Question

How do you calculate the amplitude and phase of each harmonic in a Fourier series representation?

Answer 1

To calculate the amplitude and phase of each harmonic in a Fourier series representation, you typically start with a periodic signal,

(

)
f(t), defined over a specific interval (usually a period

T or
2

2π for simplicity). The Fourier series representation of

(

)
f(t) is given by:


(

)
=

0
+
∑

=
1
∞
(


cos
⁡
(



)
+


sin
⁡
(



)
)
f(t)=a
0
    ​

+∑
n=1
∞
    ​

(a
n
    ​

cos(nωt)+b
n
    ​

sin(nωt))

Where:


0
a
0
    ​

 is the average value of the function over one period.


a
n
    ​

 and


b
n
    ​

 are the coefficients of the harmonic terms.

=
2


ω=
T
2π
    ​

 is the fundamental frequency of the signal.

To find the coefficients


a
n
    ​

 and


b
n
    ​

, you need to perform integration over one period

T. The formulas for


a
n
    ​

 and


b
n
    ​

 are as follows:



=
2

∫

0

0
+


(

)
cos
⁡
(



)
 


a
n
    ​

=
T
2
    ​

∫
t
0
    ​

t
0
    ​

+T
    ​

f(t)cos(nωt)dt



=
2

∫

0

0
+


(

)
sin
⁡
(



)
 


b
n
    ​

=
T
2
    ​

∫
t
0
    ​

t
0
    ​

+T
    ​

f(t)sin(nωt)dt

where

0
t
0
    ​

 is a reference point in the period (often set to 0 for simplicity).

To calculate the amplitude and phase of each harmonic, you can use the following relations:

The amplitude of the

nth harmonic is given by:



=


2
+


2
A
n
    ​

=
a
n
2
    ​

+b
n
2
    ​

    ​


The phase angle (


θ
n
    ​

) of the

nth harmonic can be calculated using the arctangent function:



=
arctan
⁡
(




)
θ
n
    ​

=arctan(
a
n
    ​

b
n
    ​

    ​

)

Please note that calculating Fourier series coefficients and their associated amplitudes and phases can be analytically or numerically complex, depending on the specific function

(

)
f(t). In some cases, you may need to use numerical methods or software tools to perform the integrations and extract the coefficients accurately. Many mathematical software packages (e.g., MATLAB, Python's NumPy, etc.) have built-in functions to help with Fourier analysis, making the process easier.