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capactive_mains_inputs/Makefile Normal file
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# Makefile to test and plot sections of the curve
# to test the interpolation table
#
#
#
SOURCE = capacitive_mains_inputs2
TEX_SOURCE = $(SOURCE).tex
PDF_SOURCE = $(SOURCE).pdf
#
#
# Place all .png files here as .dia targets
#
# png files mangle text now in dia
DIA = opto.dia
DIAJPG = opto.jpg
doc: $(DIAJPG)
echo $?
gnuplot reactance.gpt
pdflatex $(TEX_SOURCE)
pdflatex $(TEX_SOURCE)
acroread $(PDF_SOURCE) || evince $(PDF_SOURCE)
bib:
bibtex $(SOURCE)
#%.png:%.dia
# echo $?
# dia -t png $<
# mkdir -p images_sw_doc
# echo $< $@
# mv $@ images_sw_doc
# echo $@
%.jpg:%.dia
echo $?
dia -t jpg $<
mkdir -p images_sw_doc
echo "dia to jpg arg list and target" $< $@
mv $@ images_sw_doc
echo $@
#images.tex:
# cat begin_images.tex > images.tex
# ls images_sw_doc >> images.tex
# cat end_images.tex >> images.tex
#
clean:
rm *.pdf

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capactive_mains_inputs/capacitive_mains_inputs2.tex Normal file
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\documentclass[a4paper,10pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{lastpage}
\usepackage{color}
\definecolor{Blue}{rgb}{0.0,0.0,0.7}
\definecolor{Red}{rgb}{0.7,0.0,0.0}
\definecolor{Green}{rgb}{0.0,0.5,0.0}
\usepackage{hyperref}
\usepackage{ifthen}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{multirow}
\usepackage{textcomp}
%opening
\title{Capacitive Mains Inputs; choosing capacitor and resistor combinations for 120 V a.c. 240 V a.c and 50 and 60Hz}
\author{R.P. clark}
\begin{document}
\maketitle
\begin{abstract}
Calculations to work out capacitance values to drive an opto-coupler to detect mains voltage
for 50 to 60 Hz.
\end{abstract}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{./images_sw_doc/opto.jpg}
% opto.jpg: 854x388 pixel, 72dpi, 30.13x13.69 cm, bb=0 0 854 388
\caption{Opto-coupled mains input circuit}
\label{fig:opto}
\end{figure}
\section{Opto coupler circuit}
This circuit is used to detect mains voltage via a capacitor and a resistor
forming a potential divider so that a lower voltage can be used to drive an opto-isolator
that protects the processor reading the signal.
\section{Calculations}
A potential divider using a capacitor and a resistor is used to lower mains voltage
to levels that can drive a typical opto-coupler input ($\approx 2V$).
A potential divider using a capacitor and resistor means
using the complex identity for the capacitors reactance, $X$.
$$ X = \frac{-j}{\omega C } $$
The ${\omega C }$ term is dependent on frequency and is equivalent to $2.\pi.f$ .
Using a potential divider to determine the voltage over the resistor gives:
$$ V_{out} = V_{in} \times \frac{R}{R-\frac{j}{2.\pi.f.C}} $$
The equation above leaves a complex divisor.
To get a complex number as the numerator, the denominator and numerator must be multiplied by
the conjugate of the denominator, thus:
$$\frac{R}{R-\frac{j}{2.\pi.f.C}} \equiv \frac{R \times \Big({R+\frac{j}{2.\pi.f.C}}\Big) }{\Big({R-\frac{j}{2.\pi.f.C}}\Big) \times \Big({R+\frac{j}{2.\pi.f.C}}\Big) } $$
This leaves a real number as the denominator, i.e. $ R^2 + {\frac{1}{2.\pi.f.C}}^2$.
The resulting complex number, $X$,
$$X = \frac{R \times \Big({R+\frac{j}{2.\pi.f.C}}\Big) }{R^2 + {\frac{1}{2.\pi.f.C}}^2}$$
or,
\begin{equation}
X =\frac{R^2 + \Big({R\frac{j}{2.\pi.f.C}}\Big) }{R^2 + {\frac{1}{2.\pi.f.C}}^2}
\label{eqn:genpotdivcapres}
\end{equation}
can now be evaluated for phase and magnitude. Equation~\ref{eqn:genpotdivcapres} can be generally applied to potential dividers
in figure~\ref{fig:opto}.
\subsection{Example calculation}
At 50Hz with 240 V a.c. applied, with R at 1000 Ohms and C at 47 nF
$$\frac{1000^2 + \Big({1000\frac{j}{2.\pi.50.47e-9}}\Big) }{R^2 + {\frac{1}{2.\pi.50.47e-9}}^2}$$
$$\frac{1000^2 + \Big({1000 \times 67726j}\Big) }{1000^2 + {67726}^2}$$
$$\frac{1000^2 + \Big({67726000j}\Big) }{4.5877 \times 10^9}$$
This gives a complex number $$ \frac{1000^2 + {67726000j} }{4.5877 \times 10^9}$$ i.e. $$(216 \times 10^{-6} + 14.76\times 10^{-3} j ) \;.$$
This complex number has a magnitude of 0.0147 and an argument of 89.15 degrees (which is expected as most of the reactance comes from the capacitor).
So with 240 V a.c. applied (RMS) the opto would see a signal with $0.0147*240 = 3.54V (RMS)$
\clearpage
\section{ploting the voltage at the opto-coupler}
\begin{figure}[h]
\centering
\includegraphics[width=400pt]{./RMS_volts_to_opto.png}
% RMS_volts_to_opto.png: 640x480 pixel, 72dpi, 22.58x16.93 cm, bb=0 0 640 480
\caption{RMS voltage seen at opto-coupler for 50 to 60 Hz range}
\label{fig:rmstoopto}
\end{figure}
\clearpage
\subsection{plotting the voltage at the opto-coupler: gnuplot scripts}
{ \tiny
\begin{verbatim}
########################################################
#
p=3.14159265358979323844
#
# 47nF
C=47e-9
#
# 1k Ohms
R=1000
# define complex operator
j={0,1}
set xlabel "Hertz"
set ylabel "Resistance"
# x is the frequency
set xrange[50:60]
# z(x) is the reactance
z(x)=(j/(2*p*x*C))
# denominator
d(x)=(R*R+z(x)*z(x))
# numerator
n(x)=(R*R+R*z(x))
plot abs(z(x)) title "reactance over capacitor"
!sleep 4
set ylabel "denominator value (abs)"
plot abs(d(x))
!sleep 4
set ylabel "numerator value (abs)"
plot abs(n(x))
!sleep 4
v(x)=abs((n(x))/(d(x)))
# gives large numbers h(x)=arg((n(x))/(d(x)))
set ylabel "voltage to opto-coupler (RMS)"
plot 240*v(x) title "240 V a.c", 120*v(x) title "120 V a.c"
!sleep 4
set terminal png
set output "RMS_volts_to_opto.png"
plot 240*v(x) title "240 V a.c", 120*v(x) title "120 V a.c"
#set angles degrees
#set label "phase change in mains over opto"
#plot 240*h(x) title "240 V a.c", 120*h(x) title "120 V a.c"
#!sleep 4
#
\end{verbatim}
}
%
%
% Putting some numbers in this, 47nF for the capacitor, 1k for R and 50 Hz at 240V, means ${2.\pi.f.C} = 14.765 \times 10^{-6}$.
%
% $$ V_{out} = 240 \times \frac{1000}{ 1000 - \frac{j}{14.765 \times 10^{-6}} } $$
% or
% % $$ V_{out} = 240 \times \frac{1000}{1000 - j \times 67.726 \times 10^3 }$$
% % %
% % To get a complex number as the numerator, the denominator and numerator must be multiplied by
% % its conjugate, thus:
% % %
% % $$\frac{1000}{1000 - {j} \times 67.726 \times 10^3 } $$
% % % $$
% % %
% % \equiv \frac{ 1000 \times (1000 + {j} \times 67.726 \times 10^3) }{ (1000 - {j} \times 67.726 \times 10^3) \times (1000 + {j} \times 67.726 \times 10^3)} $$
% % $$
%
typeset in {\Huge \LaTeX} \today.
\end{document}

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capactive_mains_inputs/opto.dia Normal file
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capactive_mains_inputs/reactance.gpt Normal file
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########################################################
#
p=3.14159265358979323844
# 47nF
C=47e-9
# 1k Ohms
R=1000
# define complex operator
j={0,1}
set xlabel "Hertz"
set ylabel "Resistance"
# x is the frequency
set xrange[50:60]
# z(x) is the reactance
z(x)=(j/(2*p*x*C))
# denominator
d(x)=(R*R+z(x)*z(x))
# numerator
n(x)=(R*R+R*z(x))
plot abs(z(x)) title "reactance over capacitor"
!sleep 4
set ylabel "denominator value (abs)"
plot abs(d(x))
!sleep 4
set ylabel "numerator value (abs)"
plot abs(n(x))
!sleep 4
v(x)=abs((n(x))/(d(x)))
# gives large numbers h(x)=arg((n(x))/(d(x)))
set ylabel "voltage to opto-coupler (RMS)"
plot 240*v(x) title "240 V a.c", 120*v(x) title "120 V a.c"
!sleep 4
set terminal png
set output "RMS_volts_to_opto.png"
plot 240*v(x) title "240 V a.c", 120*v(x) title "120 V a.c"
#set angles degrees
#set label "phase change in mains over opto"
#plot 240*h(x) title "240 V a.c", 120*h(x) title "120 V a.c"
#!sleep 4
#