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feat: add analog filters

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max_richter 2022-03-23 16:31:29 +01:00
parent 808b522887
commit 6636590329
28 changed files with 1376 additions and 3 deletions
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## Positive Clamper
The output voltage is equal to the $V_{supply} + V_{capacitor}$
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
v 160 288 160 144 0 1 40 5 0 0 0.5
w 272 288 160 288 0
p 432 144 432 288 1 0 0
c 160 144 272 144 0 0.0005 -4.194136122318551 0.001
d 272 288 272 144 2 default
r 304 144 304 288 0 1000
w 272 288 304 288 0
w 272 144 304 144 0
w 304 144 432 144 0
w 304 288 432 288 0
o 0 64 0 151554 10 0.1 0 2 0 3 Power\sSupply
o 2 64 0 151554 10 0.1 1 1
```
## Negative Clamper
```circuitjs
$ 1 0.000005 9.78399845368213 50 5 43 5e-11
v 160 288 160 144 0 1 40 5 0 0 0.5
w 272 288 160 288 0
p 432 144 432 288 1 0 0
c 160 144 272 144 0 0.000049999999999999996 4.284982209935943 0.001
d 272 144 272 288 2 default
r 304 144 304 288 0 10000
w 272 288 304 288 0
w 272 144 304 144 0
w 304 144 432 144 0
w 304 288 432 288 0
o 0 64 0 151554 10 0.1 0 2 0 3 Power\sSupply
o 2 64 0 151554 10 0.1 1 1
```
## Biased Clamper
When adding a DC PowerSupply in series with the diode we can shift the amplitude of the output signal up or down.
```circuitjs
$ 1 0.000005 9.78399845368213 50 5 43 5e-11
v 160 288 160 144 0 1 40 5 0 0 0.5
w 272 288 160 288 0
p 432 144 432 288 1 0 0
c 160 144 272 144 0 0.000049999999999999996 -7.675443954087598 0.001
d 272 224 272 144 2 default
r 304 144 304 288 0 10000
w 272 288 304 288 0
w 272 144 304 144 0
w 304 144 432 144 0
w 304 288 432 288 0
v 272 288 272 224 0 0 40 1 0 0 0.5
o 0 64 0 151554 10 0.1 0 2 0 3 Power\sSupply
o 2 64 0 151554 20 0.1 1 1
```

42
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@ -0,0 +1,42 @@
## Diode Clipper
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
v 160 288 160 144 0 1 40 5 0 0 0.5
r 160 144 272 144 0 110
w 240 288 160 288 0
O 272 144 352 144 0 0
w 240 176 272 176 0
w 272 176 304 176 0
w 240 288 304 288 0
w 272 176 272 144 0
d 240 176 240 288 2 default
d 304 288 304 176 2 default
o 0 64 0 151554 10 0.1 0 2 0 3 Power\sSupply
o 3 64 0 159747 0.6389325675309103 0.0001 1 1
```
## Diode Clipper with Bias
If we want to clip the $V_{min}$ and $V_{max}$ to specific voltages we connect DC powersupplies in series with the diodes.
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
v 160 288 160 144 0 1 40 5 0 0 0.5
r 160 144 272 144 0 110
w 240 288 160 288 0
O 272 144 352 144 0 0
w 240 176 272 176 0
w 272 176 304 176 0
w 240 288 304 288 0
w 272 176 272 144 0
d 240 176 240 240 2 default
d 304 240 304 176 2 default
v 240 288 240 240 0 0 40 1 0 0 0.5
v 304 240 304 288 0 0 40 2 0 0 0.5
x 318 214 346 217 4 12 0.6Vf
x 325 267 339 270 4 12 2V
x 201 267 215 270 4 12 1V
x 194 214 222 217 4 12 0.6Vf
o 0 64 0 151554 10 0.1 0 2 0 3 Power\sSupply
o 3 64 0 12546 2.607867075802738 0.0001 1 1
```

23
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@ -0,0 +1,23 @@
The capacitor discharges in around 5RC, when we have an ac signal of 40Hz, and we want to match the frequency, we can calculate the following:
$$
\begin{flalign}
&\tau = RC &\\\
&f = 40Hz = \frac{1}{40}ds \\
&\textit{Lets put the resistor to 1k}\ohm\\
&\frac{1}{80} = 1000*C & | \div 1000 \\
&\frac{1}{80000} = C\\
&C = 12.5 \micro F\\
\end{flalign}
$$
```circuitjs
$ 1 0.000005 9.78399845368213 50 5 43 5e-11
R 80 224 0 224 0 2 40 5 0 0 0.5
c 80 224 192 224 0 0.00001 -1.936025478465976 0.001
r 192 224 192 304 0 1000
g 192 304 192 336 0 0
O 192 224 304 224 0 0
o 4 64 0 151554 10 0.1 0 1
o 0 64 0 151554 10 0.1 0 2 0 3
```
![[spike-circuit-rc-relation.png]]

5
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@ -0,0 +1,5 @@
A band pass filter only allows a certain range of frequencies. This range is sometimes called the *bandwidth*.
It is constructed by combining a high pass and low pass filter in parallel
![[rc-band-pass-example.png]]

15
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@ -0,0 +1,15 @@
High pass filters let through only signals with high frequencies and attenuate low frequency one.
Because high pass filters work exactly like low pass filters but in reverse, lets only do one example here:
**Example:**
![[rc-high-pass-example.png|300]]
Lets first calculate the cutoff frequency of this filter:
![[formulas#Cutoff Frequency for RC Filters]]
$\displaystyle f_{c} = \frac{1}{2\pi 100 * 0.00000001}$
$\displaystyle f_{c} = 159154.94 \approx 159.1kHz$

65
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@ -0,0 +1,65 @@
A low pass filter eliminates high frequency signals from an input signal.
There are two main types of passive low pass filters, a **RC LowPass Filter**
Is made from a capacitor and a resistor while a **RL LowPass Filter**
Is made from an inductor and a resistor.
![[rc-rl-lp-filter.png]]
We can also see that for the RL LowPass Filter the positions of the resistor and the reactive component are switched. This is because the inductor works in the opposite way of the capacitor, allowing low frequencies to pass and attenuating high ones.
**Example:**
Lets design a RC LowPass Filter with a [[glossary#Cutoff Frequency|Cutoff Frequency]] of $15.9kHz$. The Formular for calculating the cutoff frequency is the following:
![[formulas#Cutoff Frequency for RC LowPass]]
So, we now have the following formula:
$\displaystyle 15900 = \frac{1}{2\pi RC}$
Now we still have two unknown variables to fill in, for my personal taste that is one variable to much. Usually when designing a low pass filter we first choose the value of $R$.
We do not want $R$ to be too high, as that will limit the current flow. We also do not want it to be *too* low because then we would need a big capacitor. For now we will choose:
$\displaystyle R =1k\ohm$
With this our formula now looks something like this:
$\displaystyle 15900 = \frac{1}{2*\pi*1000*C}$
Now we only have one unknown variable, which is $C$ so we can solve the equation:
$$
\begin{flalign}
&15900 = \frac{1}{2*\pi*1000*C}& |& \textit{ lets simplify}\\\
&15900 = \frac{1}{2000*pi*C} & |&* 2000*pi*C \\
&31800000\pi C = 1 & |& \div318000000\pi \\
&C = \frac{1}{31800000\pi}\\
\\
&C = 0.00000001F = \textbf{10nF}
\end{flalign}
$$
Lets put those numbers into the simulator:
```circuitjs
$ 64 1e-8 30.13683688681966 50 5 50 5e-11
r 112 256 224 256 0 1000
c 224 256 224 368 0 1e-8 6.664641465475588 0.001
g 224 368 224 400 0 0
O 224 256 336 256 0 0
170 112 256 32 256 3 5000 44100 20 0.001
403 32 288 160 352 0 4_512_0_x8741c_0.0001_0.0001_-1_1_0.000049999999999999996_0
o 4 64 0 28686 19.99999999596653 0.0001 0 2 3 0
```
Or we can plot it
<iframe src="https://www.desmos.com/calculator/pesoftpl4g?embed" width="500" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>
# Second Order Low Pass Filter
If we place two LPF's in series

43
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@ -0,0 +1,43 @@
# Rectifiers
## Full Bridge Rectifier
```circuitjs
$ 1 0.000005 7.619785657297057 50 5 43 5e-11
v 64 240 64 144 0 1 40 230 0 0 0.5
d 208 192 256 144 2 default
d 256 144 304 192 2 default
d 208 192 256 240 2 default
d 256 240 304 192 2 default
T 112 176 160 176 0 4 0.02 0.0676206503358299 5.580371162661191e-7 0.999
w 160 144 256 144 0
w 160 240 256 240 0
w 304 192 368 192 0
w 304 304 208 304 0
w 208 304 208 192 0
O 368 192 464 192 0 0
w 64 144 112 144 0
w 64 240 112 240 0
r 368 192 368 304 0 100
c 304 192 304 304 0 0.0005 3.1394889710736953 0.001
w 368 304 304 304 0
w 160 208 160 240 0
w 160 176 160 144 0
w 112 144 112 176 0
w 112 208 112 240 0
o 14 64 0 152586 5 0.1 0 2 14 3
```
## Half Wave Rectifier
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
v 128 144 128 288 0 1 40 5 0 0 0.5
d 272 288 272 144 2 default
r 128 144 272 144 0 100
w 272 288 128 288 0
O 272 144 352 144 0 0
o 4 64 0 151554 5 0.1 0 1
o 0 64 0 151554 5 0.1 1 2 0 3
```

46
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@ -0,0 +1,46 @@
## Voltage Doubler
```circuitjs
$ 1 0.000005 3.3115451958692312 50 5 43 5e-11
v 0 192 0 272 0 1 40 5 0 0 0.5
209 208 192 208 272 0 0.00009999999999999999 4.421367380475472 1 1
209 208 272 208 368 0 0.00009999999999999999 4.343088198858901 1 1
d 128 192 208 192 2 default
d 208 368 128 368 2 default
w 128 368 128 192 0
w 0 192 128 192 0
w 0 272 208 272 0
r 288 192 288 368 0 10000
p 320 192 320 368 1 0 0
w 288 368 320 368 0
w 288 192 320 192 0
w 208 192 288 192 0
w 208 368 288 368 0
o 0 64 0 151554 10 0.1 0 2 0 3
o 9 64 0 151554 10 0.1 1 1
```
## Voltage Trippler
Basically the same, but with **three** capacitors:
```circuitjs
$ 1 0.000005 10.20027730826997 53 15 45 5e-11
d 208 160 208 256 2 default
d 208 256 288 256 2 default
w 288 256 288 160 0
c 208 160 288 160 0 0.00009999999999999999 -28.719722428729654 0.001
c 208 256 208 336 0 0.00009999999999999999 14.300474725108993 0.001
d 288 160 368 160 2 default
c 368 160 368 336 0 0.00009999999999999999 42.98991494048201 0.001
w 368 160 432 160 0
r 432 160 432 336 0 40000
g 208 336 208 352 0 0
g 368 336 368 352 0 0
g 432 336 432 352 0 0
R 208 160 160 160 0 1 120 15 0 0 0.5
o 12 64 0 4099 80 3.2 0 2 12 3
o 8 64 0 4099 80 0.0015625 1 2 8 3
```

35
Areas/electricity/formulas.md
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@ -74,3 +74,38 @@ $$
&X = X_{L} - X_{C} \\
\end{flalign}
$$
# Capacitive Reactance
$\displaystyle X_{c} = \frac{1}{2 \pi fC}$
# Inductive Reactance
$\displaystyle X_{l} = 2\pi fL$
# Filters
## Cutoff Frequency for RC Filters
$\displaystyle f_{c} = \frac{1}{2\pi RC}$
## Cutoff Frequency for RL Filters
$\displaystyle f_{c} = \frac{R}{2\pi L}$
## Signal Response of an RC Filter
Xc = [[#Capacitive Reactance]]
$\displaystyle V_{out} = V_{in}(\frac{X_{c}}{\sqrt{R^2+X_{c}^2}})$
## Cutoff Frequency for multiple Low Pass Filters
$\displaystyle f_{(-3db)} = f_{c}\sqrt{2^{(\frac{1}{n})}-1}$
Where $n$ = Number if **identical** filters
# Center Frequency for RLC Low Pass Filter
$\displaystyle f_{o} = \frac{1}{2\pi \sqrt{LC}}$

20
Areas/electricity/glossary.md
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@ -5,6 +5,10 @@ In [electrical engineering](https://en.wikipedia.org/wiki/Electrical_engineering
[[formulas#Impedance in a Circuit]]
## Input Impedance ($Z_{in}$)
## Output Impedance ($Z_{out}$)
## Voltage (V)
**Voltage**, **electric potential difference**, **electric pressure** or **electric tension** is the difference in [electric potential](https://en.wikipedia.org/wiki/Electric_potential "Electric potential") between two points.
@ -41,3 +45,19 @@ When electricity was discovered people thought the electrons flow from the posit
## Reactive Components
A component is a **reactive component** when it resists to changes in current or voltage.
## Thevenins Theorem
## Nortons Theorem
## Millmans Theorem
## Cutoff Frequency
$\rightarrow$ [[filters]]
The frequency at which the output strength of a filter is 3dB lower than the input strength. When this frequency is passed the output signal is $V_{Peak} \frac{1}{\sqrt{2}}$ the strength of the input signal.
# Pass-Band
The Pass-Band is the frequency range which is allowed to pass through a filter without changes.
# Stop-Band
The stopband is the frequency range which is attenuated by a filter.

87
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@ -0,0 +1,87 @@
When a circuit is connected to another circuit we should make sure that the output impedance of the first circuit is as low as possible compared to the input impedance of the second circuit. This makes sure that the voltage drop across the second circuits input impedance is as high as possible.
Nowadays high input impedances and low output impedances are the norm in most circuits.
The input impedance can be represented by a resistor that is connected from the input to ground.
Every circuit that has an input and an output has an input and output impedance.
![[Impedance.png|500]]
The output impedance can be represented as a resistor connected in series with the circuit.
We can also represent these impedances as seperated things:
![[input-output-impedance.png]]
# [[glossary#Input Impedance Z_ in|Input Impedance]]
Input impedance is the impedance seen by anything connected to the input of a circuit. It is the combined effect of all resistance, capacitance and inductance connected to the input side of the circuit.
Usually the input impedance **should be around ten times higher** than the output impedance.
This ensures that the circuit does not overload the source of the signal, and reduce the voltage of the signal by a substantial amount.
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
R 144 128 48 128 0 0 40 10 0 0 0.5
r 304 128 304 240 0 1000000
g 304 240 304 320 0 0
O 400 128 496 128 1 1
w 304 128 400 128 0
r 144 128 192 128 0 1000
w 192 128 304 128 0
b 251 87 432 374 0
x 252 79 287 82 4 12 Circuit
x 270 189 286 192 4 12 R2
x 162 152 178 155 4 12 R1
o 3 16 0 159746 9.990009990009991 0.0001 0 1
o 0 16 0 159746 10 0.0001 0 2 0 3
38 1 F1 0 1000 1000000 -1 Resistance
```
When we drop the input resistance, the voltage that is provided to our circuit drops.
We can use the voltage divtider equation to calculate the voltage that is available to the circuit:
![[voltage-dividers#Simple Voltage Divider#Equation]]
# [[glossary#Output Impedance Z_ out|Output Impedance]]
The output impedance is the combined effect of all resistors, capacitors and inductors connected to the output inside the circuit.
The output impedance **should be less than a thenth of the input impedance**.
```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
x 183 158 278 161 4 12 output\simpedance
w 256 128 368 128 2
r 208 128 256 128 0 1
g 368 240 368 320 0 0
r 368 128 368 240 0 1000
R 208 128 112 128 0 0 40 10 0 0 0.5
x 398 190 420 193 4 12 load
b 64 96 287 175 0
x 64 88 95 91 4 12 circuit
38 2 F1 0 1 1000 -1 Resistance
```
If we play with the resistance slider on the right side
# Impedance Matching
Normally the input impedance should be much higher than the output impedance. But in certain cases we want the two to match.
**Example:**
When we connect a Antenna with a $50\ohm$ input impedance to an amplifier circuit. When the input impedance of the antenna is equal to the output impedance of the amplifier we achieve **Maximum Power Transfer**. Power Transfer in this case is $P = VI$.
**Example2:**
![[coax-cable#Impedance Matching in Coax Cable]]

43
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@ -0,0 +1,43 @@
![[Transmission_line_schematic.svg]]
To achieve maximum power transfer over a coax cable the the characteristic impedance ($Z_{0}$) should be equal to the source ($Z_{S}$) and load ($Z_{L}$) impedance.
## Impedance Matching in Coax Cable
```circuitjs editable=false
$ 1 5e-12 1.5642631884188172 50 5 50 5e-11
171 112 192 512 192 0 1e-8 75 64 0
r 112 192 64 192 0 75
w 64 256 112 256 0
w 512 192 544 192 0
w 512 256 544 256 0
r 544 192 544 256 0 75
v 64 256 64 192 0 2 10000000 2.5 2.5 0 0.03
g 512 256 512 272 0 0
g 512 368 512 384 0 0
v 64 368 64 304 0 2 10000000 2.5 2.5 0 0.03
r 544 304 544 368 0 10000
w 512 368 544 368 0
w 512 304 544 304 0
w 64 368 112 368 0
r 112 304 64 304 0 75
171 112 304 512 304 0 1e-8 75 64 0
171 112 416 512 416 0 1e-8 75 64 0
r 112 416 64 416 0 75
w 64 480 112 480 0
w 512 416 544 416 0
w 512 480 544 480 0
r 544 416 544 480 0 10
v 64 480 64 416 0 2 10000000 2.5 2.5 0 0.03
g 512 480 512 496 0 0
g 512 592 512 608 0 0
v 64 592 64 528 0 2 10000000 2.5 2.5 0 0.03
r 544 528 544 592 0 10000
w 512 592 544 592 0
w 512 528 544 528 0
w 64 592 112 592 0
r 112 528 64 528 0 10
171 112 528 512 528 0 1e-8 75 64 0
x 221 200 381 203 4 24 input\s\q\q\soutput
x 230 314 390 317 4 24 input\s<<\soutput
x 236 426 382 429 4 24 input\s>\soutput
x 233 536 393 539 4 24 input\s<<\soutput
```

61
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@ -0,0 +1,61 @@
In some applications you may want to to eliminate signals with certain frequencies from a source signal.
You may want to use filters to remove noise from an audio signal or from the output voltage of a dc power supply.
![[signal-filter.png|400]]
In the diagram above you may want to remove all signals above 12kHz for example.
There are two different types of filters:
# Passive Filters
Passive filters use passive components to filter a signal, like resistors, capacitors or inductors.
# Active Filters
Aktive filters use active components like op-amps or transistors, active filters may be a bit better.
# Filter Types
![[filter-types.png]]
## Low Pass Filter
## High Pass Filter
## Bandpass Filter
## Notch Filter
# [[glossary#Cutoff Frequency|Cutoff Frequency]]
![[cutoff-frequency.png]]
# n'th Order Filter
Normal RC Filters have a slope of -20db/[[units#Decade|Decade]] which means a 45deg decline. When this is to flat for our needs, we can chain multiple LFP's in Series.
We can calculate the slope of nOrder Filters like so:
$slope = n *-20db/Decade$
**Example:**
We have a RC LFP which looks like this:
![[lfp-example.png|200]]
First lets calculate the [[glossary#Cutoff Frequency|Cutoff Frequency]] with the specified formula:
![[formulas#Cutoff Frequency for RC LowPass]]
$\displaystyle f_{c} = \frac{1}{2\pi300*0.000001}$
$f_{c}\approx530.516476973$
Now lets calculate the cutoff frequency when we place three of those filters in series. For this we will use the nth order fomular
![[formulas#Cutoff Frequency for multiple Low Pass Filters]]
$f_{(-3db)} = 530.51 \sqrt{2^{(\frac{1}{3})}-1}$
$\displaystyle f_{(-3db)} \approx 270.467010633$

3
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@ -0,0 +1,3 @@
![[op-amp-basic-schematic-symbol.svg]]
The operational amplifier has a very high input impedance which makes it very good for amplifying low voltage signals.

29
Areas/electricity/units.md
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@ -4,7 +4,7 @@ Measures capacitance, the ability of a body to store an electrical charge.
Term | Symbol | Weight
-----------|----|------
Picofarad | pW | $10^{-12}$
Picofarad | pF | $10^{-12}$
Nanofarad | nF | $10^{-9}$
Microfarad | $\micro$F | $10^{-6}$
Milifarad | mF | $10^{-3}$
@ -73,3 +73,30 @@ Device | Power
Arduino| 167mW
Laptop | 1.5W
House | 2.2kW
# Decibel
A decibel ($dB$) is used to describe the ratio between two values, it is calculated as $\displaystyle 10\log{10}{\frac{V1}{V2}}$
Here are a few commonly used values
| db Value | Power Ratio (10logA) | Voltage/CurrentRatio (20log(A)) |
| ---- | --- | ---- |
| -20dB | 0.01 | 0.1 |
| -10dB | 0.1 | 0.3162 |
| -6dB | 0.25 | 1/2 = 0.5 |
| -3dB |1/2 = 0.5 | $1\sqrt{2} \approx 0.707$ |
| -1dB | 0.79 | 0.89 |
| 0dB | 1 | 1 |
| 1dB | 1.26 | 1.1 |
| 3dB | 2 | $\sqrt{2} \approx 1.414$ |
| 6dB | 4 | 2 |
| 10dB | 10 | $\sqrt{10} \approx 3.162$ |
| 20dB | 100 | 10 |
| 30dB | 1000 | 31.62 |
# Decade
Related to Frequencies
One decade is a distance along the frequency line, for example $1 \rightarrow 10$ or $10 \rightarrow 100$