As you said, the given gain is constant, at least we are going to assume it is constant. Then we can ignore the input/output impedance, different gains for the different gain stages because it is the final overall gain that determines the relationship between the input voltage and output voltage at the speaker binding posts.
Edit: I should add the following short version, that is the only totally relevant part of our conversation regarding the required input voltage for an AVR to reach 400 W (obviously this is hypothetical, no AVR can reach 400 W output except for the IHF dynamic power rating thing..)
Short version:
Because of the square relationship between Voltage output and power output, to double the output from say 200 watts to 400 watts, you don't need to double the voltage, but simply multiply it by square root 2, that is,
If 1.7 V is needed to yield 200 W into 8 ohms,
1.7 X square root of 2, that is 1.7X1.414 = 2.4 V is needed to yield 400 W into the same 8 ohm load.
You are right, there is no need to reference the dB to multiple/ratio formula, not for this purpose.
Everything below the dotted line are for someone who has nothing better to do, so I am not going to hit the delete key.
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The relevant formula are:
Reference site -
http://www.sengpielaudio.com/calculatorVoltagePower.htm
Nomenclature:
Vout - Output voltage at speaker binding posts
Pout - Output power in watts, for an 8 ohm resistor load
Vin- Input voltage at the power amplifier input
Vg in dB - Voltage gain in dB
Vg in multiples, or ratio - Voltage gain in multiple, or ration form, that is, Vg = Vout/Vin
Vout = Vin X Vgain (in multiples)
Pout = (Vout^2)/Z
therefore Vout = SQRT(PoutXZ)
The formula to convert from Voltage gain in dB to multiples and vice versa are:
Vg in dB = 20Xlog10(Vg in multiples),where Vg in multiples = Vout/Vin
Vg in multiples = 10^(Vg in dB/20)
First, I am going to show if 1.7 Vin produces 200W, then the Vgain in dB is not 27 but about 27.43, and that's why I mentioned the log relationship.
Since Vout = SQRT(PoutXZ),for an 8 ohm load, and Pout = 200W,
Vout=SQRT(200X8) = SQRT(1600) = 40 V.
Vgain in multiples = Vout/Vin = 40/1.7 = 23.53
Vgain in dB = 20Xlog10(Vout/Vin) = 20Xlog10(23.53) = 27.43
Now, to calculate the input voltage Vin required to produce 400W, keeping impedance the same, i.e. 8 ohms, you are correct that we do not need to worry about whether the gain is in multiples or dB because it is assumed to be a constant.
In this case we simply use the power formula again, that is:
Pout = (Vout^2)/Z,
Vout = SQRT(PoutXZ) = SQRT (400X8) = SQRT(3200) = 56.5685
Vin = Vout/Vgain in multiples = 56.5685/23.53 = 2.4 V.
To summarize:
1) For rated output of 200 W into 8 ohms with Vin = 1.7V, and Voltage gain of 27.43 dB, the output voltage will be 40V.
2) Doubling the output to 400 W, while keeping impedance and voltage gain the same, that is, 8 ohms and 27 dB, Vin will be 2.4 V.
3) Outlaw's specified Voltage gain of 27 dB, 1.7 V sensitivity for full output (200 W 8 ohms) is an approximation. If the sensitivity is actually 1.7V for 200 W into 8 ohms, the gain should be 27.43 dB.
I always develop my own spreadsheet for quick calculations, but the linked site above seems to have all the formula one can think off, complete with calculators that always checked out fine with my spreadsheets.
