The question is: What is the effective radiated power (ERP) of a repeater with:

\begin{split} 50 \text{ watts}&\text{ transmitter power output}\\ 4\ dB &\text{ feedline loss}\\ 3\ dB &\text{ duplexer and circulator loss}\\ 6\ dB &\text{ antenna gain}\\ \end{split}

Any gain is positive, any loss is negative, so:

\begin{split} ERP &= \text{Power} * (\text{Loss} + \text{Gain})\\ ERP &= 50 \times (-4 -3 +6)dB\\ ERP &= 50 \times (-1)dB \end{split}

For simplicity, remember that 1dB is about 20%, so to get a 1dB loss we can multiply by 80% (100 - 20) which is \(0.8\):

\[50 * (.8) = 40 Watts\]

39.7 is the closest (and more precise) answer. If we use the exact value of \(1dB\) (\(0.794\)) we'd get that exactly, but it's much easier to remember 20% instead of 20.6%

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What is the effective radiated power of a repeater with
75 watts transmitter power output,
 4 dB feedline loss,
 3 dB duplexer and circulator loss, and
10 dB antenna gain?

150 watts.

From kd7bbc:

\(\text{ERP} = P \times (\text{Loss} + \text{Gain})dB\)

\(\text{ERP} = 75 \times (-4 -3 +10)dB\)

\(\text{ERP} = 75 \times (3)dB\)

A \(3 db\) gain is approximately double the power, or \(2\)

\(\text{ERP} = 75 \times 2 = 150 \text{ Watts}\)

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What is the effective radiated power of a repeater with
75 watts transmitter power output,
 5 dB feedline loss,
 4 dB duplexer and circulator loss, and
 6 dB antenna gain?

37.6 watts

From kd7bbc:

\(\text{ERP} = P \times (\text{Loss} + \text{Gain})\)

\(\text{ERP} = 75 \times (-5 -4 +6)\)

\(\text{ERP} = 75 \times (-3)\)

A \(-3 db\) loss cuts power in half or \(0.5\)

\(\text{ERP} = 75 \times 0.5 = 37.5 \text{ Watts}\),
so choose \(37.6 \text{Watts}\)

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What is the effective radiated power of a repeater with
100 watts transmitter power output,
4 dB feedline loss,
3 dB duplexer and circulator loss, and
7 dB antenna gain?

100 watts.

From wp2ahg:

You start with 100 watts and lose 7 dB in the feedline & duplexer/circulator, but you gain those 7 dB back in the antenna gain, so it's a wash.

100 watts with 0 dB net loss/gain = 100 watts.

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What is the effective radiated power of a repeater with
100 watts transmitter power output,
   5 dB feedline loss,
   4 dB duplexer and circulator loss, and
10 dB antenna gain?

126 watts.

From seanst:

\[ERP = P * (Loss + Gain)\]

\[ERP = 100 * (-5 -4 +10)\]

\[ERP = 100 * (+1)\]

A 1 db gain = 1.26 power ratio so

\[ERP = 100 * 1.26 \text{ or } 126 \text{ watts}\]

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What is the effective radiated power of a repeater with
50 watts transmitter power output,
 5 dB feedline loss,
 4 dB duplexer and circulator loss, and
 7 dB antenna gain?

31.5 watts.

From seanst:

ERP = P * (Loss + Gain)

ERP = 50 * (-5 -4 +7)

ERP = 50 * (-2)

Since we have a -2 db loss, the answer must be below 50 watts. There is only one answer so, pick 31.5 watts.

If you're a glutton for punishment, this is the actual math:

To calculate the total power transmitted by the antenna we need to convert the feedline loss, duplexer and circulator loss, and antenna gain from decibels to linear scale using the formula \(10^{(dB/10)}\), which gives us \(10^{(-5/10)} = 0.316\), \(10^{(-4/10)} = 0.398\), and \(10^{(7/10)} = 5.01\) respectively.

Not as tough as it looks - on a TI-30Xa calculator (allowed at almost any exam because it's non-programmable), follow these steps, using the first equation, \(10^{(-5/10)}\) as an example:

1. Press the [5] button.
2. Press the [(+-)] button to make that \(5\) a negative.
3. Press the [÷] button.
4. Press the [1] and [0] buttons to enter the number \(10\).
5. Press the [=] button to calculate the result of \(-5/10\).
6. Press the [2nd] button to access the secondary functions.
7. Press the [LOG] button to calculate \(10^{x}\), where \(x\) is the result of \(-5/10\).

The result should be displayed as \(0.316227766\).

Handle the other two equations the same way.

Then, we can calculate the total power transmitted to the antenna by multiplying the transmitter power output by the feedline loss and duplexer and circulator loss, which gives us \(50 × 0.316 × 0.398 = 6.3\) watts. Finally, we can calculate the ERP by multiplying the total power transmitted by the antenna by the antenna gain, which gives us \(6.3 × 5.01 = 31.56\) watts.

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