# How Does the Transfer Function Affect Sinusoidal Inputs?

• noobish
In summary, the conversation discusses the frequency response topic and provides a transfer function G(s) = 4/(s+1) and input as a sinusoidal function. The magnitude of G(s) is found to be 4/sqrt(101) with a phase angle of -84 degrees. The output is calculated as (4/sqrt(101) + 2) sin (10t - 74) and it is clarified that in order to obtain the magnitude of the output, the gain of the transfer function must be multiplied by the amplitude of the input, not added to it.
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)

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.

noobish said:

## 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)

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.

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.

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.

## 1. What is a frequency response dilemma?

A frequency response dilemma is a situation where a system or device is unable to accurately reproduce or transmit a wide range of frequencies, resulting in distorted or incomplete signals. This can occur in electronic devices, audio systems, and communication networks.

## 2. How does a frequency response dilemma affect audio quality?

A frequency response dilemma can significantly impact audio quality by causing certain frequencies to be overemphasized or underrepresented in the sound output. This can lead to a lack of clarity, muffled sound, or unwanted distortion.

## 3. What factors can contribute to a frequency response dilemma?

There are several factors that can contribute to a frequency response dilemma, including limitations in the design or components of a system, improper calibration, interference from other signals, and changes in temperature or humidity.

## 4. How can a frequency response dilemma be measured and evaluated?

A frequency response dilemma can be measured and evaluated using specialized equipment such as an audio frequency analyzer or an oscilloscope. These tools can measure the amplitude and frequency of signals and identify any discrepancies or irregularities.

## 5. How can a frequency response dilemma be addressed or corrected?

To address a frequency response dilemma, it is important to identify the root cause and make appropriate adjustments or improvements. This may involve modifying the design or components of a system, calibrating equipment, or implementing filters and equalizers to balance out the frequency response.

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