How Does the Transfer Function Affect Sinusoidal Inputs?

[noobish]

Homework Statement

This is related to the frequency response topic.
Given transfer function G(s) = 4/(s+1)
and input is sinusoidal i.e. 2 sin (10t + 10)

Homework Equations

None.

The Attempt at a Solution

I have found out that the magnitude of the transfer function G(s) is 4/sqrt(101) with phase angle -84 degree.

Is the output
(4/sqrt(101) + 2) sin (10t - 74) ?
because that is what i was told.

All these while I thought that the magnitude of G(s) have to be multiplied with the magnitude of the input sinusoidal to obtain the magnitude of the output. But in this case, he just add both of them together.
Please clarify. =)

 

[CEL]

Homework Statement

This is related to the frequency response topic.
Given transfer function G(s) = 4/(s+1)
and input is sinusoidal i.e. 2 sin (10t + 10)

Homework Equations

None.

The Attempt at a Solution

I have found out that the magnitude of the transfer function G(s) is 4/sqrt(101) with phase angle -84 degree.

Is the output
(4/sqrt(101) + 2) sin (10t - 74) ?
because that is what i was told.

All these while I thought that the magnitude of G(s) have to be multiplied with the magnitude of the input sinusoidal to obtain the magnitude of the output. But in this case, he just add both of them together.
Please clarify. =)

First of all the 10 in sin(10t + 10) is radians, so you can not add it to -84 degree and obtain 74 radians.
And yes, the gain of the transfer function must be multiplied by and not summed to the amplitude.

 

[piercas]

I would like to clarify the concept of frequency response in this situation. The magnitude of the transfer function G(s) represents the gain or amplification of the input signal at a certain frequency. In this case, the input signal is a sinusoidal with a frequency of 10 Hz. The magnitude of the transfer function is 4/sqrt(101), which means that the amplitude of the output signal will be amplified by this factor at a frequency of 10 Hz.

However, the output signal will also have a phase shift of -84 degrees compared to the input signal. This means that the output signal will be shifted in time, which can be seen in the equation provided: (4/sqrt(101) + 2) sin (10t - 74). The output signal is not simply the sum of the magnitudes of the transfer function and the input signal, but rather a combination of both the magnitude and phase shift.

It is important to note that the magnitude and phase shift of the transfer function may vary for different frequencies. This is known as the frequency response of a system. In this case, we have only looked at the response at a frequency of 10 Hz. To fully understand the frequency response of a system, we need to analyze the response at different frequencies.

In summary, the output signal is not simply the sum of the magnitudes of the transfer function and the input signal, but rather a combination of both the magnitude and phase shift. It is important to consider both factors when analyzing the frequency response of a system.