Question E9A02:
What is the effective radiated power (ERP) of a repeater station with 150 watts transmitter power output, 2 dB feed line loss, 2.2 dB duplexer loss, and 7 dBd antenna gain?  

Choices:  
A: 469 watts  
B: 78.7 watts  
C: 420 watts  
D: 286 watts    

Correct Answer is  D: 286 watts  

To find ERP, add gains and subtract losses in dB, then convert the net dB to a linear power multiplier and apply it to the transmitter power.

Step-by-step:
- Net gain in dB:
  $$G_{\text{net}} = 7\ \text{dBd} - 2.0\ \text{dB} - 2.2\ \text{dB} = 2.8\ \text{dB}$$
- Convert 2.8 dB to a linear factor:
  $$\text{Factor} = 10^{(2.8/10)} \approx 10^{0.28} \approx 1.905$$
- Compute ERP:
  $$\text{ERP} = 150\ \text{W} \times 1.905 \approx 286\ \text{W}$$

Why 286 W (D) is correct:
- It correctly uses 7 dBd (relative to a dipole) for ERP, subtracts the 2 dB and 2.2 dB losses, and converts the net 2.8 dB to a linear multiplier.

Why the other choices are incorrect:
- 469 W (A): Assumes EIRP by adding 2.15 dB (converting to dBi), giving net 4.95 dB: $150 \times 10^{0.495} \approx 469$ W. That’s EIRP, not ERP.
- 78.7 W (B): Uses the wrong sign for gain/loss, effectively applying −2.8 dB: $150 \times 10^{-0.28} \approx 78.7$ W.
- 420 W (C): Treats 2.8 dB as a linear factor of 2.8 instead of converting from dB: $150 \times 2.8 = 420$ W.

Useful formulas and tips:
- ERP in watts:
  $$\text{ERP} = P_T \times 10^{\left(\frac{G_{\text{dBd}} - \sum L_{\text{dB}}}{10}\right)}$$
- dB arithmetic: add/subtract dB for gains/losses; convert to linear only at the end.
- Quick conversions: 3 dB ≈ ×2; 7 dB ≈ ×5.0; 2 dB loss ≈ ×0.63; 2.2 dB loss ≈ ×0.60.
- ERP uses dBd; EIRP uses dBi with $dBi = dBd + 2.15$.