A calculator is a valuable tool when it comes to circuit design or modification. With it, calculating low pass and high pass filters, amplifier gain, resistors or capacitors in series/parallel becomes the simplest of tasks.
I’m sure everybody can use the basic functions (add, subtract, multiply and divide) of a calculator, so this tutorial will concentrate on the “other” buttons found on a calculator that will make the job of designing or modding easier.
The type of calculator you will need is a scientific calculator, something like this:
If you don’t have such a calculator, you can use the calculator on your computer – it’s basically the same.
All of the formulae used in this tutorial are the ones you will meet in other tutorials. There will be no complex maths or complex formulae involved, no detailed discussion of how we arrive at a formula, and hopefully nothing to frighten the average beer-drinking, woman-chasing, guitar-playing pedal builder away.
Taking a look at the buttons we need:
The EXP – exponential button is one of the most pressed buttons, a quick look at what the exponential button does:
The first two columns should be obvious to everyone who has used resistors (M?, k?), calculated gain (dB) or used capacitors (mF, µF, nF and pF). The third and fourth coulumns are how different calculators display the exponential number and the last column is the “straight” decimal value. One tip, the number of zeros in the decimal value (including the zero before the decimal point) is the same as the exponential number – 4.7µF is 4.7 x 10
-6 F which is 0.0000047F (six zeros),
22k? is 22 x 10
3 ? which is 22000 ? (three zeros).
Where does all this super techie information get us?
When we come to calculate high and low pass filters, part of the formula requires us to muliply a resistance value (in Ohms) by a capacitance value (in Farads) for example:
220 k? x 15nF
I suppose we could enter 220000 x 0.0000000015 in the calculator, but that's a lot of work to make sure we have hit the zero button the correct amount of times. Using the EXP button and the + / - button:
we can make life a lot easier. The + / - button converts a positive number (or in this case the exponential) to a negative number. Try entering the following for the above example. Normal numbers are in black, standard operation buttons (+, -, x, ÷, =) are in blue and other operators are in red.
220
EXP 3
x 15
EXP 9
+/- =The result should be either 0.0033 or 3.3e-3, depending on how your calculator displays the results. You can change the reading from 0.0033 to 3.3e-3 by hitting the
ENG button, or the
F-E button in the Windows calculator.
Hopefully you can see how this works.
220
EXP 3 (which is 220 x 10
3 or 220k)
15
EXP 9
+/- (which is 15 x 10
-9 or 15n)
The only thing you have to remember is the correct exponential value:
M is 6
k is 3
m is -3
µ is -6
n is - 9
p is -12
(notice how everything moves in threes)
Moving on to the next button:
The reciprocal or "inverting" button gives the reciprocal value of whatever is displayed by the calculator
Simply put, it turns the value upside down:
Punch in the number 2 and hit the reciprocal button and the answer is 0.5
(2, which is the same as 2/1 is inverted to give 1/2, which is 0.5)
Punch in the number 0.4 and hit the reciprocal button and the answer is 2.5
(0.4, which is the same as 0.4/1 is inverted to give 1/0.4, which is 2.5)
The next button is the ? or "pi" button:
? is a mathematical constant (the ratio of a circle's circumference to it's diameter) that is used in many forumulae. It's exact value can never be calculated so a calculator will display ? with something between seven and twenty decimal places which is a bit of over-kill. A value of 3.1416 for ? is good enough for our calculations.
With the buttons already discussed, we can move on to what will probably be the hardest formula you will come across in this forum - high and low pass filters. As this is a tutorial on calculators, I don't propose to go into the how and why of filters (this will be done in another tutorial), it is suffcient to say that the cut-off frequency for a simple filter is calculated by:
f = 1 / (2 x ? x R x C)
f is the frequency in Hz
R is the resistance in ?
C is the capacitance in Farads
An example:
What is the cut-off frequency of the op amp input ?
f = 1 / (2 x ? x 470 k? x 1nF).
Doing the part in brackets first:
2
x ? x 470
EXP 3
x 1
EXP 9
+/- = (at this point your calculator should read 2.9530971e-3 or 0.002953097)
now hit the reciprocal (
1/x) button and the answer should be:
338,627 which is the answer; 339 Hz - to the nearest whole number.
You can change the formula around to find either the value of the capacitor or resistor for a given cut-off frequency. Just move whatever is to the left of the = sign into the bracket and the value you are interested in from the bracket to the left of the = sign:
C = 1 / (2 x ? x f x R)
R = 1 / (2 x ? x f x C)
For the above diagram, the input resistance stays at 470 k?, what capacitor value makes the cut-off frequency 80 Hz ?
C = 1 / (2 x ? x 80 Hz x 470 k?)
Doing the part in brackets first:
2
x ? x 80
x 470
EXP 3
= (at this point your calculator should read 236247767.55 or 2,3624776755e+8)
now hit the reciprocal (
1/x) button and the answer should be:
4.232844231e-9 which is about 4.23e-9 (remember e-9 is nano), so the answer is:
4.23 nF or 4.7nF, which is the nearest standard value.
That concludes the hardest bit of maths and formuale you will probably ever use.
The next part deals with converting "straight" gain into dB (deciBel) and vice versa. Why you would want to / need to convert gain into dB and logarithms are covered in another tutorial, here we deal with the "how to convert" part.
Since we are primarily concerned with voltage gain in our circuits, we can define the gain of an amplifier as the output voltage divided by the input voltage: if the input signal to an amplifier is 15mV (0.1V) and the output signal is 1.6V, the gain of the amplifier is:
1.6V / 15mV = 106.7
If the input signal to a passive tone control is 600mV and the output signal is 200mV, the gain of the tone control circuit is:
0.2V / 0.6V = 0.33. Obviously this isn't a gain because any value under 1 (unity gain) must be considered a loss. I'm only getting a third of the input signal at the output.
To convert the above examples into dB we need the formula:
Gain in dB = 20 log (Output / Input)
First the bracket calculation:
1.6
÷ 15
EXP 3
+/- = (at this point your calculator reading should be 106.67)
Then the logarithim
log (at this point your calculator reading should be 2.028)
Then multiply by 20
x 20 The answer is 40.56dB
In the calculation above, I could have used 1.6
÷ 0.015, but I like using the
EXP button.
The second example:
First the bracket calculation:
0.2
÷ 0.6
= (at this point your calculator reading should be 0.33)
Then the logarithim
log (at this point your calculator reading should be -0.4771)
Then multiply by 20
x 20 The answer is -9.54dB
While dB gain is a positive number, a negative number means there is a loss.
I could, of course, calculate the reverse i.e. from dB gain to normal gain by changing the formula a bit:
Gain = antilog (dB/20)
There are different methods for locating the antilog button on different calculators. On my calculator you would have to push the
SHIFT button and then the
LOG button, on other calculators there is a
10x button, on the Windows calculator you would first have to tick the
INV box and then push the
LOG button. I'll do an example using the Windows calculator:
tonmann's treble titilator has the following frequency response:
12dB at 100 Hz, 35dB at 1 kHz, -3dB at 5 kHz
First the bracket calculation:
12
÷ 20 = (at this point your calculator reading should be 0.6)
Then the antilogarithm:
Inv log (at this point your calculator reading should be 3.98)
At 100 Hz there is a gain of about 4.
First the bracket calculation:
35
÷ 20
= (at this point your calculator reading should be 1.75)
Then the antilogarithm:
Inv log (at this point your calculator reading should be 56.23)
At 1 kHz there is a gain of about 56.
First the bracket calculation:
3
+/- ÷ 20
= (at this point your calculator reading should be -0.15)
Then the antilogarithm:
Inv log (at this point your calculator reading should be 0.708)
At 5 kHz there is a loss of about 30% as I'm only getting 70% of my signal.
The rest of this tutorial concerns the use of brackets in calculations.
Consider the simple formula for two resistors in parallel, (or two capacitors in series):
R
T = R1 x R2 / R1 + R2
If you didn't know that this formula is the product of the resistors divided by the sum of the resistors you could have problems interpreting what the formula is trying to tell you to do. Do you multiply R1 by R2, divide this answer by R1 and then add R2 ?
Do you multiply R2 by R1 and divide the answer by R1 plus R2 ? If you use brackets the meaning becomes clearer:
R
T = R1 x R2 / (R1 + R2)
Now you can see that it is R1 multiplied by R2 (the product) divided by R1 + R2 (the sum). As an example we'll calculate the total resistance of a 100 ? resistor in parallel with a 500 ? resistor.
First without brackets:
R
T = 100 x 500 / 100 + 500 = 1000. This can't be correct as the total resistance can't be larger than the smallest resistance. The calculator has multiplied the first number by the second number, divided this answer by the third number and then added the fourth number.
Using brackets:
100 x 500 ( 100 + 500 ) = 83.3
This looks much better - here you are telling the calculator to multiply the first number by the second number and divide the answer by the sum of the third and fourth number.
Another example is when we have, for example, three resistors connected in series with a battery connected across the resistors and we need to calculate the voltage across one of the resistors (let's use the second resistor, R2, as an example):
The voltage across R2 (V
R2) is equal to the power supply voltage (V
S) multiplied by the resistance of the resistor under examination (R2) divided by the total resistance (R1 + R2 + R3).
V
R2 = V
S x R2 / R1 + R2 + R3
If you punch the above formula into your calculator you will get an incorrect answer; if you use brackets:
V
R2 = V
S x R2 / (R1 + R2 + R3 )
The answer will be correct.
This is the end of the tutorial on effective use of a calculator. If there are any questions or problems, I will be happy to try and answer them publicly in "Beginner Questions" or privately via PM
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