The main aim of mathematics is to solve problems easily. That's the reason why we have formulae. Consider the series of cos x, sin x and e^x.
cos x= 1 - x^2/2 + x^4/24 - x^6/720 ...
sin x=x - x^3/6 + x^5/120 - x^7/5040...
e^x= 1 + x + x^2/2 + x^3/6 + x^4/24....
Now coming back to electronics. In electronics we frequently encounter waveforms like sin x and cos x. We multiply them sometimes.
For eg: sin x and cos x may be multiplied in some equations. Sometimes we also find the differentiation of the multiplication (as in inductors and capacitors). In such cases we can clearly see that we lose our temper while solving such problems which involves differentiation and integration. To make it worse consider this. If you have a damped sinusoidal waveform we have an exponential term in addition to sin x and cos x. If someone asks you to find the derivative of this term we will see some kind of formula that we would have learnt in school ( I believe you know that formula and how lengthy it is). When you want to do this as a program in your computer then it becomes very lengthy. Our hero Complex numbers and Frequency domain enters here.
Let's come back to those series of cos x , sin x and e ^x. Many mathematicians were pondering how to represent cos x, sin x in terms of exponentials mainly because of the complexity involved with the cosine and sine functions as I had described in the previous paragraph. It was Euler who first found this- i.e., representing cos and sine in terms of exponential.
Consider these series-
cos x= 1 - x^2/2 + x^4/24 - x^6/720 ...
sin x=x - x^3/6 + x^5/120 - x^7/5040...
e^x= 1 + x + x^2/2 + x^3/6 + x^4/24....
It's very clear that if we substitute sqrt(-1) * x in the place of x in the equation of e^x we will get the following relationship (Try yourself. It's very simple. Just direct substitution).
e^(sqrt(-1)*x) = cos x + sqrt(-1) sin x;
This is usually known as Euler's identity.
What's sqrt(-1) ? Come on, it has no solution at all. It's funny rather. But you should remember you are living in the realm of mathematics where anything is possible. I believe that complex numbers do exist. They are indeed real in the world of mathematics. I am not going to talk about the origin of the complex numbers (it's in wikipedia. If you like you can refer there). But I can say one thing. When people were considering about this sqrt(-1) they rather thought it was absurd to include this in Mathematics because it's not true at all. But after seeing the simplicity involved after including the sqrt(-1) in equations such as e^x (which I have described above) it became inevitable to include sqrt(-1) and they named it as complex numbers. Complex numbers has its own rules which are similar to normal algebra. Sqrt(-1) is denoted by "i" or electrical engineers use "j" so as not to confuse with the term current.
How can this be used for simplifying equations?
From the Euler's identity we can say that
cos x = [e^(jx) + e^(-jx)]/2;
sin x=[ e^(jx) - e^(-jx)]/2j;
So now if we see any equations such as (e^x)cos x and if we are asked the differentiation we need not use the formula that we had studied before (I have forgotten that formula though). We frequently encounter these kind of terms in electronics especially capacitors and inductors. In those places we can easily use these conversions and solve those easily by just having in terms of powers of exponential e. Now we can easily differentiate the term because it involves powers of e only. This reduces time and if you are writing some kind of C program for this it reduces time to do the operation. In other words we have increased the efficiency. (Besides that's the reason why we have mathematics- To increase the speed of calculation). So after representing everything in terms of exponential we can indeed say many things about the signal (whether it's stable or unstable) easily in the domain of complex numbers instead of time domain which you might feel difficult. Laplace transforms involve conversion of time domain to complex domain (also known as frequency domain) and it's very easy to perform operations on this domain. In other words Laplace transform reduces the complexity further.
For eg: the Laplace transform of e^(-at) is equal to 1/ (s+ a);
The reason for using Laplace transform is very simple. It reduces the calculation time further. That is, after expressing everything in terms of exponential we have to see characteristics of the signal (whether it is keeping on increasing (unstable) or decreasing (stable)). This can done by seeing the powers in the exponential term. In other words the term "e" does not have that much significance when compared to it's power. So Laplace transform is the best place to see the system characteristics which clearly states about the power. In the above example the pole is at -a which in turn is the power of the exponential term.
You can perform any operation on the Laplace transform of the signal such as involving exponential, cosine, sine etc. and get the original term back in the time domain easily. It reduces the time taken to compute the answer. In the above example e^(-at) has been converted to a somewhat easier term 1/s+a. It does not involve any exponential term and hence becomes easy to perform many operations such as normal addition and so on. If you want to know much about Laplace transforms see some book on it. Better see "Engineering Circuit Analysis" by Hayt. There you can clearly see how mathematics helps you in solving complex ( I mean difficult) electricity problems easily.
But don't forget- The main use of the complex numbers and Laplace transforms (which is an easy version of representing complex numbers) is to increase the speed of calculations by working in the frequency domain rather than the time domain.
Anyway some computer algorithms are written in time domain because it takes no time to solve those. Eg: Convolution is done in time domain in image processing tool of the matlab. It's because if you work in the frequency domain then you have to find the inverse which is somewhat difficult to find out.
cos x= 1 - x^2/2 + x^4/24 - x^6/720 ...
sin x=x - x^3/6 + x^5/120 - x^7/5040...
e^x= 1 + x + x^2/2 + x^3/6 + x^4/24....
Now coming back to electronics. In electronics we frequently encounter waveforms like sin x and cos x. We multiply them sometimes.
For eg: sin x and cos x may be multiplied in some equations. Sometimes we also find the differentiation of the multiplication (as in inductors and capacitors). In such cases we can clearly see that we lose our temper while solving such problems which involves differentiation and integration. To make it worse consider this. If you have a damped sinusoidal waveform we have an exponential term in addition to sin x and cos x. If someone asks you to find the derivative of this term we will see some kind of formula that we would have learnt in school ( I believe you know that formula and how lengthy it is). When you want to do this as a program in your computer then it becomes very lengthy. Our hero Complex numbers and Frequency domain enters here.
Let's come back to those series of cos x , sin x and e ^x. Many mathematicians were pondering how to represent cos x, sin x in terms of exponentials mainly because of the complexity involved with the cosine and sine functions as I had described in the previous paragraph. It was Euler who first found this- i.e., representing cos and sine in terms of exponential.
Consider these series-
cos x= 1 - x^2/2 + x^4/24 - x^6/720 ...
sin x=x - x^3/6 + x^5/120 - x^7/5040...
e^x= 1 + x + x^2/2 + x^3/6 + x^4/24....
It's very clear that if we substitute sqrt(-1) * x in the place of x in the equation of e^x we will get the following relationship (Try yourself. It's very simple. Just direct substitution).
e^(sqrt(-1)*x) = cos x + sqrt(-1) sin x;
This is usually known as Euler's identity.
What's sqrt(-1) ? Come on, it has no solution at all. It's funny rather. But you should remember you are living in the realm of mathematics where anything is possible. I believe that complex numbers do exist. They are indeed real in the world of mathematics. I am not going to talk about the origin of the complex numbers (it's in wikipedia. If you like you can refer there). But I can say one thing. When people were considering about this sqrt(-1) they rather thought it was absurd to include this in Mathematics because it's not true at all. But after seeing the simplicity involved after including the sqrt(-1) in equations such as e^x (which I have described above) it became inevitable to include sqrt(-1) and they named it as complex numbers. Complex numbers has its own rules which are similar to normal algebra. Sqrt(-1) is denoted by "i" or electrical engineers use "j" so as not to confuse with the term current.
How can this be used for simplifying equations?
From the Euler's identity we can say that
cos x = [e^(jx) + e^(-jx)]/2;
sin x=[ e^(jx) - e^(-jx)]/2j;
So now if we see any equations such as (e^x)cos x and if we are asked the differentiation we need not use the formula that we had studied before (I have forgotten that formula though). We frequently encounter these kind of terms in electronics especially capacitors and inductors. In those places we can easily use these conversions and solve those easily by just having in terms of powers of exponential e. Now we can easily differentiate the term because it involves powers of e only. This reduces time and if you are writing some kind of C program for this it reduces time to do the operation. In other words we have increased the efficiency. (Besides that's the reason why we have mathematics- To increase the speed of calculation). So after representing everything in terms of exponential we can indeed say many things about the signal (whether it's stable or unstable) easily in the domain of complex numbers instead of time domain which you might feel difficult. Laplace transforms involve conversion of time domain to complex domain (also known as frequency domain) and it's very easy to perform operations on this domain. In other words Laplace transform reduces the complexity further.
For eg: the Laplace transform of e^(-at) is equal to 1/ (s+ a);
The reason for using Laplace transform is very simple. It reduces the calculation time further. That is, after expressing everything in terms of exponential we have to see characteristics of the signal (whether it is keeping on increasing (unstable) or decreasing (stable)). This can done by seeing the powers in the exponential term. In other words the term "e" does not have that much significance when compared to it's power. So Laplace transform is the best place to see the system characteristics which clearly states about the power. In the above example the pole is at -a which in turn is the power of the exponential term.
You can perform any operation on the Laplace transform of the signal such as involving exponential, cosine, sine etc. and get the original term back in the time domain easily. It reduces the time taken to compute the answer. In the above example e^(-at) has been converted to a somewhat easier term 1/s+a. It does not involve any exponential term and hence becomes easy to perform many operations such as normal addition and so on. If you want to know much about Laplace transforms see some book on it. Better see "Engineering Circuit Analysis" by Hayt. There you can clearly see how mathematics helps you in solving complex ( I mean difficult) electricity problems easily.
But don't forget- The main use of the complex numbers and Laplace transforms (which is an easy version of representing complex numbers) is to increase the speed of calculations by working in the frequency domain rather than the time domain.
Anyway some computer algorithms are written in time domain because it takes no time to solve those. Eg: Convolution is done in time domain in image processing tool of the matlab. It's because if you work in the frequency domain then you have to find the inverse which is somewhat difficult to find out.
