IE1206 Embedded Electronics Le1 Le2 PIC-block Documentation, Seriecom Pulse sensors I , U , R , P , serial and parallel Le3 Ex1 KC1 LAB1 Pulse sensors, Menu program • Start of programing task • • • Ex2 Le4 Kirchhoffs laws Node analysis Two-terminals R2R AD Two-terminals, AD, Comparator/Schmitt Ex3 Le5 KC2 LAB2 Transients PWM Le6 Ex4 Le7 KC3 LAB3 Step-up, RC-oscillator Phasor j ω PWM CCP CAP/IND-sensor Ex5 Le8 Le9 LC-osc, DC-motor, CCP PWM Le11 KC4 LAB4 Ex6 Le10 LP-filter Trafo Display Le12 Ex7 • • Display of programing task • • Written exam Le13 Trafo, Ethernet contact William Sandqvist william@kth.se
A signal in reality … Actual signals are difficult to interpret. They are often disturbed by noise and hum. Hum is our 50Hz network induced into the signal lines. Noise is random disturbances from amplifiers (or even resistors). William Sandqvist william@kth.se
Maybe a slow DC … Perhaps the signal is a slowly increasing direct A LP-filter voltage from eg. a (=LowPass) temperature filters away the sensor? interference and In this case, the removes the interference interferences from consist of 50 Hz the signal. hum and high frequency noise. William Sandqvist william@kth.se
Maybe a sine wave … Maybe the signal is a sine wave? In this case, the interference consist of the DC voltage level A BP-filter slowly changing, (BandPass) will offst, and that block the offset and noise is added. filters out the noise. William Sandqvist william@kth.se
Maybe rapid variations … Perhaps the signal is the rapid variations? In this case, the interference consist of the DC voltage level slowly changing, and that hum has A HP-filter been added. (HighPass) removes the interferences from the signal. William Sandqvist william@kth.se
Filter With R L and C one can build effective filters . Inductors are more complicated to manufacture than capacitors and resistors, therefore, is typically only combination R and C used. Fast computers can filter signals digitally. Calculating a signal’s moving average can for example correspond to the LP filter. Nowadays dominates the digital filter technology over the analog. Simple RC filter are naturally in most measuring instruments, or even arising from "itself" when linking equipment. This is the reason that one must know and be able to calculate on simple RC-links, even though they regarded as filters are very incomplete. William Sandqvist william@kth.se
LP HP BP BS LP lowpass HP highpass BP bandpass BS bandstop BP and BS filters can be seen as different combination of LP and HP filters. William Sandqvist william@kth.se
William Sandqvist william@kth.se
Voltage divider, Transfer function Simple filters are often designed as a voltage dividers.A filter transfer function , H ( ω ) or H ( f ), is the ratio between output voltage and input voltage. This ratio we get directly from the voltage divider formula! Z U Z = � ω = = U U H 2 2 2 ( ) + + 2 1 Z Z U Z Z 1 2 1 2 1 William Sandqvist william@kth.se
William Sandqvist william@kth.se
RC LP-filter, vectors Phasor diagram: R and C has the current I in common. Volyage over resistor and voltage over capacitors kondensatorn therefore becomes perpendicular. Pythagorean theorem can be used: = + U U U 2 2 2 1 3 2 U ϕ = ϕ 5 2 3 | | arctan U 1 4 William Sandqvist william@kth.se
RC LP-filter, j ω 1 ω U ω C C j j 1 = ⋅ = 2 ω + ω U C RC 1 j 1 j + R 1 ω C j U 1 = 2 U + ω RC 2 1 ( ) 1 � � ω U RC � � ( ) � � ϕ = = − + ω = − = − ω RC � � RC 2 arg arg( 1 ) arg( 1 j ) 0 arctan arctan � � U � � � � 1 1 William Sandqvist william@kth.se
RC LP-filter, H ( ω ) 1 ( ) 1 ( ) ( ) = = = = − ω H H H H RC abs arg arctan + ω RC + ω 1 j RC 2 1 ( ) At the angular frequency when ω RC = 1 , will the numerator real part and imaginary part be equal. This is the filter cutoff frequency. William Sandqvist william@kth.se
LP-magnitude function R = 1 k Ω C = 1 µ F 1 = f π ⋅ ⋅ ⋅ ⋅ − G 3 6 2 1 10 1 10 ≈ 160 Hz 1 1 1 = ω = = H f π G RC G RC + ω RC 2 2 1 ( ) William Sandqvist william@kth.se
LP-Phase function ( ) ϕ = = − ω H RC arg arctan( ) William Sandqvist william@kth.se
Graphs with Mathematica Mathematica has commands for complex absolute value (abs []) and argument (arg [], in radians). <<Graphics r=1*10^3; c=1*10^-6; w=2*Pi*f; u2u1[f_]=1/(1+I*w*r*c); LogLinearPlot[Abs[u2u1[f]],{f,1,10000},PlotRange->All,PlotPoints->100]; LogLinearPlot[Arg[u2u1[f]],{f,1,10000},PlotRange->All,PlotPoints->100]; Press SHIFT + ENTER to start the calculation and the plot. Amount plot Phase plot [rad] William Sandqvist william@kth.se
RC Two sides of the same coin 1 ω = τ = RC G RC Low cut off frequency ω G will supresses interference good, but it will also mean that the time constant τ is long so it takes time until U UT reaches its final value and can be read. William Sandqvist william@kth.se
( AD-converter LP-filter ) • It is necessary to compromise! C In order to remove noise from the input signal to the AD converter one usually add a capacitor C . • R S must have no bigger value than 10k Ω – otherwise you risk losing accuracy because of the leakage current I LEAKAGE . • When the sample charge from C is taken to sampling capacitor C HOLD . C should therefore be at least 1024 times greater than C HOLD (10pF) if you do not want to lose accuracy. • C ⋅ R S gives the cutoff frequency of how fast signals AD converter can follow. William Sandqvist william@kth.se
William Sandqvist william@kth.se
RC HP-filter, j ω ω ω U R C RC j j = ⋅ = 2 ω + ω U C RC 1 j 1 j + R 1 ω C j ω U RC = arccot() = 2 U + ω RC 2 1 ( ) 1 � � ω � � U RC � � 1 � � � � = ω − + ω = ° − = RC RC � � 2 arg arg( j ) arg( 1 j ) 90 arctan arctan � � � � ω U RC � � � � 1 � � 1 William Sandqvist william@kth.se
RC HP-filter, H ( ω ) ω ω � � RC RC j 1 ( ) ( ) � � = = = H H H abs arg arctan � � + ω ω RC RC + ω RC � � 1 j 2 1 ( ) At the angular frequency when ω RC = 1 , will the numerator real part and imaginary part be equal. This is the filter cutoff frequency. William Sandqvist william@kth.se
HP-magnitude function R = 1 k Ω C = 1 µ F 1 = f G π ⋅ ⋅ ⋅ ⋅ − 3 6 2 1 10 1 10 ≈ 160 Hz ω RC ( ) = H abs + ω RC 2 1 ( ) William Sandqvist william@kth.se
HP-phase function � � 1 ( ) � � ϕ = = H arg arctan � � ω RC � � William Sandqvist william@kth.se
William Sandqvist william@kth.se
Wienbridge (14.5) Was investigated by Max Wien 1891 For a certain frequency U 1 and U 2 are in phase. What frequency? William Sandqvist william@kth.se
Wienbridge 1 = + Z R 1 ω C j 1 ⋅ R ω ω C R C j j = ⋅ = Z 2 ω + ω C RC 1 j 1 j + R ω C j U 1 and U 2 are in phase if the transferfunction imaginary part is 0! R + ω RC ( 1 j ) U + ω RC 1 j 1 1 R = ⋅ = = 2 + ω R RC U 1 ( 1 j ) 1 1 + + + ω + R RC 3 j + ω − RC 1 3 j ( ) ω + ω ω C RC R RC j 1 j j ω RC = 0 William Sandqvist william@kth.se
Wienbridge U 1 = � 2 U 1 + ω − 1 RC 3 j ( ) ω RC 1 1 = � ω = ω − 0 RC 0 RC ω RC William Sandqvist william@kth.se
Wienbridge 1 1 ω = = f 0 0 π RC RC 2 Magnitude plot Phase plot Wienbridge is a band pass filter. William Sandqvist william@kth.se
William Hewlett’s master thesis Master thesis 1930. Wienbridge with lamp! William Sandqvist william@kth.se
William Hewletts master thesis Hewlett constructed a tone generator. Wien bridge attenuates the signal to 1/3 so he needed a amplifier with the gain exactly three times. In phase! The bulb stabilizes the signal. If the amplitude becomes too large the lamp will glow and then the signal is attenuated in the voltage divider at the amplifier input. William Sandqvist william@kth.se
The Palo Alto garage the birthplace of Silicon Valley Which global business will you start with your thesis? William Sandqvist william@kth.se
William Sandqvist william@kth.se
When are filters used? One speaker alone can not cope with all frequencies! William Sandqvist william@kth.se
Cross over filter The crossover filter split the frequencies between the speakers. William Sandqvist william@kth.se
Passive/Active speaker • Analog crossoverfilter R L C • Digital crossoverfilter = computer program When the amplifier is built in the speaker it becomes possible to use digital crossovers. ( XOVER = crossover filter ) William Sandqvist william@kth.se
William Sandqvist william@kth.se
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