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Chapter 9: Problem 6

Two voltage measurements made across the same resistance are \(V_{1}=110 \mathrm{~V}\) and \(V_{2}=220 \mathrm{~V}\). Calculate the power gain in decibels of the second reading over the first reading.

Short Answer

Expert verified
The power gain in decibels of the second reading over the first reading is approximately 6.0206 decibels.

Step by step solution

01

Calculate the power ratios

To find the power gain, first, calculate the power of the individual voltage readings. We know that power is proportional to the square of the voltage over the same resistance. Therefore, we have: \n\[ P_2 = k\cdot V_2^2 = k\cdot (220)^2 \]\n and \n\[ P_1 = k\cdot V_1^2 = k\cdot (110)^2 \]
02

Calculate power gain

After obtaining the power ratios, plug the values into the formula for power gain in decibels: \[ G_{dB} = 10 \log(\frac{P_2}{P_1})\] \nwhere \(\log\) is the logarithm to the base 10. Substitute \(P_2\) and \(P_1\) with their respective expressions: \n\[G_{dB} = 10 \log(\frac{k\cdot (220)^2}{k\cdot (110)^2})\]
03

Simplify the power gain expression

In Step 3, note that the constant \(k\) can be cancelled from the numerator and denominator. Also, recall that the property of logarithms states that \(\log(a^b) = b\cdot \log(a)\). Applying this property, we can simplify the expression as follows: \n\[G_{dB} = 10 \log((220/110)^2)\] \nFurther simplifying, the final expression is: \n\[G_{dB} = 10 \log(4)= 10 \cdot 2 \log(2)\] \nGiven that \(\log(2) \approx 0.3010\), the power gain is approximately \(10 \cdot 2 \cdot 0.3010 = 6.0206 \) decibels.

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Most popular questions from this chapter

If the applied ac power to a system is \(5 \mu \mathrm{W}\) at \(100 \mathrm{mV}\) and the output power is \(48 \mathrm{~W}\). determine: a. The power gain in decibels. b. The voltage gain in decibels if the output impedance is \(40 \mathrm{k} \Omega\). c. The input impedance. d. The output voltage.

Calculate the overall upper \(3-\mathrm{dB}\) frequency for a four-stage amplifier having an individual stage value of \(f_{2}=2.5 \mathrm{MHz}\)

Calculate the overall voltage gain of four identical stages of an amplifier, each having a gain of 20 .

a. The total decibel gain of a three-stage system is \(120 \mathrm{~dB}\). Determine the decibel gain of each stage if the second stage has twice the decibel gain of the first and the third has \(2.7\) times the decibel gain of the first. b. Determine the voltage gain of each stage.

a. What frequency is one octave above \(5 \mathrm{kHz}\) ? b. What frequency is one decade below \(10 \mathrm{kHz}\) ? c. What frequency is two octaves below \(20 \mathrm{kHz}\) ? d. What frequency is two decades above \(1 \mathrm{kHz}\) ?
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