Chapter 3, Problem 8PP

(Electrical eng.) a. The voltage gain of an amplifier is given by this formula:

$voltagegain={\left[\frac{275}{\sqrt{{23}^2+0.5f^2}}\right]}^n$

f is the frequency in Hz.

n is the number of stages in the amplifier.

Using this formula, write, compile, and run a C++ program to determine the value of the voltage gain for a four-stage amplifier operating at a frequency of 120 Hz. Your program should produce the following display:

At a frequency of xxxxx hertz, the voltage gain is yyyyy

Your program should replace xxxxx with the frequency and yyyyy with the voltage gain.

b. Manually check the value your program produces. After verifying that your program is working correctly, modify it to determine the voltage gain of a 12-stage amplifier operating at a frequency of 9500 Hz.

Program Plan Intro

Program Plan:

• The variable vgis usedto store the voltage gain
• The variable nis used to storethe number of stages
• The variable fis used to storethe frequency (in Hertz)
• A pre-defined function sqrt() is used to find the square root of an argument given.
• A pre-defined function pow() is used to give a mathematical power to a variable.

Program Description:

The C++ program finds the voltage gain using following formulae:

  $\begin{array}{l}voltagegain={\left[\frac{275}{\sqrt{{23}^2+0.5\times {120}^2}}\right]}^4\\ ={\left[\frac{275}{\sqrt{{23}^2+0.5\times 14400}}\right]}^4\\ ={\left[\frac{275}{\sqrt{529+7200}}\right]}^4\\ ={\left[\frac{275}{\sqrt{7729}}\right]}^4\\ ={\left[\frac{275}{87.91}\right]}^4\\ ={\left[3.12\right]}^4\\ =95.802\end{array}$

A formula of a voltage gain is used to find the voltage gain at the stage of four and a frequency of 120 Hz.

Explanation of Solution

Program:

#include <iostream>
#include <math.h>
using namespace std;

int main()
{
float vg; // voltage gain
int n = 4; // number of stages
float f = 120; // frequency(Hz)

// formula
    vg = pow((275/(sqrt(pow(23,2)+(0.5*(f * f))))),4);

    cout <<"At a frequency of "<< f
<<" hertz, the voltage gain is "<< vg;

return 0;
}

Sample Output:

When f = 120 Hz and number of stages = 4

Program Plan Intro

To check manually if the above program works correctly using the given values.

Explanation of Solution

Given:

Frequency (f) = 120 Hz

Stages (n) = 4

Explanation:

  $\begin{array}{l}voltagegain={\left[\frac{275}{\sqrt{{23}^2+0.5\times {120}^2}}\right]}^4\\ ={\left[\frac{275}{\sqrt{{23}^2+0.5\times 14400}}\right]}^4\\ ={\left[\frac{275}{\sqrt{529+7200}}\right]}^4\\ ={\left[\frac{275}{\sqrt{7729}}\right]}^4\\ ={\left[\frac{275}{87.91}\right]}^4\\ ={\left[3.12\right]}^4\\ =95.802\end{array}$

The answer is close to the above output produced by the program.Hence the program is working correctly.