main.jl

### A Pluto.jl notebook ###
# v0.17.1

using Markdown
using InteractiveUtils

# ╔═╡ e30aad7a-1a47-49d9-9142-441290446140
using Plots, Unitful, Measurements, PlutoUI, DifferentialEquations, CSV, DataFrames

# ╔═╡ 15b4171f-9a0c-4fd9-94ca-9e28fcffe818
md"""
# Time and frequency domain analysis of an electronic low pass filter
"""

# ╔═╡ ef223005-81dc-4717-9e3f-ef8f06870614
HTML("""
<center><h6>Krish Shah <br> Lab Group 37 <br> Girton College</h6></center>
""")

# ╔═╡ 8d9f40ef-21a4-4a9c-988d-4fb3a936009e
md"""
This report details the theoretical and experimental time and frequency domain analysis of a first order Butterworth low pass filter.
Using Ohms and Kirchoffs Laws, we derive equations for the gain and phase of the filter, as well as the charging response in DC current.
Experimental data must be scrutinized due to the large uncertainties of electrical components.
We use sophisticated error propogation tools to show that almost all experimental data lies well within the tolerances of the components used.
"""

# ╔═╡ 8652ef03-7cb3-4b34-9ff4-7ba3cc62188e
md"""
### Introduction and Objectives

Low pass filters act to block the high frequency component of an arbitary signal.
This property makes them very useful in acoustics and electronics, where they have applications ranging from subwoofer electronics to telecommunications equipment.

This report has the following aims and objectives:

- To compare the measurements obtained from the picoscope readings with expected theoretical calculations of the nominal value of the RC time constant of the components.
- To understand the impact of the resistor tolerance (5%) and the capacitator tolerance (20%) on the value of the RC time constant and the expected performance of the components.
- To investigate the attenuation vs frequency behaviour of a low pass filter in terms of dB/decade.

### Experimental Method

Two experiments were completed for this report, following the guidance in the lab handouts provided.
Both experiments involved the analysis of a $100\,\mathrm k\Omega$ resistor in series with a $1 \mu \mathrm F$ capacitor.
The circuit diagram below labels the input voltage, $v_{\text{in}}(t)$ and the output voltage, $v_{\text{out}}(t)$ as functions of time.

$( Show(MIME"image/png"(), read("img/CircuitDiagram.png")) )

> **Figure 1:** The circuit diagram used throughout this experiment

IEP Exercise B involved the time domain analysis of the circuit.
This experiment measured the charging response of the capacitator, with $$v_{\text{in}}(t)$$ equal to 

$$v_{\text{in}}(t)= \left\{ 
    \begin{array}{cc} 
        0 & t<0 \\ 
        V_{max} & t \geq 0 
    \end{array}
\right.$$

IEP Exercise C involved the frequency domain analysis of the circuit.
In this experiment the frequency of the input voltage was varied and the output voltage was measured.

### Theory


First, we consider the time domain analysis.
The experiment begins with the voltage across all components set to zero, so $$v_{\text{in}} = v_{\text{out}} = 0$$ for $$t \lt 0$$.

At $$t\ge 0$$, $$v_{\text{in}}(t)$$ is equal to $$V_{max}$$.
Applying Ohm's Law and the definition of capacitance, we can derive a differential equation for $$v_{\text{out}}(t)$$ for when $$t\ge 0$$.

$$I=\frac{V_{max}-v_{\text{out}}(t)}{R}=C \frac{d\left[ v_{\text{out}}(t) \right]}{d t}$$

This can be readily solved to give an equation for $$v_{\text{out}}(t)$$.
Putting everything together, we have:

$$v_{\text{out}}(t)= \left\{ 
    \begin{array}{cc} 
        0 & t<0 \\ 
        V_{max} \left[1-\exp \left(\frac{-t}{R C}\right) \right] & t \geq 0 
    \end{array}
\right.$$

In practice, our charging response data is not perfectly zeroed.
To counteract this, we shift the input times by a fixed amount, $$t_d$$, such that the data is closer the theoretical line of best fit.


For the second experiment, we begin by using Kirchoffs laws to get an expression for the ratio of $$v_{in}$$ and $$v_{out}$$.

$$\frac{v_{\text {out }}}{v_{\text {in }}}
= \frac{\frac 1 R }{\frac 1 R +j \omega C}
= \frac{1}{\sqrt{1+(\omega R C)^{2}}} \exp \left[-j \tan ^{-1}(\omega R C) \right]$$


From this equation we can inferr that the phase shift between $$v_{in}$$ and $$v_{out}$$ is $$\phi = -\tan ^{-1}(\omega R C)$$.

Next, we find the theoretical attenuation in decibels.

$$20 \log _{10}\left(\frac{V_{o}}{V_{i}}\right)
= 20 \log _{10}\left[\frac{1}{\sqrt{1+(\omega R C)^{2}}}\right]
= -10 \log _{10}\left[1+(\omega R C)^{2}\right]$$

At very low frequencies, this equation is approximated by a straight line of zero slope.
For large frequencies, $$\log \left(1+(\omega R C)^{2}\right)
\approx 2 \log \left(\omega R C\right)$$, so the attenuation is approximated by a straight line of −20 dB per decade slope.

### Results

Uncertainties are a decisive factor in our analysis.
The resistor and capacitor have tolerances of 5% and 20% respectively, making the value of $$RC$$ differ by up to 25%.
We need error propagation tools in order to determine whether experiment differs significantly from theory.

In light of these issues, the analysis is conducted in the Julia programming language using the `Measurements.jl` module.
The module manages uncertainties in a comprehensive manner and has functional correlation built in, so it understands that `sec²(x) - tan²(x) ≈ 1 ± 0`.

Entering values for $$R$$ and $$C$$ into the program, we can see that the uncertainty of $$\frac{1}{RC}$$ given by the program agrees with the answer when calculated by hand:
"""

# ╔═╡ f6f1ecaf-228d-4ac9-963c-229b14ea8242
begin
	R    = 100e3 ± 1000    # Resistance values from lab handout
	C    = 1e-6  ± 2e-7    # Capacitance values from lab handout
	(1/(R * C))u"s^-1"     # Display the value of 1/RC (with units)
end

# ╔═╡ 1d70a4fa-c5e2-44b1-9d13-1d788709c754
md"""
Using this software package, we can plot our analysis comparing theoretical quantities with measured ones.
In each case the values of $$R$$ and $$C$$ used are the same as above.
We represent the area of uncertainty on each graph below with a blue shaded region.
"""

# ╔═╡ 9c561430-28e0-408e-9665-0c904c2ba893
md"""
> **Figure 2:** A graph showing the output voltage as a function of time. The theoretical voltage curve used ``V_{max} = 5.102 ± 0.0002 \mathrm V``, obtained from the picoscope measuring software. The measured voltage was shifted by ``t_d = 0.018 \mathrm s`` in the time domain to zero the picoscope readings.

----
"""

# ╔═╡ 1f759aa4-7e8f-4746-a2f5-20141db658bd
md"""
> **Figure 3:** A Bode Magnitude plot showing gain as a function of frequency.

----
"""

# ╔═╡ d4b2c2db-5267-4de7-9d0a-3f11872aacd8
md"""
> **Figure 4:** A Bode Phase plot showing phase as a function of frequency.

"""

# ╔═╡ 7a0b8b81-a8f6-48f1-9ed1-7d7b8ae73778
md"""
### Discussion

Time domain analysis agrees almost perfectly with theoretical data.
- In figure 2, all points are within the shaded blue region, which shows that the measurement is consistent with the resistance and capacitance within the tolerances given.

Frequency domain analysis agrees strongly with theoretical data, but hints strongly towards human error.
- In figure 3, all points are within the shaded blue region, so the measurements are consistent with the resistance and capacitance within the tolerances given.

- Figure 4 required the phase to be recorded manually. This likely made the measurements far less accurate and more prone to human error. Indeed, the author particularly regrets their lack of discipline in measuring the last few data points.

"""

# ╔═╡ 8f4aa2aa-e860-4857-8f32-7b79fabf82c1
md"""
### Conclusions 

1. Time and frequency domain analysis was performed on a low pass filter with a ``1\mu \mathrm F`` capacitor and a ``1\mathrm k \Omega`` resistor to obtain a value for the RC time constant and plot Bode plots for gain and phase as functions of frequency.
2. Time domain showed that the RC time constant measured was very accurate, virtually identical to the expected value
3. The measured data for the phase as a function of frequency was less accurate, almost certainly due to the manual process in which the phase was determined.
4. The Julia programming language presents a unique set of tools that could be useful in the Engineering Department

"""

# ╔═╡ 35c3fd8a-bb8d-471e-b057-3f02913ca13b
html"""
<script>
let cell = currentScript.closest("pluto-cell");
cell.style.pageBreakAfter = "always";
</script>
"""

# ╔═╡ b75150d2-43b9-4d16-a197-d72f5dc1317e
md"""
### Appendix: Data

"""

# ╔═╡ 6f395a82-a7c2-4238-82cc-f0d820a7dcbb
begin
	df = DataFrame(CSV.File("data/B_2.csv", skipto=3))
	t = df[!, 1] .+ 0.018
    vₒᵤₜ = df[!, 2]
	
    # Data collected in Part C
    vᵢₙ_rms  = [723.7, 703.4, 680.9, 687.7, 686.2, 682.8, 683.2, 682.9, 682.6, 682.7]
    vₒᵤₜ_rms = [714.8, 689.8, 642.9, 573.1, 423.1, 205.8, 110.4, 57.3, 22.66, 13.46]
    freq     = [0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100]
    delay    = [175.6, 133.5, 106.1, 96.2, 75.2, 47.9, 27.7, 12.0, 5.117, 2.354]
    gain     = -20log10.(vᵢₙ_rms ./ vₒᵤₜ_rms)
    phase    = -0.36(delay .* freq)
	
	DataFrame(
        Vᵢₙ=vᵢₙ_rms * u"mV", Vₒᵤₜ=vₒᵤₜ_rms * u"mV", 
		Frequency=freq * u"Hz", Delay=delay * u"ms",
		Gain=gain * u"dB", Phase=phase * u"°"
	)
end

# ╔═╡ ac0f240a-9a4b-49f5-8912-0073e128ab83
begin
    function vₒᵤₜmodel_B(t)
        if t < 0;
            return 0
        end
        return Vₘₐₓ * (1 - ℯ^(-t/(R*C)))
	end

	Vₘₐₓ = 5.102 ± 0.0002  # Limiting Voltage values from picoscope
	x = -0.1:0.01:1.0
	y = vₒᵤₜmodel_B.(x)
    plot(x, 
        Measurements.value.(y), 
        ribbon=Measurements.uncertainty.(y),
        label="Theoretical Voltage",
        ylabel="Voltage (V)",
        xlabel="Time (s)",
        title="Output voltage as a function of time",
        legend=:right,
        xlims=(-0.02, 0.65),
        linewidth=3,
    )
	plot!(t, vₒᵤₜ,
        linewidth=3,
        label="Measured Voltage"
    )
end

# ╔═╡ 9fc1d884-baf9-4510-bc4e-a1178f232bcb
begin
    gain_model(f) = -10log10(1 + (2π * f * R * C)^2)
    dummy_freq  = [10^y for y in range(log10(minimum(freq)), log10(maximum(freq)), length=200)]
    theoretical_gain = gain_model.(dummy_freq)
	plot(
        dummy_freq,
        Measurements.value.(theoretical_gain), 
        ribbon=Measurements.uncertainty.(theoretical_gain),
        linewidth=3,
	    xaxis=:log,
        label="Theoretical gain",
        ylabel="Gain (dB)",
        xlabel="Frequency (Hz)",
        title="Gain as a function of Frequency",
    )
	scatter!(freq,
	    gain,
        label="Measured gain",
	)
end

# ╔═╡ 1f9e12f3-f8be-49c4-87d1-fb3ec92769ac
begin
    phase_model(f) = -atan(2π * f * R * C) * 180 / π
    theoretical_phase = phase_model.(dummy_freq)
	plot(
        dummy_freq,
        Measurements.value.(theoretical_phase), 
        ribbon=Measurements.uncertainty.(theoretical_phase),
        linewidth=3,
	    xaxis=:log,
        label="Theoretical phase",
        ylabel="Phase (°)",
        xlabel="Frequency (Hz)",
        title="Phase as a function of Frequency",
    )
	scatter!(freq,
	    phase,
        label="Measured phase",
	)
end

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version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[WeakRefStrings]]
deps = ["DataAPI", "InlineStrings", "Parsers"]
git-tree-sha1 = "c69f9da3ff2f4f02e811c3323c22e5dfcb584cfa"
uuid = "ea10d353-3f73-51f8-a26c-33c1cb351aa5"
version = "1.4.1"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─e30aad7a-1a47-49d9-9142-441290446140
# ╟─15b4171f-9a0c-4fd9-94ca-9e28fcffe818
# ╟─ef223005-81dc-4717-9e3f-ef8f06870614
# ╟─8d9f40ef-21a4-4a9c-988d-4fb3a936009e
# ╟─8652ef03-7cb3-4b34-9ff4-7ba3cc62188e
# ╠═f6f1ecaf-228d-4ac9-963c-229b14ea8242
# ╟─1d70a4fa-c5e2-44b1-9d13-1d788709c754
# ╟─ac0f240a-9a4b-49f5-8912-0073e128ab83
# ╟─9c561430-28e0-408e-9665-0c904c2ba893
# ╟─9fc1d884-baf9-4510-bc4e-a1178f232bcb
# ╟─1f759aa4-7e8f-4746-a2f5-20141db658bd
# ╟─1f9e12f3-f8be-49c4-87d1-fb3ec92769ac
# ╟─d4b2c2db-5267-4de7-9d0a-3f11872aacd8
# ╟─7a0b8b81-a8f6-48f1-9ed1-7d7b8ae73778
# ╟─8f4aa2aa-e860-4857-8f32-7b79fabf82c1
# ╟─35c3fd8a-bb8d-471e-b057-3f02913ca13b
# ╟─b75150d2-43b9-4d16-a197-d72f5dc1317e
# ╟─6f395a82-a7c2-4238-82cc-f0d820a7dcbb
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002