## Usefull Equations

### Usefull Equations

Ok, as requested a place to put things of use. The one that suggested the thread is how to calculate a voltage difference as a decibel value.

Vdecibel = 20log(v1/v2)

or to help the calculation and a sort of RP form

((v1/v2) log ) * 20

This is for voltage, for power change the 20 to a 10.

To go the other way, to get a voltage ratio from a dB value the inverse is

VRatio = 10 ^ ( VDecibel / 20 )

So to help with that (and how to do it using the windows calculator) to convert a -23dB (for example value to a ratio)

( -23 / 20 )

Press = to get the result of that, then check the Inv (for inverse) box and click the Log button. That gives the ratio, so the ratio of

1v to 0.0707v

Just to add to this, calculate the ration of -20dB

-20dB = 1 : 0.1

Then calculate the ratio of -3dB

-3dB = -0.707

And notice how adding the log value

-20 + -3 = -23

gives the same result as multiplying the voltage ratio

0.1 * 0.707 = 0.0707

You can always do that with decibels (and any logarithmic value).

So if you have a amplifying stage that has a gain of 10dB followed by a stage that has a gain of 5dB, the gain of the two combined is simply 15dB

Vdecibel = 20log(v1/v2)

or to help the calculation and a sort of RP form

((v1/v2) log ) * 20

This is for voltage, for power change the 20 to a 10.

To go the other way, to get a voltage ratio from a dB value the inverse is

VRatio = 10 ^ ( VDecibel / 20 )

So to help with that (and how to do it using the windows calculator) to convert a -23dB (for example value to a ratio)

( -23 / 20 )

Press = to get the result of that, then check the Inv (for inverse) box and click the Log button. That gives the ratio, so the ratio of

1v to 0.0707v

Just to add to this, calculate the ration of -20dB

-20dB = 1 : 0.1

Then calculate the ratio of -3dB

-3dB = -0.707

And notice how adding the log value

-20 + -3 = -23

gives the same result as multiplying the voltage ratio

0.1 * 0.707 = 0.0707

You can always do that with decibels (and any logarithmic value).

So if you have a amplifying stage that has a gain of 10dB followed by a stage that has a gain of 5dB, the gain of the two combined is simply 15dB

Last edited by Nick on Sat Jun 19, 2010 12:05 pm, edited 1 time in total.

Resistance isn't futile it's V / I.

is this the right place to put this? :

http://www.sengpielaudio.com/calculator-db-volt.htm

it's the one I mentioned in the thd discussion

http://www.sengpielaudio.com/calculator-db-volt.htm

it's the one I mentioned in the thd discussion

There's nowhere you can be that isn't where you're meant to be

- Dave the bass
- Amstrad Tower of Power
**Posts:**10494**Joined:**Tue May 22, 2007 4:36 pm**Location:**NW Kent, Darn Sarf innit.

Adding to this thread of Formula's and equations, I found most of these on the 'Choke loading Anodes' thread ( http://www.audio-talk.co.uk/phpBB2/view ... sc&start=0 ). If any are wrong let me know and I'll edit the post if thats OK.

Ohms law.

V= IR

I= V/R

R= V/I

To calculate power in Watts

VI = W

V2/R = W

I2R = W

The -3dB point of a LR filter is

f-3dB = R / L * 2 * pi

The relationships between gm, u and ra.

gm = Âµ / ra

Âµ = gm * ra

ra = Âµ / gm

Gain from Âµ is:

G = Âµ * (RL / (Ri + RL))

And from gm is:

G = (gm * Ri * RL) / (1000000 * (Ri + RL))

Where 'gm' is written in Âµmhos (specifically) and Ri is internal anode resistance (Ohms).

WARNING gm/mu and Ri change with DC bias conditions so really need the average characteristics graph that plots these against anode current.

To calculate cathode resistance.

rk = ( ra + Rl ) / ( 1 + Âµ )

Rearanging this to give C for a target f-3

C = 1 / ( 2 * Pi * R * f-3 )

Where f-3 is in hertz, R is in ohm, Pi is 3.1415 etc and C is in Farad.

The equation for the -3dB point of an RC filter

f-3 = 1 / ( 2 * Pi * R * C )

The formula for 2 resistors in parallel is

1/r+1/r=1/R

or

r1xr2/r1+r2 = R

To calculate Miller Capacitance.

Cmiller = (A + 1) x Cag

Cag is the capacitance from anode to grid from the datasheet.

RL formula

f-3 = R / ( 2 * Pi * L )

Where in this case R = ra, and L is the size of the choke in Henry.

Understanding Interstage Ratio's.

"The interstage ratio 1:1 1:1.5 and so on is the turns ratio of the transformer, and so the voltage ratio.

Assuming no losses, if you pu a 10v RMS sine wave across the primary of a 1:1 interstage, you would expect 10v RMS on the secondary, If you did the same with a 1:1.5 you would expect 15v on the secondary. And so on.

As always the reflected impedance ratio of a transformer is the square of the voltage ratio.

So if you put a 24k resistor on the secondary, and drive the primary with a triode, that triode will see a 24k load. Likewise if the anode resistance of a triode is 900R, then the load sees its being driven by 900R, its reflected both ways.

If you do the same with a 1:1.5 transformer, the impedance is transformed by 1: 1.5*1.5 or 1:2.25, so 24k on the secondary would be reduced by the ratio to 10k6, likewise the 900R anode resistance will be seen by the load as 2k."

"Summary ~

Turns ratio = Volts ratio.

Impedance ratio = turns ratio ^2.

Turns ratio = square root of impedance ratio.

When you double up two identical windings in series, you don't get the Henries of one x 2, it's actually x 4, because it's the turns ratio that is increased x 2.

Also, current is affected, 2:1 turns ratio gets you double the current on the secondary, 1:2 gets you half the current. The way to think about that is that always the power is the same."

"OK equations ~

impedance ratio:

IR = primary impedance / secondary impedance.

turns ratio:

TR = square root of IR above.

Other way round:

If turns ratio is known then:

IR = TR^2 (to the power of 2, or times itself once. e.g. 2:1 would be 2 * 2, = 4.

If stepping down, divide the primary impedance by 4, to get the secondary; if stepping up, multiply by 4.

So if you had a 2:1, and you wanted the primary to have an impedance of 100k, by loading the secondary, you would have to put 25k on it, not 50k.

Yes it took me a while to get my head round it as well

Real example ~ 5k o/p transformer for 8 Ohms speaker

5000 / 8 = 625

That's the impedance ratio.

From that you can see 8 x 625 gets 5,000

Square root of 625 = 25. Now we know what the turns ratio is. 25:1

From that you can work out what the anode Volts needs to be for a certain speaker Volts, and vice versa."

Useful REF thread for learning about loadlines and O.P.'s etc http://www.audio-talk.co.uk/phpBB2/view ... es&start=0

And another from a guitar amp perspective. http://www.freewebs.com/valvewizard1/Co ... _Stage.pdf

Ohms law.

V= IR

I= V/R

R= V/I

To calculate power in Watts

VI = W

V2/R = W

I2R = W

The -3dB point of a LR filter is

f-3dB = R / L * 2 * pi

The relationships between gm, u and ra.

gm = Âµ / ra

Âµ = gm * ra

ra = Âµ / gm

Gain from Âµ is:

G = Âµ * (RL / (Ri + RL))

And from gm is:

G = (gm * Ri * RL) / (1000000 * (Ri + RL))

Where 'gm' is written in Âµmhos (specifically) and Ri is internal anode resistance (Ohms).

WARNING gm/mu and Ri change with DC bias conditions so really need the average characteristics graph that plots these against anode current.

To calculate cathode resistance.

rk = ( ra + Rl ) / ( 1 + Âµ )

Rearanging this to give C for a target f-3

C = 1 / ( 2 * Pi * R * f-3 )

Where f-3 is in hertz, R is in ohm, Pi is 3.1415 etc and C is in Farad.

The equation for the -3dB point of an RC filter

f-3 = 1 / ( 2 * Pi * R * C )

The formula for 2 resistors in parallel is

1/r+1/r=1/R

or

r1xr2/r1+r2 = R

To calculate Miller Capacitance.

Cmiller = (A + 1) x Cag

Cag is the capacitance from anode to grid from the datasheet.

RL formula

f-3 = R / ( 2 * Pi * L )

Where in this case R = ra, and L is the size of the choke in Henry.

Understanding Interstage Ratio's.

"The interstage ratio 1:1 1:1.5 and so on is the turns ratio of the transformer, and so the voltage ratio.

Assuming no losses, if you pu a 10v RMS sine wave across the primary of a 1:1 interstage, you would expect 10v RMS on the secondary, If you did the same with a 1:1.5 you would expect 15v on the secondary. And so on.

As always the reflected impedance ratio of a transformer is the square of the voltage ratio.

So if you put a 24k resistor on the secondary, and drive the primary with a triode, that triode will see a 24k load. Likewise if the anode resistance of a triode is 900R, then the load sees its being driven by 900R, its reflected both ways.

If you do the same with a 1:1.5 transformer, the impedance is transformed by 1: 1.5*1.5 or 1:2.25, so 24k on the secondary would be reduced by the ratio to 10k6, likewise the 900R anode resistance will be seen by the load as 2k."

"Summary ~

Turns ratio = Volts ratio.

Impedance ratio = turns ratio ^2.

Turns ratio = square root of impedance ratio.

When you double up two identical windings in series, you don't get the Henries of one x 2, it's actually x 4, because it's the turns ratio that is increased x 2.

Also, current is affected, 2:1 turns ratio gets you double the current on the secondary, 1:2 gets you half the current. The way to think about that is that always the power is the same."

"OK equations ~

impedance ratio:

IR = primary impedance / secondary impedance.

turns ratio:

TR = square root of IR above.

Other way round:

If turns ratio is known then:

IR = TR^2 (to the power of 2, or times itself once. e.g. 2:1 would be 2 * 2, = 4.

If stepping down, divide the primary impedance by 4, to get the secondary; if stepping up, multiply by 4.

So if you had a 2:1, and you wanted the primary to have an impedance of 100k, by loading the secondary, you would have to put 25k on it, not 50k.

Yes it took me a while to get my head round it as well

Real example ~ 5k o/p transformer for 8 Ohms speaker

5000 / 8 = 625

That's the impedance ratio.

From that you can see 8 x 625 gets 5,000

Square root of 625 = 25. Now we know what the turns ratio is. 25:1

From that you can work out what the anode Volts needs to be for a certain speaker Volts, and vice versa."

Useful REF thread for learning about loadlines and O.P.'s etc http://www.audio-talk.co.uk/phpBB2/view ... es&start=0

And another from a guitar amp perspective. http://www.freewebs.com/valvewizard1/Co ... _Stage.pdf

Last edited by Dave the bass on Tue Dec 28, 2010 10:19 pm, edited 3 times in total.

"'Occasionally phenomenal'"

- Mike H
- Amstrad Tower of Power
**Posts:**17429**Joined:**Sat Oct 04, 2008 5:38 pm**Location:**The Fens-
**Contact:**

and 2 * Pi * f may be written as a lowercase omega (a 'squiggly' w)

To add to that, Volts from power across Ohms is:

V = square root of (W * R)

Â

To add to that, Volts from power across Ohms is:

V = square root of (W * R)

Â

Last edited by Mike H on Tue Aug 24, 2010 1:33 pm, edited 1 time in total.

"The beer was so flat it could have been served in an envelope...."

had a look through and couldn't find any previous reference to this: its the formula xls that I'm referring to....maybe it should have a heading(thread) of its own.....

http://www.vt52.com/diy/tips/tips_software.htm

I found it in a very old 2a3 post by James D on diyaudio...where is James D????anybody heard from him???

http://www.vt52.com/diy/tips/tips_software.htm

I found it in a very old 2a3 post by James D on diyaudio...where is James D????anybody heard from him???

There's nowhere you can be that isn't where you're meant to be

- Mike H
- Amstrad Tower of Power
**Posts:**17429**Joined:**Sat Oct 04, 2008 5:38 pm**Location:**The Fens-
**Contact:**

Resonant Frequency of a LC tuned circuit is:

freq. = 1 / (2 * pi * sqrt(L * C))

L in Henries, C in Farads, freq. in Hertz

To get the value of C or L from the other known value of C or L and the frequency:

omega = freq. x (pi * 2)

num = 1 / (omega ^ 2)

then:

L = num / C

or:

C = num / L

' Self tuning' tuned circuit for LTspice ~ adjusts value of C or L to given frequency and value of known C or L:

L1 1 2 {Lval} Rser=1

C1 1 2 {Cval}

.param Freq = [whatever] ; Hertz

.param Cval = [whatever] ; Farads

.param omega = {F * (pi * 2)}

.param num = {1 / (omega ** 2)}

.param Lval = {num / Cval}

Or:

.param Freq = [whatever] ; Hertz

.param Lval = [whatever] ; Henries

.param omega = {F * (pi * 2)}

.param num = {1 / (omega ** 2)}

.param Cval = {num / Lval}

freq. = 1 / (2 * pi * sqrt(L * C))

L in Henries, C in Farads, freq. in Hertz

To get the value of C or L from the other known value of C or L and the frequency:

omega = freq. x (pi * 2)

num = 1 / (omega ^ 2)

then:

L = num / C

or:

C = num / L

' Self tuning' tuned circuit for LTspice ~ adjusts value of C or L to given frequency and value of known C or L:

L1 1 2 {Lval} Rser=1

C1 1 2 {Cval}

.param Freq = [whatever] ; Hertz

.param Cval = [whatever] ; Farads

.param omega = {F * (pi * 2)}

.param num = {1 / (omega ** 2)}

.param Lval = {num / Cval}

Or:

.param Freq = [whatever] ; Hertz

.param Lval = [whatever] ; Henries

.param omega = {F * (pi * 2)}

.param num = {1 / (omega ** 2)}

.param Cval = {num / Lval}

"The beer was so flat it could have been served in an envelope...."

Just spotted this, its not correct, the angula frequency w isMike H wrote:and 2 * Pi may be written as a lowercase omega (a 'squiggly' w)

Â

w = 2 * pi * f

Where f is in Hertz.

http://en.wikipedia.org/wiki/Angular_frequency

Resistance isn't futile it's V / I.

- Dave the bass
- Amstrad Tower of Power
**Posts:**10494**Joined:**Tue May 22, 2007 4:36 pm**Location:**NW Kent, Darn Sarf innit.

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