), much less than 1mA and much far less than 1A. Analogically if you a resistor your effort to increase from 1M ohm to 1G ohm (1,000 times higher) will very little rewarded as the current decreases only by 1 / 1000 or 0.999 uA.

On the other hand,  when R is small (not very small), say from 1 ohm to 0.1 ohm (differs only 0.9 ohm), the current changes from 1A to 10A (fairy large current). If  R is 0.1 Ohm and  0.01 Ohm, the difference is only 0.09 Ohm but the current increases from 10A to 100A (huge current), which will make a common consumer use battery die in a very short period. Again analogically if you a resistor and you decrease (yourself) from 1 ohm to 0.01 ohm (only by 0.99 ohm) you will get a big reward of 100A from 1A.

This relates with the following fundamentally very important relation is

Ref

y = sin (1/x)

The graph of this relation becomes



Copy from https://samjshah.com/2009/10/03/sin1x/

This article is easy to read.

The sine function has the max value 1 (one) and the minimum value is -1 (minus one) but the cycle becomes extremely frequent as x reaches 0 (zero), which the chart cannot show.

＜to becomes extremely frequent＞can be expressed as ＜to become more frequent exponentially＞. Reciprocal and exponential may have some relations.


Looks similar but different - sinc function (wiki)









Another Ref


R, C, L Circuits

In Parallel Circuit of Resistors and Inductors (Coils) and the Series Circuit of Capacitors we can find reciprocals. Simple one is a Parallel Circuit of Resistors. The conclusion is

The Reciprocal of the total resistance = the sum of the Reciprocal of each Resistance.

Generally 

IT  =  I1 + I2 + I3.

I = V/R

Therefore

VT/RT  =  V1/R1 + V2/R2 + V3/R3

Since VT =  V1 = V2 = V3

Therefore 1/RT  = 1/R1 + 1/R2 + 1/R3

Intuitively to bundle or twist the three wires together more current flows in them than the current flow in one single wire. But why <The Reciprocal of the total resistance = the sum of the Reciprocal of each Resistance> ?  we need the above calculation. And remember <The total resistance is not the sum of the Reciprocal of each Resistance> but <The Reciprocal of the total resistance is the sum of the Reciprocal of each Resistance>, I would say double Reciprocal.

Not RT  = 1/1 Ohm + 1/10 Ohm  + 1/100 Ohm   = 111/100 Ohm

But 

1/RT  = 1/1 Ohm + 1/10 Ohm  + 1/100 Ohm   = 111/100 Ohm

RT  = 100/111 Ohm  (Less than 100 Ohm)

I would say "Double Reciprocal " as  we usually calculate like

                        1
RT  =   -----------------------------
              1/R1 + 1/R2 + 1/R3

And I think many people do this though some strangeness remains.
 

Contrary to the Parallel arrangement Series Circuit is straight forward or very simple. The total Resistance = the sum of each Resistance.


R = R₁ + R₂ + R₃ + ...

Generally 

VT  = V1 + V2 +V3.

IR = V

Therefore

ITRT  =  I1R1 + I2R2 + I3R3

Since IT =  I1 = I2 = I3


Therefore  RT  = R1 + R2 + R3

Inductors act like resistors as XL (which is regarded as resistance) = 2πfL. On the other and Capacitors act as reciprocal of Resistors as XC (which is regarded as resistance) = 1/2πfC. 


More interesting reciprocal nature will be found in the combination of  RC, RL, LC and LCR Circuits, which I will review later.



sptt