=======================ELECT_SIGNAL========================= _|_ Z(S)= 1/s*C ___ | _ _ _ /*\/ \/ \ Z(S) =s*L ___ | () () | ___ | |__| |__| | |___| |___| 1/(s^2 +a*s +b) =1/(( s +alpha +jwo )*( s +alpha -jwo )) 1/(s^2 +a*s +b) =1/(s^2 +(wo/Q)*s +wo^2) IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | PULSE RESPONSE _ _ _ | | | | | | |_|_|_|_|_|_____\ | | | | | | / Time | |_| |_| | IMAGINARY ^ /|\ | X ______|____\ REAL | / X _ PULSE RESPONSE | | _ | | | | _ |_|_|_|_|_|_____\ | | | |_| / Time | |_| IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | _ PULSE RESPONSE _ | | _ | | | | |_|_|_|_|_|_____\ | |_| | | | / Time |_| | IMAGINARY ^ /|\ | | X ______|____\ REAL | / | X PULSE RESPONSE _ - _ - ________________\ / Time IMAGINARY ^ /|\ | | ______|_X__\ REAL | / | _ PULSE RESPONSE - _ - _ ________________\ / Time IMAGINARY ^ /|\ | | ___X__|____\ REAL | / | on imaginary axes When poles are exactly on the imaginary axis it says you will have a constant sine wave at frequency 2*pi*w _ _ _ | | | | | | |_|_|_|_|_|_____\ | | | | | | / Time PULSE RESPONSE | |_| |_| | IMAGINARY ^ /|\ | X ______|____\ REAL | / X Normal poles When poles are slightly in the left half plane it says you will have a sine wave at frequency 2*pi*w which is decaying at the a rate defined the dsitance from the axis. _ | | _ | | | | _ |_|_|_|_|_|_____\ | | | |_| / Time | |_| PULSE RESPONSE IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | Left Half Plane When poles are in left half plane you will have a decay at the a rate defined the dsitance from the axis. _ PULSE RESPONSE - _ - _ ________________\ / Time IMAGINARY ^ /|\ | | ___X__|____\ REAL | / | Right Half Plane When poles are in Right half plane you are unstable PULSE RESPONSE _ - _ - ________________\ / Time IMAGINARY ^ /|\ | | ______|_X__\ REAL | / | _ PULSE RESPONSE _ | | _ | | | | |_|_|_|_|_|_____\ | |_| | | | / Time |_| | IMAGINARY ^ /|\ | | X ______|____\ REAL | / | X ======================SYSTEMS_TRANSFER_FUNCTIONS============================ TYPES NONLINEAR incidental and intentional nonlinearities. Incidental nonlinearities Saturation, dead zone, and backlash may inaccurate or even instability Intentional nonlinearities NONLINEAR SYSTEMS ^ OUT /|\ Limiting | ____ Saturation | / | / _______|/________\ IN All transistor input stages do this / / /| ____/ | ^ OUT Deadzone /|\ threshold | | / _______|___/________\ IN / | / / | ^ OUT On/off switch /|\ coulomb friction |_____ | _______|____________\ IN | / ______| ^ OUT preload /|\ | / |/ _______|____________\ IN | / /| / ^ OUT spring /|\ hardening | / | _/ _______|-____________\ IN _-| / / | / ^ OUT linear /|\ rectification | / | / _______|/____________\ IN | / | ^ OUT relay hysteresis /|\ | ____/_ | | | \ | v ^ _______|__|__|______\ IN | | | / v ^ | _\_|__| | / ^ OUT Toggle /|\ --|<----- | | ^ | | | ____v__|__^__________\ IN | | | / | | | ------>-- ^ OUT backlash /|\ -|<----_ / | /| /<-|--- / ____/___|___/__________\ IN / ---|->/ / |/_ | / ----->- Describing Function the following basic assumptions. A) input to nonlinear element n is sinusoidal only the fundamental component of output of n. contributes to the input. The output response of nonlinear element consists of the fundamental frequency and harmonics Generally, harmonic smaller than fundamental most control systems the system a low-pass filter and the higher harmonics are attenuated. If harmonics small can be neglected B) All nonlinearities lumped into one single nonlinear element n C) output of nonlinear element is a function only of the present value and past history of the input, i D) n is not a function of time. The describing function of a nonlinear element __ + / \ e ____ ____ ___\__/ \/ \__\__| n |___\__| G |________\ C r / \ /\ / / |____| / |____| | / \__/ | - ^ | |______________________________| diagram of nonlinear closed-loop system. Limit Cycle oscillation of nonlinear systems characterized by a constant am1diiude and frequency determined by the nonlinear properties of system. output approaching the amplitude of that limit cycle regardless of initial condition and forcing function. Hewlet Packard used a lightbulb to define the amplitude of their oscilator. As the amplitude of the output got larger, it would lower in resistance and such that the positive feedback was exactly one. _ | _\| /| __-->|--__ / _ | \ | / /| | _\| \ V _____|______|______|______ ^ | _ | | | | \ |\ | |/_ / V \ | / <--__|_<-- |/_ nonlinear system output of nonlinear device contains harmonic and subbarmonic frequencies (except for HP¹s oscillator) soft self-excitation limit cycle in very small excitation or disturbances. hard self-excitation requiring forced excitation above a certain minimum amplitude o Undervoltage Lockout Hysteresis Switcher on _|_ _____________________ | | <-- | | | | ^ | | | v --> | | Switcher off |_____|_______________|______\ Input Voltage V(turn off) V(turn on) / DeltaV V(turn on) No load voltage from transfomer @ lO8Vac V(turn off) Full load voltage from transformer @ lO5Vac Delta_V V(turn on) - V(turn off) ; Hysteresis using undervotage circuit with hysteresis, we can prevent the power supply from oscillating on and off. V(turn on) the no load voltage from the transformer. This is very close to 108*sqrt(2)*Ns/Np V(turn off) the full load voltage from the transformer at IO5Vac. This value is sensitive to load conditions for each design, and should be measured on the bench during design. delta_V the difference between these two voltages, is value use for the undervoltage lockout hysteresis. Hysteresis Multivalued functions exist when two or moore function values correspond to the value of the variable. hysteresis curves of magnetic materials and the backlash of a gear train. ROOT_LOCUS ___ R(s) + / _ \ E _____ C _____\/ \ \_______| G(s)|_________\ /\ /_ / |_____| | / \___/ | - ^ _____ | |___| H(s)|_________| B |_____| For stabilty G(s)*H(s)=(K*(s-z)*../((s-p)..)=>gain< 1 @ 180deg IMAGINARY ^ _ /|\ How loop poles move /| | as loop gain is increased / | <--X X->|<-X |____\ REAL \ | / _\| IMAGINARY ^ _ /|\ /| | / | \ | <--X X->Z X->|<-X |____\ REAL / | / \ _\| IMAGINARY ^ _ /|\ /| | / | \ | <--X X->Z X->|<-X |____\ REAL / | / \ _\| Complex_Poles 1/(s^2 +a*s +b) =1/(( s +alpha +jwo )*( s +alpha -jwo )) 1/(s^2 +a*s +b) =1/(s^2 +(wo/Q)*s +wo^2) IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | MARGINAL STABLE 20dB|................................................ | . . . . | . b Q=10 . . | . . . . | . b . . . | . b .b . . 0dB|..........a..a...a.............................. | . a . b . . | . . . . | . a b . . | . . a . . | . . a . . -20dB|.............................a.................. | . . . | . . . . | . . . . | . . . . | . . . . |______________________________________________ . 100KHz 1MHz 10MHz 100MHz 1GHz CONDITIONALY STABLE GAIN_PHASE CONDITIONALY STABLE can only oscilate by changing loop gain during start up,sat.lower VCC .. delicate 60dB |................................................ | P . . . . | P . . . . |G G G <== Open Loop Gain . . | P G . . . . | P G . . . . 40dB|........P..G.P..................................90 | . G . . . | . G . . . | . G P . . . | . G . . . | . G . . . 20dB |....................P..G.........P..............45 | . . G P P . | . . G .P 38deg . | . P. GP . P Phase . | . . G . P Margin . | . .P P G . P . 0dB |0000000000000000000000000000P000000V0000P0000000 0 | . . P . G ^ Gain . | . . . G| Margin | . . . V . | . . . G . | . . . . -20dB|______________________________________________ . 10KHz 100KHz 1MHz 10MHz 100MHz ======================COMPONENTS_NOISE========================================== Rules of thumb 1K is 4nV/sqrt(Hz) 100R is 10 time noise Noise_figure Total_noise/noise_R pink noise equal noise per octave red noise 1/f^2 Excess_Noise 1/f called noise current, flicker noise Due to microarc between carbon granules for dc current. Noise_Index_dB 20*log( Excess_noise_uV/Vdc ) 0 dB = 1uV of noise per 1volt DC current. Noise_Index_dB carbon composition -20 to +10 carbon granular mixed with binder carbon comp_molded -20 to -8 deposited carbon -25 to -8 C film on ceramic rod,cut spiral grooves metal film -35 to -20 M film on ceramic rod,cut spiral grooves wirewound -40 to -10 Zeners have shot noise (7V or less ) below 3V .. very little noise 3V = 30uV_per_rt_Hz at 250uA 30V = 40uV_per_rt_Hz at 250uA Battery quiet until almost dead. Caps quiet because they low pass filter. Pop Corn Noise pops msec to sec 1/F_Noise about three times value at 1Hz 2.7 times 1Hz over 1 second 4 times 1Hz over 1 hour 4.4 times 1Hz over 1 day transient Noise Bandwidth = 1/2 sample period RMS=stand_dev=Flat_nV_per_sqrtHz*sqrt(Bandwidth) 6 sd about 1ppm Q number of cycles to settle ratio of real resistance to X 6 time constants in a cycle (2pi) __ / \ each section a RC \__/ noise/offset You can treat the same Shot noise statical, not physcial Thermal noise physical, not stasitical negative feedback one pole must dominate for stabilty Write down all assumptions, calculations Tolerance Think in terms of signal to noise/interferrence Know errors that can be cancelled out If Linear Use the SuperPosition Theory... offset/noise drops as the square root.. ======================SIGNAL_3D_SPECTRUM================================ 3D_Spectrum Tayler discovered that an alternative ways to express a function was to use a power series. This new way of looking at sine and cosine functions lead to the discovery that they were related to the exponental function. COS(X) = 1 - X^2/2 + X^4/(4*3*2) +..etc SIN(X) = + X + X^3/(3*2) +..etc EXP(X) = 1 + X + X^2/2 + X^3/(3*2) + X^4/(4*3*2) +..etc By using complex numbers, Euler was able to find the following relationship. EXP(j*X) = COS(X) -j*SIN(X) In the form below , this relationship effectly translates a time function into a spectrum function. COS(X) = EXP(j*X)/2 +EXP(-j*X)/2 SIN(X) = j*EXP(j*X)/2 -j*EXP(-j*X)/2 A sine wave really needs only three data points to completely define it. By looking at a spectrum, if you know the amplitude and frequency, you still can't relate it to time. You need to know when the sinewave crosses zero in terms of time. In other words, you need to know phase information. Suppose you have a 5Hz signal that has a peak amplitude of 3volts. That is saying that the X term will be 2*PI for every 1/5th of a second. This is expressed below. function(time) = 3*SIN(2*PI*5Hz*time) <--ignoring phase It is traditional to replace a the 2*PI*Freq term with a single unit call Omega or "w". w = 2*PI*Freq If the amplitude is expressed as the term AMP, then.. function(time) = AMP*SIN(2*PI*5Hz*time) <--ignoring phase ^ /_\ | __-----__ | __/ \ _ | _/ \_ _ _|/ _ _ _ _ _ _ _ _ _ \_ _ _ _ _ _ _ _ _ _ _ _ \ | \_ _/ / | \__ __/ | \__ ___/ | ----- To know phase, one has to know in time when the sinewave crosses zero. If at time "zero" the signal is zero, you have a "SINE" wave. Now with the three data points..(Frequency, Amplitude, Phase), you know the signal in terms of voltage versus time. function(time) = AMP*SIN(2*PI*5Hz*time) It is common to express the magnitude and phase of something in terms of a complex number. The ratio to the real to imaginary values of give the phase while the combine distance of the real and imaginary parts give the magnitude. REAL ^ X = valueR +jvalueI /_\ | valueR .|.. __ Magnitude | /| | / : | / __-- Phase | /_--: IMAGINARY |/-___:____________\ : / jvalueI By adding one more dimension to represent frequency, the Euler relationship shows an alternative way to view Frequency, Amplitude, and Phase. Remmenber SIN(X) = j*EXP(j*X)/2 -j*EXP(-j*X)/2 So the function of time can be viewed as a function of frequency. The sine wave is mapped as two vectors along the jw axis. function(time) = AMP*SIN(2*PI*5Hz*time) AMP*SIN(w*t) = j*AMP*EXP(j*w*t)/2 - j*AMP*EXP(-j*w*t)/2 REAL MAGNITUDE ^ /_\ | \ | \ | \ | \ | \ | /______\ <--|- -j*AMP*EXP(-j*w*t)/2 \ \ | \ | \ | IMAGINARY MAGNITUDE \|__________________\ \ / \ \ j*AMP*EXP(j*w*t)/2---> \______\ <--Magnitude = AMP \ / <--Direction = Phase \ <--Position = Freq \ _V J_OMEGA These vectors always come in pairs ( jw and -jw). Where a vector is, how large it is, and which way it is pointed tells frequency, amplitude, and phase. If the phase of the signal happened to be a COSINE, then the 3D spectrum would look like... (Amplitude) REAL MAGNITUDE ^ /|\ | \ | \ | \ ^ | \ /|\ | \ | | \| | AMP*EXP(-j*w*t)/2 \ | \ | (90degree Amplitude) \ | IMAGINARY MAGNITUDE \|__________________\ \ ^ / \ /|\ \ | \| AMP*EXP(j*w*t)/2 \ \ \ \ _V J_OMEGA (frequency) 3D_Amplitude_Modulation_COS ________________\_ _ _\_ _ _\ 1001KHz at 0 degrees / / / 1000KHz at 0 degrees 999KHz at 0 degrees _ /| / 1001KHz leads 45 degree ________________\/ 1000KHz at 0 degrees /\ 999KHz lag by 45 degrees \ _\| ^ /|\ | 1001KHz leads 90 degree ________________\| 1000KHz at 0 degrees /| 999KHz lag by 90 degrees | \|/ V _ /\ \ 1001KHz leads 135 degree ________________\ 1000KHz at 0 degrees / 999KHz lag by 135 degree / |/_ 1001KHz leads 180 degree ______/ _ _/ _ _\ 1000KHz at 0 degrees \ \ / 999KHz lag by 180 degree ^ (Amplitude) /|\ REAL MAGNITUDE \ | ^ \ ^ | /|\ \ /|\| | \ | | | \| | | \ | ^ | \|/|\ | \ | | \| | \ | ~etc~| ^ \ | /|\ IMAGINARY \|_____|____________\ \ ^ | / \ /|\| MAGANITUDE ~etc~| | \| | \ | ^ \|/|\ \ | \| \ _V J_OMEGA (frequency) 3D_Amplitude_Modulation_SIN ^ /_\ | __--|--__ __-----__ Sine lags 90 cosine __/ | _\/_ \__ _/ | _/ \_ \_ _ _ / _ _ _ _ |/_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \ | \_ \ _/ / | \__ __/ | \__ ___/ ----- _ _ _/| | | | | _ _ | | | | | | | \ / | | | | | | | | | _ | | | | | | | | | | | \ _ | | | | | | | | | | | | _ / | | | | | | | | | | | | | | | | __ _/| | | | | | | | | | | | | | | | | | | | --- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | -- |_| | | | | | | | | | | | | | | | | | |/ |_| | | | | | | | | | | | | | |_/ | | | | | | | | | | | | | \_| | | | | | | | | |_/ \_| | | | | | | |_| |_| |/ ^ /|\ | 1001KHz lags 90 degree ________________\| 1000KHz at 0 degrees /| 999KHz leads by 90 degrees | \|/ V _ /| / 1001KHz lags 45 degree ________________\/ 1000KHz at 0 degrees /\ 999KHz leads by 45 degrees \ _\| ________________\_ _ _\_ _ _\ 1001KHz at 0 degrees / / / 1000KHz at 0 degrees 999KHz at 0 degrees ^ (Amplitude) /|\ REAL MAGNITUDE \ | ^ \ | /|\ \ | | \ | | /____\ | | \ \ | | \| | \ | \____|\ \ |/ ~etc~| ^ \ | /|\ IMAGINARY \|_____|____________\ \ | / \ | MAGANITUDE ~etc~ | /____\ | \ \ | \| \ \____\ \ / _V J_OMEGA (frequency) ---------------3DS_Phase_Modulation--------------------- ^ /|\ | __--|--__ __-----__ Sine lags 90 cosine __/ | _\/_ \__ _/ | _/ \_ \_ _ _ / _ _ _ _ |/_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \ | \_ \ _/ / | \__ __/ | \__ ___/ ----- _________________\ /| 1000KHz at 0 degrees | 1001KHz lags 90 degrees \|/ 999KHz lags 90 degrees V | | \|/ V __________________\ 1000KHz at 0 degrees /\ 1001KHz lags 45 degrees / \ 999KHz lags 135 degrees |/_ _\| ___________/_ _ _\_ _ _\ 1000KHz at 0 degrees \ / / 1001KHz lags 0 degrees 999KHz lags 180 degree _ _ |\ /| \ / 1001KHz leads 45 degrees ________________\/ 1000KHz at 0 degrees / 999KHz lags 225 degree ^ /|\ | 1001KHz leads 90 degrees | 1000KHz at 0 degrees ^ 999KHz lags 270 degree /|\ | ________________\| / ^ (Amplitude) /|\ REAL MAGNITUDE \ | ^ \ | /|\ \ | | \ | | /____\ | | \ \ | | \| | \ | /____\ | \ \ | ~etc~| ^ \ | /|\ IMAGINARY \|_____|____________\ \ | / \ | MAGANITUDE ~etc~ | \__|_\ \ | / \| \ \____\ \ / _V J_OMEGA (frequency) ----------------FM_2_PM------------------- _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | Pure sign wave (no FM) _ _ __ __ _ _ _ _ _ _ | | | | | | | | | | | | | | [|[| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | |_| |_| |__| |_| |_| |_| |_|L||_| |_| |_| A Moving sign wave Frequency Modulate a 1kHz carrier with a 1Hz signwave Freq(t) = [ 1000 + sin(2*PI*t) ] Signal= sin( (Freq(t))*2*PI*t) <--FM format at time = 1/4 The Frequency is 1001 Hz at time = 3/4 The Frequency is 999 Hz Another way to think of it is Phase Modulation... Signal= sin( (1000)*2*PI*t +Phase(t)) <-- PM format the phase will be Integral of FM ... _ / Phase(t) = | 2*PI*sin(2*PI*t)*dt = -cos(2*PI*t) _/ So 1Hz peak FM over 1Hz will generate 1radian peak of PM And to convert FM to PM, just integrate and One Hz FM at One Hz equals One radian PM The PM will lag the FM which causes it. ^ /|\ Freq=1001Hz| Phase= +1radian __--|--__ __-----__ __/ | _\/_ \__ _/ | _/ \_ \_ _ _ / _ _ _ _ |/_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \ _/| \_ \_ _/ / __/ | \__ \__ __/ TIME __/ | \__ ___/___ ---- ----- ---- Phase= -1radian Freq=999Hz FFT_SINE Using Excel to FFT signwaves ^ /|\ Signal = SIN(2*P()*Num/8) | __--S--__ | __S \S_ | _/ \_ _ _S/ _ _ _ _ _ _ _ _ _ \S _ _ _ _ _ _ _ _ _ _ _ \ | \_ _/ / | \__ __/ | S__ ___S | --S-- Num Samp FFt 0 0 0 1 .71 -4i 2 1 0 3 .71 0 4 0 0 5 -.71 0 6 -1 0 7 -.71 4i REAL MAGNITUDE ^ /|\ | \ | \ | /______\ | \ \ | \ | Nyq\__ | \ | \ | \_|____\ \|____/_____________\ -4j /______\ / @bin1 \ \ IMAGINARY MAGNITUDE \ Nyq\__ \ \ \______\ +4j \ / @bin7 _V J_OMEGA FFT_NYQ ^ /|\ Signal = SIN(2*P()*Num/8) S_ S S S | \ / \ / \ / \ | \ / \ / \ / \ _ _|_ _|_ _|_ _|_ _|_ _|_ _| _ \ _ _ _ \ | | | | | | | | | / | \ / \ / \ / \ / | \S/ \S/ \S/ \S/ | Num Samp FFt 0 1 0 Real \ 1 -1 0 Complex | 2 1 0 Complex | 8 data points in/out 3 -1 0 Complex | 4 1 8 Real / 5 -1 0 6 1 0 7 -1 4 ^ REAL MAGNITUDE ^ /|\ /|\ | | | \ | | \ | | \ | | \ | | \ | | Nyq\| | ^ \ | /|\ \ | | 8 at @bin4 \ | | \|___|_________________\ \ | / \ | IMAGINARY MAGNITUDE \ | Nyq\| \ \ \ \ _V J_OMEGA FM_2_PM _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | Pure sign wave _ _ __ __ _ _ _ _ _ _ | | | | | | | | | | | | | | [|[| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | |_| |_| |__| |_| |_| |_| |_|L||_| |_| |_| A Moving sign wave Frequency Modulate a 1kHz carrier with a 1Hz signwave Freq(t) = [ 1000 + sin(2*PI*t) ] Signal= sin( (Freq(t))*2*PI*t) <--FM format at time = 1/4 The Frequency is 1001 Hz at time = 3/4 The Frequency is 999 Hz Another way to think of it is Phase Modulation... Signal= sin( (1000)*2*PI*t +Phase(t)) <-- PM format the phase will be Integral of FM ... _ / Phase(t) = | 2*PI*sin(2*PI*t)*dt = -cos(2*PI*t) _/ So 1Hz peak FM over 1Hz will generate 1radian peak of PM And to convert FM to PM, just integrate and One Hz FM at One Hz equals One radian PM The PM will lag the FM which causes it. ^ /|\ Freq=1001Hz| Phase= +1radian __--|--__ __-----__ __/ | _\/_ \__ _/ | _/ \_ \_ _ _ / _ _ _ _ |/_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \ _/| \_ \_ _/ / __/ | \__ \__ __/ TIME __/ | \__ ___/___ ---- ----- ---- Phase= -1radian Freq=999Hz Noise Floor_dB Theoretical FFT Points 12Bits 14Bits 16Bits 1024 101 113 125 2048 104 116 128 4096 107 119 131 8192 110 122 134 16384 113 125 137 32768 116 128 140 SNR_dB 74.0 86.0 98.1 ------------------------------------------------------------- FFT _ _ _ _ _ _ _ _ _ _ _ _ OdB Spurious | ^ Free Dyn | | Fundamental | SNR Range | | at_max_signal v |__\ |--|---------------------------RMS_Noise_Level / | | | 3rd Harmonic | _| | __| 5rd Harmonic |/ \/\__|/\/ |_/\/ -------Average Noise Level |______________________________\ (Noise Floor) ^ ^ ^ ^ ^ ^ / Bin_width = Sample_rate/Number_Points_FFT F_max =Sample_rate/2 RMS Signal A/SQRT(2) = (FSR/2)/SQRT(2) = 2^(n-1)*q/SQRT(2) RMS Noise = Qn = q/SQRT(12) SNR= RMS_Signal/RMS_Noise = 2^(n-1)*SQRT(6) SNR_pos_dB = 20*log(2^(n-1)*SQRT(6)) = 6.02*n + 1.76 NOISE_FLOOR_neg_dB = 6.02*n + 1.76 + 10*log(N/2) NOISE_FLOOR_neg_dB = 6.02*n +10*log(3*N/(PI*ENBW)) THD_neg_dB= 20*log(SQRT(sum_of_Harm_squared) ) Note: HAR (-dB) SINAD_pos_dB = -20*log( SQRT( SNR^2+THD^2 ) ) SINAD = Singal to noise ratio and distortion ENOB (SNR+Distortion-1.76+20*log(Amp_FS/Amp_Actual))/6.02 ENOB= Effective number of bits fs = Sampling Rate (Hz) fin = Input Signal Frequency (Hz) FSR = Full Scale (Input) Range of Sampling A/D Converter FS = FSRi2 = Full Scale input of Sampling A/D Converter A = Input Signal Amplitude = FS = FSR/2= OdB n = Numberof Bitsof Resolution q = LSB Size LSB = Least Significant Bit N = Numberof FFT Points N/2 = Numberof Frequency Bins (real component) ENBW = Equivalent noise bandwidth of the window function. (For a four-term Blackman-Harris window, ENBW= 2.) --------------------FFT_COS------------------------------ ^ /|\ Signal = SIN(2*P()*Num/8) S--__ __-- | \S_ __S | \_ _/ | \S S/ | \_ _/ | \__ __/ | S__ ___S |_ _ _ _ _ _ _ _ _ _--S-- _ _ _ _ _ \ | / Num Samp FFt 0 2 8 1 1.71 4 2 1 0 3 .29 0 4 0 0 5 .29 0 6 1 0 7 1.71 4 REAL MAGNITUDE ^ /|\ | \ | \ ^ 8 at @bin0 \ /|\ \ ^ | Nyq\_ /|\| \ | | \ | | \| | ^ 4 at @bin1 \ |/|\ \|_|_________________\ \ | / \| IMAGINARY MAGNITUDE \ \ ^ Nyq\_ /|\ 4 at @bin7 \ | \ | \| \ _V J_OMEGA ----------------FM_2_PM------------------- _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | Pure sign wave _ _ __ __ _ _ _ _ _ _ | | | | | | | | | | | | | | [|[| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | | | | | | | | | | | | | | |||| | | | | | | |_| |_| |__| |_| |_| |_| |_|L||_| |_| |_| A Moving sign wave Frequency Modulate a 1kHz carrier with a 1Hz signwave Freq(t) = [ 1000 + sin(2*PI*t) ] Signal= sin( (Freq(t))*2*PI*t) <--FM format at time = 1/4 The Frequency is 1001 Hz at time = 3/4 The Frequency is 999 Hz Another way to think of it is Phase Modulation... Signal= sin( (1000)*2*PI*t +Phase(t)) <-- PM format the phase will be Integral of FM ... _ / Phase(t) = | 2*PI*sin(2*PI*t)*dt = -cos(2*PI*t) _/ So 1Hz peak FM over 1Hz will generate 1radian peak of PM And to convert FM to PM, just integrate and One Hz FM at One Hz equals One radian PM The PM will lag the FM which causes it. ^ /|\ Freq=1001Hz| Phase= +1radian __--|--__ __-----__ __/ | _\/_ \__ _/ | _/ \_ \_ _ _ / _ _ _ _ |/_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \ _/| \_ \_ _/ / __/ | \__ \__ __/ TIME __/ | \__ ___/___ ---- ----- ---- Phase= -1radian Freq=999Hz ======================SIGNAL_DEFINITIONS================================ exp(j*X) cos(x) +j*sin(x) Maxwell's Equations __ \/ dot J = -delta_p/dt __ \/ cross E = -delta_B/dt V=delta_Phi/dt __ \/ cross H = J+ delta_D/dt H =I*N __ \/ dot D = p __ \/ dot B = 0 Theoretical Noise Floor_dB (Table 1) FFT Points 12Bits 14Bits 16Bits 1024 101 113 125 2048 104 116 128 4096 107 119 131 8192 110 122 134 16384 113 125 137 32768 116 128 140 SNR_dB 74.0 86.0 98.1 FFT _ _ _ _ _ _ _ _ _ _ _ _ OdB Spurious | ^ Free Dyn | | Fundamental | SNR Range | | at_max_signal v |__\ |--|---------------------------RMS_Noise_Level / | | | 3rd Harmonic | _| | __| 5rd Harmonic |/ \/\__|/\/ |_/\/ ----------- Ave Noise Level |______________________________\ (Noise Floor) ^ ^ ^ ^ ^ ^ / Bin_width Sample_rate/Number_Points_FFT F_max Sample_rate/2 RMS Signal A/SQRT(2) (FSR/2)/SQRT(2) 2^(n-1)*q/SQRT(2) RMS Noise Qn q/SQRT(12) SNR RMS_Signal/RMS_Noise 2^(n-1)*SQRT(6) SNR_pos_dB 20*log(2^(n-1)*SQRT(6)) 6.02*n + 1.76 (See Table 1) NOISE_FLOOR_neg_dB 6.02*n + 1.76 + 10*log(N/2) (See Table 1) 6.02*n +10*log(3*N/(PI*ENBW)) THD_neg_dB 20*log(SQRT(sum_of_Harm_squared) ) Note: HAR (-dB) SINAD_pos_dB -20*log( SQRT( SNR^2+THD^2 ) ) SINAD Singal to noise ratio and distortion ENOB Effective number of bits (SNR +Distort -1.76 +20*log(Amp_FS/Amp_Actual) )/6.02 fs Sampling Rate (Hz) fin Input Signal Frequency (Hz) FSR Full Scale (Input) Range of Sampling A/D Converter FS FSRi2 = Full Scale input of Sampling A/D Converter A Input Signal Amplitude FSR/2= OdB n Numberof Bitsof Resolution q LSB Size LSB Least Significant Bit N Numberof FFT Points N/2 Numberof Frequency Bins (real component) ENBW Equivalent noise bandwidth of window function. Blackman-Harris four-term Blackman-Harris window, ENBW= 2. Sample Commmand SAMPLE_HOLD | |<-----> Acquistion Time | | _ |_|_ _ /_\_ _ _ _ _ Specified Error Band |_|___/___\__--__________________________ Final Value |_|_ _|_ _ \/ _ _ _ Specified Error Band | | | | | | <-- cap charging | | / |_|_/_____________________ ^ ^ Switching time delay (Aperture delay) Hold Commmand SAMPLE_HOLD | |<---> Hold mode settle Time | | |_____|__ _ _ _ _ _ _ Specified Error Band | | \ /\ __________________________ Final Value | |_ _\/_ -- _ _ _ Specified Error Band | | | | _ Start Convert Pulse |_____|___| |______________________ Anti_Aliasing Filters Butterworth Flattest response near dc moderately fast roll off attenuation rate = 6dB/octave stable phase shift fc at -3dB overshoot on step response constant amplitude>>time delay or phase response Bessel Optimum flat phase response avoids overshoot/undershoot important for pulses amplitude not as flat roll off is slow gain rool off modifiy amplitude moderate attentuation rate fc at phase shift is 1/2 rate Chebyshev Rapide attenutation above cut off some bandpass ripple squarer amplitude response than butterworth less desireable phase and time delay fc at attenuation exceeds specified ripple Cauer (Elliptical) Surpasess for critical amplitude application ver sharp roll offwith some ripple squarest possible amplitude response porr phase and transient responses. ----------------------SIGNAL_TRANSFER_FUNCTIONS---------------------- TYPES NONLINEAR incidental and intentional nonlinearities. Incidental nonlinearities Saturation, dead zone, and backlash may inaccurate or even instability Intentional nonlinearities NONLINEAR SYSTEMS ^ OUT /|\ Limiting | ____ Saturation | / | / _______|/________\ IN / / /| ____/ | ^ OUT Deadzone /|\ threshold | | / _______|___/________\ IN / | / / | ^ OUT On/off switch /|\ coulomb friction |_____ | _______|____________\ IN | / ______| ^ OUT preload /|\ | / |/ _______|____________\ IN | / /| / ^ OUT spring /|\ hardening | / | _/ _______|-____________\ IN _-| / / | / ^ OUT linear /|\ rectification | / | / _______|/____________\ IN | / | ^ OUT relay hysteresis /|\ | ____/_ | | | \ | v ^ _______|__|__|______\ IN | | | / v ^ | _\_|__| | / ^ OUT Toggle /|\ --|<----- | | ^ | | | ____v__|__^__________\ IN | | | / | | | ------>-- ^ OUT backlash /|\ -|<----_ / | /| /<-|--- / ____/___|___/__________\ IN / ---|->/ / |/_ | / ----->- Describing Function the following basic assumptions. A) input to nonlinear element n is sinusoidal only the fundamental component of output of n. contributes to the input. The output response of nonlinear element consists of the fundamental frequency and harmonics Generally, harmonic smaller than fundamental most control systems the system a low-pass filter and the higher harmonics are attenuated. If harmonics small can be neglected B) All nonlinearities lumped into one single nonlinear element n C) output of nonlinear element is a function only of the present value and past history of the input, i D) n is not a function of time. The describing function of a nonlinear element __ + / \ e ____ ____ ___\__/ \/ \__\__| n |___\__| G |________\ C r / \ /\ / / |____| / |____| | / \__/ | - ^ | |______________________________| diagram of nonlinear closed-loop system. Limit Cycle a phenomenon of oscillation peculiar to nonlinear systems. oscillatory behavior characterized by a constant am1diiude and frequency determined by the nonlinear properties of system. the output approaching the amplitude of that limit cycle regardless of the initial condition and forcing function. _ | _\| /| __-->|--__ / _ | \ | / /| | _\| \ V _____|______|______|______ ^ | _ | | | | \ |\ | |/_ / V \ | / <--__|_<-- |/_ easily recognized in phase plane as an isolated closed path soft self-excitation limit cycle in presence of very small excitation or disturbances. hard self-excitation systems requiring forced excitation above a certain minimum amplitude or appropriate initial Undervoltage Lockout Hysteresis Switcher on _|_ _____________________ | | <-- | | | | ^ | | | v --> | | Switcher off |_____|_______________|______\ Input Voltage V(turn off) V(turn on) / DeltaV V(turn on) No load voltage from transfomer @ lO8Vac V(turn off) Full load voltage from transformer @ lO5Vac Delta_V V(turn on) - V(turn off) ; Hysteresis using undervotage circuit with hysteresis, we can prevent the power supply from oscillating on and off. V(turn on) the no load voltage from the transformer. This is very close to 108*sqrt(2)*Ns/Np V(turn off) the full load voltage from the transformer at IO5Vac. This value is sensitive to load conditions for each design, and should be measured on the bench during design. delta_V the difference between these two voltages, is value use for the undervoltage lockout hysteresis. nonlinear system output of nonlinear devicec ontains harmonic and subbarmonic frequencies of the input signal Hysteresis Multivalued functions exist when two or moore function values correspond to the value of the variable. examples hysteresis curves of magnetic materials and the backlash of a gear train. ------------------------------------------------------------- ROOT_LOCUS ___ R(s) + / _ \ E _____ C _____\/ \ \_______| G(s)|_________\ /\ /_ / |_____| | / \___/ | - ^ _____ | |___| H(s)|_________| B |_____| For stabilty G(s)*H(s)=(K*(s-z)*../((s-p)..)=>gain< 1 @ 180deg ------------------------------------------------------------- IMAGINARY ^ _ /|\ How loop poles move /| | as loop gain is increased / | <--X X->|<-X |____\ REAL \ | / _\| IMAGINARY ^ _ /|\ /| | / | \ | <--X X->Z X->|<-X |____\ REAL / | / \ _\| IMAGINARY ^ _ /|\ /| | / | \ | <--X X->Z X->|<-X |____\ REAL / | / \ _\| ----------------------SIGNAL_STABLITY---------------------- Complex_Poles 1/(s^2 +a*s +b) =1/(( s +alpha +jwo )*( s +alpha -jwo )) 1/(s^2 +a*s +b) =1/(s^2 +(wo/Q)*s +wo^2) IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | 20dB|................................................ | . . . . | . b Q=10 . . | . . . . | . b . . . | . b .b . . 0dB|..........a..a...a.............................. | . a . b . . | . . . . | . a b . . | . . a . . | . . a . . -20dB|.............................a.................. | . . . | . . . . | . . . . | . . . . | . . . . |______________________________________________ . 100KHz 1MHz 10MHz 100MHz 1GHz CONDITIONALY STABLE GAIN_PHASE CONDITIONALY STABLE can only oscilate by changing loop gain during start up,sat.lower VCC .. delicate 60dB |................................................ | P . . . . | P . . . . |G G G <== Open Loop Gain . . | P G . . . . | P G . . . . 40dB|........P..G.P..................................90 | . G . . . | . G . . . | . G P . . . | . G . . . | . G . . . 20dB |....................P..G.........P..............45 | . . G P P . | . . G .P 38deg . | . P. GP . P Phase . | . . G . P Margin . | . .P P G . P . 0dB |0000000000000000000000000000P000000V0000P0000000 0 | . . P . G ^ Gain . | . . . G| Margin | . . . V . | . . . G . | . . . . -20dB|______________________________________________ . 10KHz 100KHz 1MHz 10MHz 100MHz on imaginary axes When poles are exactly on the imaginary axis it says you will have a constant sine wave at frequency 2*pi*w _ _ _ | | | | | | |_|_|_|_|_|_____\ | | | | | | / Time PULSE RESPONSE | |_| |_| | IMAGINARY ^ /|\ | X ______|____\ REAL | / X Normal poles When poles are slightly in the left half plane it says you will have a sine wave at frequency 2*pi*w which is decaying at the a rate defined the dsitance from the axis. _ | | _ | | | | _ |_|_|_|_|_|_____\ | | | |_| / Time | |_| PULSE RESPONSE IMAGINARY ^ /|\ | X | ______|____\ REAL | / X | Left Half Plane When poles are in left half plane you will have a decay at the a rate defined the dsitance from the axis. _ PULSE RESPONSE - _ - _ ________________\ / Time IMAGINARY ^ /|\ | | ___X__|____\ REAL | / | Right Half Plane When poles are in Right half plane you are unstable PULSE RESPONSE _ - _ - ________________\ / Time IMAGINARY ^ /|\ | | ______|_X__\ REAL | / | _ PULSE RESPONSE _ | | _ | | | | |_|_|_|_|_|_____\ | |_| | | | / Time |_| | IMAGINARY ^ /|\ | | X ______|____\ REAL | / | X