# Converting Gain to dB

Discussion in 'Homework Help' started by Maintenance Man, Mar 16, 2011.

1. ### Maintenance Man Thread Starter New Member

May 12, 2010
25
0
I have a question about converting gain in an amplifier (Av, Ai, and Ap) to dB. Is there a formula used for converting. I have an example that shows Av = 1 divided by 0.01 = 100, Ai =10 divided by 1 = 10, and Ap = 100 x 10 = 1,000 then to convert amplifier gain to Decibels I have the example av = 20 log Av = 20 log 100 = 40 dB and ai = 20 log Ai = 20 log 10 = 20 dB and ap = 10 log Ap = 10 log 1000 = 30 dB. I was wondering how they get the 40dB, 20dB, and 30 dB. I have looked up about logarithms on line, but it really is not making sense to me on how they converted from Gain to Decibels. Is there a formula for this? Your help would be greatly appreciated in helping me understand the conversion process. Thank you very much for your help.

Sincerely:
Maintenance man

Apr 17, 2010
790
186
3. ### Papabravo Expert

Feb 24, 2006
10,340
1,850
It's pretty simple:

For Voltage and Current Gain you multiply the logarithm to the base 10 by 20.
Code ( (Unknown Language)):
1.
2. Gv(dB) = 20 * log(Av)
3. Gi(dB) = 20 * log Ai)
4.
Keep in mind that both Av and Ai are ratios, that is one thing divided by another. Expressing things in dB is ALWAYS about ratios.

For Power Gain you multiply the logarithm to the base 10 by 10
Code ( (Unknown Language)):
1.
2. Gp(dB) = 10 * log(Ap)
3.
That's it

Maintenance Man likes this.
4. ### Maintenance Man Thread Starter New Member

May 12, 2010
25
0
I understand when the gain is 10, 100, and 1000 but how do you find the dB when the gain is a number like 2, 3, 4, 20, 30, 40, 55, 56 etc. This is where I am having the trouble is all the other numbers between 10, 100, 1000. Thank you all for your help.

5. ### justtrying Active Member

Mar 9, 2011
329
405
keeping in mind that Av is Vout/Vin (as can be seen in the e-book), the ratio will not always work out to a nice power of 10. If your gain happens to be 1.53, you just plug it into the same formula where Gv(dB)=20logAv=20log(1.53) and using the magic of calculator will get an answer that is not so nice = 3.69dB.

Maintenance Man likes this.
6. ### Maintenance Man Thread Starter New Member

May 12, 2010
25
0
Thank you for your help, but is there a formula to figure Av of 1.53 in which the log of that number would be multiplied by 20 for the dB. I was just looking for the long hand way to do these conversions without a calculator.

7. ### Maintenance Man Thread Starter New Member

May 12, 2010
25
0
I guess the question I really need to ask is how you figure the logarithm (how many times 10 is multiplied to get a certain number). 10 x 10 is 100 in which the log is 2. This is easy to do but how do you find the logarithm of a number such as 50 in which the log would be 1.7 (found this number by dividing 34 by 20 using a conversion web page). I would like to know how you get a log of 1.7 for the number 50. 1.7 would be multiplied by 20 to get the 34 dB. I would like to learn to do this without using a calculator. The key is to find the log of a number then multiply it by 20 to get the dB of a Av. Thank you very much.

Last edited: Mar 17, 2011
8. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
You could memorise a reduced table of logarithms from 1 to 10 in steps of 0.1 - but why bother.

Better to take your pick .....

• Electronic Calculator - a problem when the batteries run out.
• Slide Rule - if you can find one. I had one for many years.
• Table of Logarithms - which is what I used at school.
• Infinite Series approximation - not recommended - you'll probably need a calculator!

Maintenance Man likes this.
9. ### Audioguru New Member

Dec 20, 2007
9,411
896
In my entire audio career I never used logarithms to calculate voltage gain to dB or dB to gain.
I simply remember:
+3db is 1.414 times and -3dB is 0.707 times.
+6db is double and -6dB is half.
+10dB is 3.16 times and -10dB is 0.316 times.
+20dB is 10.0 times and -20dB is 0.1 times.
Then it is easy to use simple arithmatic to find higher values like +70dB is (10 x 10 x 10 x 3.16)= 3160 times.

Maintenance Man likes this.
10. ### Papabravo Expert

Feb 24, 2006
10,340
1,850
An apocryphal tale from my undergraduate days in 1965 was of the poor engineering freshman at U of M who could not afford \$25.00 for a slide rule and reputedly memorized the complete table of four place logarithms. I never met the guy but he was a legend.

Maintenance Man likes this.
11. ### Maintenance Man Thread Starter New Member

May 12, 2010
25
0
Thank you all for your help, I appreciate it a lot. I am going to just use the good old LOG key on my calculator. Hope you all have a great day.

Sincerely:
Maintenance Man

Last edited: Mar 17, 2011
12. ### Audioguru New Member

Dec 20, 2007
9,411
896
Some people are brilliant at doing something but don't have the smarts to tie their shoes.