# Power vs. Voltage in a 50 Ω System

A lot of people have a hard time converting between power and voltage in 50 Ω systems. The full derivation requires a little extra work because of the use of the root-mean-square value of a sinewave, based on its peak voltage value. Here's how it's done.

The Table of Equivalent Voltages & Powers was generated using an Excel spreadsheet. Equations for the cells are shown at the bottom of the table.

Voltage to Power (for a sinewave) in a 50 Ω System

where v is in volts
Power to Voltage (for a sinewave) in a 50 Ω System

Volts
Table of Equivalent Voltages & Powers in a 50 Ω System
 Vpeak(V) Vrms(V) Vavg(V) P(W) P(mW) P(dBm) 10 7.07107 6.36620 1.00000 1000.00000 30.00000 9 6.36396 5.72958 0.81000 810.00000 29.08485 8 5.65685 5.09296 0.64000 640.00000 28.06180 7 4.94975 4.45634 0.49000 490.00000 26.90196 6 4.24264 3.81972 0.36000 360.00000 25.56303 5 3.53553 3.18310 0.25000 250.00000 23.97940 4 2.82843 2.54648 0.16000 160.00000 22.04120 3.162278 2.23607 2.01317 0.10000 100.00000 20.00000 3 2.12132 1.90986 0.09000 90.00000 19.54243 2 1.41421 1.27324 0.04000 40.00000 16.02060 1 0.70711 0.63662 0.01000 10.00000 10.00000 0.9 0.63640 0.57296 0.00810 8.10000 9.08485 0.8 0.56569 0.50930 0.00640 6.40000 8.06180 0.7 0.49497 0.44563 0.00490 4.90000 6.90196 0.6 0.42426 0.38197 0.00360 3.60000 5.56303 0.5 0.35355 0.31831 0.00250 2.50000 3.97940 0.4 0.28284 0.25465 0.00160 1.60000 2.04120 0.316228 0.22361 0.20132 0.00100 1.00000 0.00000 0.3 0.21213 0.19099 0.00090 0.90000 -0.45757 0.2 0.14142 0.12732 0.00040 0.40000 -3.97940 0.1 0.07071 0.06366 0.00010 0.10000 -10.00000 0.09 0.06364 0.05730 0.00008 0.08100 -10.91515 0.08 0.05657 0.05093 0.00006 0.06400 -11.93820 0.07 0.04950 0.04456 0.00005 0.04900 -13.09804 0.06 0.04243 0.03820 0.00004 0.03600 -14.43697 0.05 0.03536 0.03183 0.00003 0.02500 -16.02060 0.04 0.02828 0.02546 0.00002 0.01600 -17.95880 0.031623 0.02236 0.02013 0.00001 0.01000 -20.00000 0.03 0.02121 0.01910 0.00900 -20.45757 0.02 0.01414 0.01273 0.00400 -23.97940 0.01 0.00707 0.00637 0.00100 -30.00000 0.009 0.00636 0.00573 0.00081 -30.91515 0.008 0.00566 0.00509 0.00064 -31.93820 0.007 0.00495 0.00446 0.00049 -33.09804 0.006 0.00424 0.00382 0.00036 -34.43697 0.005 0.00354 0.00318 0.00025 -36.02060 0.004 0.00283 0.00255 0.00016 -37.95880 0.003162 0.00224 0.00201 0.00010 -40.00000 0.003 0.00212 0.00191 0.00009 -40.45757 0.002 0.00141 0.00127 0.00004 -43.97940 0.001 0.00071 0.00064 0.00001 -50.00000 Formulas Vpk/SQRT(2) Vpk*2/PI() V2pk/(2*50) PW*1000 10*log(PmW)
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