Today’s discussion focuses on Euler’s number and the utility of the related natural logarithm or “ln” function as they pertain to technician level circuit analysis. Once we have discussed the nature of Euler’s number and the related base “e” natural log function, we will move on to take a look at an example of their practical application for technicians when performing circuit analysis tasks.
What is Euler’s Number ?
Euler’s number is also known as the exponential constant and is a non terminating “irrational” number. It is derived from the following infinite power series:
e = 1 + ( 1/1! ) + ( 1/2! ) + ( 1/3! ) + ( 1/4! ) + ( 1/5! ) + ...
e = 1 + ( 1/1 ) + ( 1/2 ) + ( 1/6 ) + ( 1/24 ) + ( 1/120 ) + ...
e = 2.718281828459 approx.
This mathematical constant is widely used in instances where quantities of interest exhibit exponential growth or decay. For the practical technician, an everyday example of such a relationship would be the charging and discharging of capacitors and inductors in series RC and RL circuits with respect to time.
What is the Natural Logarithm (ln) Function
Logarithms in general allow you to operate on, or isolate, exponents used with various “base” values. A well known example is the “Common” or “base 10” logarithm used by technicians for power calculations. The “ ln ” function is the base “e” natural logarithm function. The natural logarithm or "ln" of the value “ex” is equal to “x”.
ln (e x) = x
A Typical Natural Log Application for Technicians
As with all logarithm functions, the “ln” function is very useful in instances where we wish to isolate or manipulate exponents used, in this instance, with base “e” values. An example of such an instance arises whenever we need to know the time it takes the voltage across a capacitor to reach a desired level during the capacitor charging curve. We begin with the commonly used relationship for calculating the voltage across the plates of a capacitor as it charges.
Vc = E * (1 - e –t/Tau )
Vc = Capacitor Voltage in volts
E = Applied DC Voltage in volts
e = Euler’s Number (exponential constant)
t = Time in seconds
Tau = Time constant ( R*C) in seconds
The equation defining capacitor voltage uses Euler’s number. This is due to the fact that the capacitor charging curve exhibits exponential growth. We rearrange this relationship, isolating Euler’s number and its exponent on one side of the equation, and everything else on the other side. Once we have isolated Euler’s number and its exponent, we need to bring the exponent of “e” down to the main line so that we can operate on it as the variable we wish to isolate, “t” for time in seconds, is part of the exponent. This type of mathematical manipulation is accomplished by using logarithms. As we are dealing with a base “e” value, we apply the natural logarithm or “ln” function to both sides of the equation.
e –t/Tau = 1 – Vc/E
ln (e –t/Tau) = ln (1 – Vc/E)
Given that ln (e x) = x, the left side of the equation results in the desired “t” variable being moved to main line. It can then be isolated to solve for time given applied voltage and desired capacitor voltage level.
-t/Tau = ln (1 – Vc/E)
t = -Tau * ln (1 – Vc/E)
The above relationship provides the time it takes for the voltage on the plates of a capacitor to reach a specified value in a series RC circuit. A series RC circuit example along with the step by step solution using the natural log function is provided in the video animation below.
Online Tutorial Capacitor Transient Response – Series RC Circuits
We hope this has been helpful to you as a practicing or student technician. We are looking for other ideas for this continuing Circuit Analysis Tips series. Please let us know what you would like us to write about by sending us your thoughts and questions at firstname.lastname@example.org.