A two-times increase or decrease in power results in a change of approximately 3 decibels.

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The logarithmic decibel scale is used to measure changes in power or signal strength.

Note: Formula to calculate change in Power in Decibels:

\[\Delta\text{ dB} = 10 \times \log_{10} \bigg ( \frac{P_2}{P_1} \bigg )\]

Where:
\(P_1 =\) reference power
\(P_2 =\) power being compared.

If we plug in the values given in this question we get:

\begin{align} \Delta\text{ dB} &= 10 \times \log_{10} \bigg ( \frac{P_2}{P_1} \bigg )\\ &= 10 \times \log_{10} \bigg ( \frac{2}{1} \bigg )\\ &= 10 \times \log_{10} (2)\\ &= 10 \times 0.3\\ &= 3\text{ dB}\\ \end{align}

Hint: Watch out for (C). Amateur radio uses an S scale to measure signal strength, and a change of 1 S unit corresponds to a four-times increase in power or a 6 dB change, but this is NOT what they are asking for in this question. Don't get fooled.

For more info see Wikipedia: Decibel (dB)

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Current doesn't get lost. It gets split up across all of the branches of the circuit, but all the currents in all the branches add up to the total current.

The equation for the total current of a parallel circuit is:

\[\text{I}_{total} = \text{I}_1 + \text{I}_2 + \ldots + \text{I}_n\]

For an excellent, intuitive explanation of this, see the video from the Khan Academy on Kirchhoff's current law.

Hint: The question asks about "individual currents" and while two answers mention current, only one answer--the correct one--uses the plural "currents".

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To solve for power use Ohm's Law \(I = E / R\) and Watt's Law \(P = I * E\), where \(I\) = current in amperes, \(E\) = voltage, \(R\) = resistance in ohms, and \(P\) = power in watts.

We know \(E = 400V\), and \(R = 800Ω\).

\[\text{Method 1:}\]

First solve for \(I\), using \(I = E/R\): \[400/800 = .5A\] Then solve for \(P\), using \(P = I*E\): \[.5 * 400 = 200\text{ watts.}\]

\[\text{Method 2:}\]

Combining the two equations gives us \(P = (E/R)* E\), or \(P = E^2/R\).

Solving for the given values:

\[400^2 / 800 = 200\text{ watts.}\]

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The amount of power (watts) used by a 12-VDC (volts direct current) light bulb that draws 0.2 amperes is 2.4 watts. The power circle equation based on Ohm's law shows that P = I x E, where P is power in Watts, I is current in Amperes, and E is energy in Volts.

So for this question: P = I x E

P = (0.2 amperes) x (12 volts) = 2.4 watts

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Approximately 61 milliwatts are dissipated when a current of 7.0 milliamperes flows through 1.25 kilohms.

By combining the Ohm's Law equations and the power circle equations we can solve the problem.

• \(P\) = Power in Watts
• \(I\) = Current in Amperes
• \(E\) = Energy in Volts
• \(R\) = Resistance in Ohms

We are given:

\(I_\text{current}\) \(=\) \(7.0 \text{milliamperes}\) \(=\) \(0.007 \text{Amperes}\)

\(R_\text{resistance}\) \(=\) \(1.25 \text{kilohms}\) \(=\) \(1250 \text{Ohms}\)

To solve for \(P\), use the power equation \(P = I \times E\) and combine it with the equation from Ohm's Law \(E = I \times R\).

This gives us: \(P = I \times (I \times R)\) \(=\) \(I^2 \times R\)

So for this question:

\(P = 0.007 A \times (0.007 A \times 1250 Ω)\) \(=\) \(0.061 \text{Watts}\) \(=\) \(61 \text{milliwatts}\), or

\(P = (0.007 A)^2 \times 1250 Ω\)=\(0.061 \text{Watts}\)=\(61 \text{milliwatts}\)

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Alternate hint: \(\bigg(P = \frac{I^2 \times R}{1,000}\bigg)\) (divide by 1,000 because it’s measured in milliamperes)

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The output Peak Envelope Power (PEP) is 100 watts.

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This problem can be solved using Joule's Law (also known as the power law):

\[P = E \times I\]

There are two problems here, though.

First, the question provides values for voltage and resistance, but not current. That's OK, though, since the current can be derived using Ohm's Law.

Second, the question provides a value for peak-to-peak voltage (\(V_{PP}\)) and assumes an alternating current (AC) in the form of sine wave. \(V_{PP}\) can't be plugged straight into the power law as is. It will first have to be converted to peak voltage (\(V_{peak}\)) and then to the root mean square voltage (\(V_{RMS}\)) value.

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To get a version of the power law that will solve this problem, take Ohm's law, rearranged to define current:

\[I = \frac{E}{R}\]

and plug this into the power law in place of I:

\[P = E \times \frac{E}{R}\]

\(E \times E\) can be written as \(E^2\):

\[P = \frac{E^2}{R}\]

This is the power equation used for direct current (DC). But that brings us to the second problem. Because we're working with AC, in order to plug a value of \(E\) into this equation, the voltage needs to be expressed in terms of \(V_{RMS}\).

The peak voltage is simply the peak-to-peak voltage divided by two:

\begin{align} V_{peak}& = \frac{V_{PP}}{2}\\ &=\frac{200\text{ V}}{2}\\ &=100\text{ V}\\ \end{align}

Next, the root mean square voltage for a sine wave can be calculated from the peak voltage as follows:

\begin{align} V_{RMS} & = \frac{V_{peak}}{\sqrt{2}}\\ & = \frac{100\text{ V}}{\sqrt{2}}\\ \end{align}

Now we can calculate the PEP using the power law, as derived above:

\begin{align} P & = \frac{E^2}{R}\\ &= \frac{\Big (\frac{100}{\sqrt{2}} \Big )^2}{50\ \Omega}\\ &= \frac{\frac{10000}{2}}{50}\\ &= \frac{5000}{50}\\ &= 100\text{ W}\\ \end{align}

Hint: If you calculate \(V_{RMS}\) directly for 100 volts you will get a value for E of approximately 70.7 volts. Squaring that and then dividing it by R (in our example, 50 ohms) will give you 99.9698 watts, which is approximately 100 watts.

Silly hint: Even a dummy knows that 50 percent of 200 is 100.

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The RMS (root mean square) value of an AC signal results in the same power dissipation as a DC voltage of the same value. The RMS value is useful as an average value for the voltage in an AC circuit throughout the alternating wave cycle, as if the current were a constant as in DC.

For more info see Wikipedia: RMS, or root mean square

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The RMS voltage of a sine wave is approximately \(0.7 \times \text{peak voltage}\).

Looking at the possible answers, which are all peak-to-peak: divide them in half to know the peak voltage, and only one answer is still high enough to be correct -- \(\frac{339.4}{2} = 169.7\). Multiply that by \(0.7\) to get approximately \(120V\).

Alternatively, for a sine wave the peak-to-peak voltage: \[2\sqrt{2} \times \text{RMS voltage} = 2.828 \times \text{RMS voltage}\] So: \(120 \times 2.828 = 339.4\)

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Another way to look at it:

You may know \(\frac{1}{\sqrt{2}} = 0.707\) which is the factor used to multiply times peak voltage to obtain RMS voltage. To reverse the process, divide the RMS voltage by the already memorized 0.707 to obtain the peak voltage and double it to obtain peak-to-peak. \((\frac{120V}{0.707} \times 2) = 339.4\)

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Another way to look at it: \[V_\text{rms} = V_\text{peak} \times (.707)\] therefore: \[V_\text{peak} = \frac{V_\text{rms}}{.707} = \frac{120}{.707} = 169.73\] therefore: \[V_p-p = V_\text{peak} \times 2 = 169.73 \times 2 = 339.46\]

Silly hint: From RMS "to Peak to Peak" think "2 peak 2 peak", "2 8 2 8".

Silly hint 2: Remember The Steve Miller band song “Jet Airliner” as you board the Boing 707.

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12 volts is the RMS (i.e. Root Mean Square) voltage of a sine wave with a value of 17 volts peak.

To find the RMS voltage, multiply the peak voltage by 0.707, which is the same as dividing by the square root of 2.

\[0.707 =1/\sqrt{2}\]

RMS voltage = peak voltage x 0.707

So for this problem RMS voltage = 17 volts x 0.707 = 12 volts.

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A percentage power loss of 20.6% would result from a transmission line power loss of 1 dB.

Given

\begin{align} P_i &= Initial\ Power\\ P_f &= Final\ Power \end{align}

The generalized formula to compute the power change (i.e. loss or gain) in decibels (dB) is:

\[\Delta\ \text{dB} = 10 \times \log_{10} \bigg (\frac{P_f}{P_i} \bigg )\]

In this problem, the power change is given as "a transmission line loss of 1 dB", which we can plug into our equation:

\[-1\ \text{dB} = 10 \times \log_{10} \bigg (\frac{P_f}{P_i} \bigg )\]

Let's isolate the variables \(P_i\) and \(P_f\). First, divide both sides by 10:

\[\frac{-1}{10}\ \text{dB} = \log_{10} \bigg (\frac{P_f}{P_i} \bigg )\]

Exponent (i.e. raise) 10 to the power of both sides of the equation:

\[10^\frac{-1}{10} = 10^{\log_{10} \Big (\frac{P_f}{P_i} \Big )}\]

Simplify:

\[10^\frac{-1}{10} = \bigg (\frac{P_f}{P_i} \bigg )\]

The problem asks us to calculate the "percentage of power loss", which will be the absolute value of % power change.

\begin{align} \%_{power\ change} &= 100 \times \frac{P_f - P_i}{P_i}\\ &= 100 \times \bigg (\frac{P_f}{P_i} - 1 \bigg ) \end{align}

Plug in the calculated value \(\Big (\frac{P_f}{P_i} = 10^\frac{-1}{10} \Big )\) :

\begin{align} \%_{power\ change} &= 100 \times (10^\frac{-1}{10} - 1 )\\ &= -20.6\ \% \end{align}

So:

\begin{align} \%_{power\ loss} &= | \%_{power\ change} |\\ &= |-20.6\ \%|\\ &= 20.6\ \%\\ \end{align}

Hint: You can check the answer if you know that -3dB halves power.

Ex: \begin{align} 100 \times (100\%-20.6\%) &= 79.4\\ 79.4 * (100\%-20.6\%) &= 63\\ 63 * (100\%-20.6\%) &= 50 \end{align}

50 is half of 100. Put another way: \[(1-.206)^3 \approx .5\]

(Note also that the suggested answer 25.9 is the percentage increase from a gain of 1db.)

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The key word here is "unmodulated."

A modulation envelope shows how a carrier wave's amplitude changes over time, the envelope power is the power value of that envelope, and the peak envelope power is the highest power value within some timeframe.

However, if a carrier is unmodulated, its power isn't changing, which means its modulation envelope is flat. And as with any unchanging value, its peak equals its average, and the ratio of two equal values is \[1:1 = 1/1 = 1.00\]

(Note that as power varies directly with amplitude\(^2\), it's convenient to think of the envelope as representing both.)

(Note also that the question asks about the ratio of two power values, not the values themselves, so RMS calculations aren't involved here.)

Hint: There’s only one (1.00) answer.

Useful further reading:

• https://www.rfglobalnet.com/doc/basics-of-power-measurement-average-or-peak-0001
• https://resources.pcb.cadence.com/blog/2020-peak-envelope-power-and-your-rf-signal-chain-simulations

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Tags: arrl chapter 4 arrl module 11

The RMS voltage across a 50-ohm dummy load dissipating 1200 watts is 245 volts.

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You are given:
\begin{align} Power (P) &= 1200\text{ watts}\\ Resistance (R) &= 50\ \Omega \end{align}

Combining the power circle equation \(P = I \times E\) and the Ohm's Law equation \(I = \frac{E}{R}\) gives the equation: \[P = \frac{ E^2 }{ R\ \ }\]

Rearranging to solve for \(E\):

\[E = \sqrt{ P \times R }\]

Using the values in this question:

\begin{align} E &= \sqrt{ 1200\text{ W} \times 50\ \Omega }\\ &= \sqrt{ 60000 }\text{ V}\\ &= 100\sqrt{ 6 }\text{ V}\\ &= 245\text{ V} \end{align}

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Alternate Hint: \(E= \sqrt{P \times R}\). \(1200 \times 50 = 60,000\); (\(\sqrt{60,000} = 245\))

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Silly Hint: The correct answer is the only one that has both a 2 (as in 1200 watts) and a 5 (as in 50 ohms).
Or Think about a bathroom heater "240 Volts"

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1060 watts is the output PEP (Peak Envelope Power) of an unmodulated carrier if an average reading wattmeter connected to the transmitter output indicates 1060. The average power and the PEP are the same value for an unmodulated carrier, so the values are the same.

Silly hint: 1060 is both in the question and answer

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625 watts is the output PEP from a transmitter if an oscilloscope measures 500 volts peak-to-peak across a 50-ohm resistor connected to the transmitter output.

Use the combined power and Ohm's law equation:

\[P = \frac{E^2}{R}\]

where: \begin{align} P &= \text{Peak Envelope Power}\\ E &= \text{Root Mean Square (RMS) Voltage} \end{align}

First find the \(E_{RMS}\):

\[E_{RMS} = \frac{\text{Peak Envelope Voltage}}{\sqrt{2}}\]

To find the Peak Envelope Voltage (PEV): \begin{align} \text{PEV} &= \frac{\text{Peak-to-Peak Voltage}}{2}\\ &= \frac{500\ \text{V}}{2}\\ &= 250\ \text{V} \end{align}

So now the Peak Envelope Power (PEP) output in this case can be calculated as:

\begin{align} P_{PEP} &= \frac{(E_{RMS})^2}{R}\\ &= \frac{ \Big ( \frac{ 250\ \text{V}}{ \sqrt{2} } \Big ) ^2}{50\ \Omega}\\ &= 625\ \text{W} \end{align}

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