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Chap 3. signal processing elemnt part three

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An analog-to-digital converter (ADC) converts a continuous analog signal to a discrete digital signal. It uses two main processes: quantization, which breaks down the analog value into a set of finite states, and encoding, which assigns a digital word to each state. An ADC uses sampling to take discrete voltage measurements of a signal over time. A digital-to-analog converter (DAC) performs the opposite conversion, reconstructing an analog signal from a digital code. DACs use techniques like binary weighted resistors or R-2R ladders to generate weighted voltage levels corresponding to each bit in the digital input code.

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Chapter Three Signal Processing and Conversion Dept. of Electrical and Computer Eng., AASTU Addis Ababa By Biruk T.
Analog to Digital Converter (ADC) • An analog signal is a continuous signal that contains time-varying quantities, such as temperature or speed, with infinite possible values in between. which are directly measurable quantities. • An analog signal can be used to measure changes in some physical phenomena such as light, sound, pressure, or temperature. • ADC is an electronic integrated circuit which transforms a signal from analog (continuous) to digital (discrete) form. • ADC Provides a link between the analog world of transducers and the digital world of signal processing and data handling. Examples: • Thermometer – mercury height rises as temperature rises • Car Speedometer – Needle moves farther right as you accelerate
There are two step Process :- • Quantizing - breaking down analog value is a set of finite states • Encoding - assigning a digital word or number to each state and matching it to the input signal Quantizing: the number of possible states that the converter can output is: N=2n The number of possible states that the converter can output is: Sampling :- • It is a process of taking a sufficient number of discrete values at point on a waveform that willdefine the shape of waveform. • The more samples you take, the more accurately you will define the waveform. • It converts analog signal into series of impulses, each representing amplitude of the signal atgiven point.
ADC SAMPLED AND QUANTIZED WAVEFORM DAC RECONSTRUCTED WAVEFORM ADC DAC DSP MemoryChannel Analog Digital timetime Analog Digital Amplitude Value “Real World” Sampled Data Systems Consist of ADCs and DACs Nyquist Rule:  Use a sampling frequency at least twice as high as the maximum frequency in the signal to avoid aliasing. maxsamples 2 ff 
Example: Quantizing You have 0-10V signals. Separate them into a set of discrete states with 1.25V increments. (How did we get 1.25V?) The number of possible states that the converter can output is: N=2n where n is the number of bits in the AD converter Example: For a 3 bit A/D converter, N=23=8. Analog quantization size: Q=(Vmax-Vmin)/N = (10V – 0V)/8 = 1.25V Output States Discrete Voltage Ranges (V) 0 0.00-1.25 1 1.25-2.50 2 2.50-3.75 3 3.75-5.00 4 5.00-6.25 5 6.25-7.50 6 7.50-8.75 7 8.75-10.0
Encoding • Here we assign the digital value (binary number) to each state for the computer to read. There are two ways to best improve accuracy of A/D conversion: • increasing the resolution which improves the accuracy in measuring the amplitude of the analog signal. • increasing the sampling rate which increases the maximum frequency that can be measured. Output States Output Binary Equivalent 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111  Resolution (number of discrete values the converter can produce) = Analog Quantization size (Q) = Vrange / 2^n, where Vrange is the range of analog voltages which can be represented limited by signal-to-noise ratio (should be around 6dB) In example: Q = 1.25V, this is a high resolution. A lower resolution would be if we used a 2-bit converter, then the resolution would be 10/2^2 = 2.50V.
Types of A/D convertor:- • FlashADC • Dual slope/Counter slopeADC • Successive ApproximationADC Flash ADC: series of comparators, each one compares input to a unique reference voltage. • comparator outputs connect to a priority encoder circuit  produces binary output Fast – but more expensive : Single cycle - Uses many Comparators in parallel with different reference voltages How Flash Works  As the analog input voltage exceeds the reference voltage at each comparator, the comparator outputs will sequentially saturate to a high state.  The priority encoder generates a binary number based on the highest-order active input, ignoring all other active inputs.
Figure: Parallel, simultaneous flash A/D conversion. Analog Digital• 2N-1 comparators for N-bits • Each reference voltage equivalent to a quantization level • Encoding logic produces word Advantages • Simplest in terms of operational theory, Most efficient in terms of speed, very fast, limited only in terms of comparator and gate propagation delays Disadvantages • Lower resolution, Expensive, For each additional output bit, the number of comparators is doubled, i.e. for 8 bits, 256 comparators needed
9 Integrating or Single & Dual Slope • Accumulate the input current on a capacitor for a fixed time and then measure the time (T) to discharge the capacitor at a fixed discharge rate. 1) S1->V1:Integrate the input on the cap. For N clock ticks 2) S1-> -Vref: restart clock (S2->counter) discharge C at know rate(governed by -Vref and R) 3) When the cap. is discharged to 0 voltage, the comparator will stop the counter. problem --very slow 0 1 arg _ / constant T fixed ch e held Q Idt I V R       Discharge time for stopping counter by S2 depends on RC and Q
Successive Approximation: continuously convert analog signal to discreet digital signal. • Assume that • Vin= 0.6 volts • Vref=1volts • Find the digital value of Vin Divided Vref by 2 Compare Vref /2 with Vin If Vin is greater than Vref /2 , turn MSB on (1) If Vin is less than Vref /2 , turn MSB off (0) Vin =0.6V and V=0.5 Since Vin>V, MSB = 1 (on)
• Next Calculate MSB-1 (bit 8) • Compare Vin=0.6 V to V=Vref/2 + Vref/4= 0.5+0.25 =0.75V • Since 0.6<0.75, MSB is turned off • Calculate MSB-2 (bit 7) • Go back to the last voltage that caused it to be turned on (Bit 9) and add it to Vref/8, and compare with Vin • Compare Vin with (0.5+Vref/8)=0.625 • Since 0.6<0.625, MSB is turned off • Calculate the state of MSB-3 (bit 6) • Go to the last bit that caused it to be turned on (In this case MSB-1) and add it to Vref/16, and compare it to Vin • Compare Vin to V= 0.5 + Vref/16= 0.5625 • Since 0.6>0.5625, MSB-3=1 (turned on)
• This process continues for all the remaining bits.
ADC example 2. using successive approximation • Given an analog input signal whose voltage should range from 0 to 15 volts, and an 8-bit digital encoding, calculate the correct encoding for 5 volts. Then trace the successive- approximation approach to find the correct encoding. • Assume M = 2n – 1, a / Vmax = d / M, 5 / 15 = d / (256 - 1), d = 85 or binary 01010101 Step 1-4: determine bits 0-3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 ½(Vmax – Vmin) = 7.5 volts Vmax = 7.5 volts. ½(7.5 + 0) = 3.75 volts, Vmin = 3.75 volts. ½(7.5 + 3.75) = 5.63 volts, Vmax = 5.63 volts ½(5.63 + 3.75) = 4.69 volts, Vmin = 4.69 volts. 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 ½(5.63 + 4.69) = 5.16 volts, Vmax = 5.16 volts. ½(5.16 + 4.69) = 4.93 volts, Vmin = 4.93 volts. ½(5.16 + 4.93) = 5.05 volts, Vmax = 5.05 volts. ½(5.05 + 4.93) = 4.99 volts
Constructing SA ADC SARComparator DAC Vmax Vmin Analog input SAR BUF Digital output State machine SAR: Successive approximation register Timing control
Digital to Analog Converter (DAC) • Signals are easily stored and transmitted in digital form, but a DAC is needed for the signal to be recognized by human senses or other non-digital systems. • A digital-to-analog converter (DAC or D-to-A) is a device that converts a digital (usually binary) code signal to an analog signal (current, voltage, or electric charge). • Digital signals is a type of signal that can take on a set of discrete values (a quantized signal) • It can represent a discrete set of values using any discrete set of waveforms .. And we can represent it like (0 or 1) ,( on or off )….. etc Examples:  Light switch can be either on or off
ANALOG OUTPUT DIGITAL INPUT REFERENCE INPUT Digital Input Analog Output = x Reference Input (2N - 1) What is a Digital-Analog Converter? • Produces a Quantized (Discrete Step) Analog Output (Voltage or Current) in Response to Binary Digital Input Code • A reference quantity (either voltage or current) is accurately divided into binary and/or linear segments. • The digital input drives switches that connect an appropriate number of segments to the output. • Finite Number of Discrete Values : 2N Resulting in Quantization Uncertainty • Sampling and Quantization Impose Fundamental yet Predictable Limitations RESOLUTION = N BITS
10111001 10100111 10000110010101000011001000010000 AnalogOutputSignal Digital Input Signal Two Types of DAC: • Binary Weighted Resistor • R-2R Ladder
Terms & Equations Offset: minimum analog value Span (or Range): difference between maximum and minimum analog values Max - Min n: number of bits in digital code (sometimes referred to as n-bit resolution) Bit Weight: analog value corresponding to a bit in the digital number Step Size (or Resolution): smallest analog change resulting from changing one bit in the digital number, or the analog difference between two consecutive digital numbers; also the bit weight of the Span / 2n (Assume M = 2n) Let AV be Analog Value; DN be Digital Number: AV = DN * Step Size + Offset = (DN / 2n )* Span + Offset DN = (AV - Offset) / Step Size = (AV - Offset) * 2n / Span
Bit Weight Each bit is weighted with an analog value, such that a 1 in that bit position adds its analog value to the total analog value represented by the digital encoding. Digital Bit Bit Weight (V) 7 10/2 = 5 6 10/4 = 2.5 5 10/8 = 1.25 4 10/16 = 0.625 3 10/32 = 0.313 2 10/64 = 0.157 1 10/128 = 0.078 0 10/256 = 0.039 Binary Weighted Resistor • Utilizes a summing op-amp circuit • Weighted resistors are used to distinguish each bit from the most significant to the least significant • Transistors are used to switch between Vref and ground (bit high or low) Example: -5 V to +5 V analog range, n=8 Solution Vref(pp)=10v
Binary Weighted Resistor • Assume Ideal Op-amp • No current into op-amp • Virtual ground at inverting input • Vout= -IRf - + R 2R 4R 2nR Rf Vout I Vref Voltages V1 through Vn are either Vref if corresponding bit is high or ground if corresponding bit is low V1 is most significant bit, Vn is least significant bit I - + R 2R 4R 2n-1R Rf Vout Vref V1 V2 V3 Vn        R V R V R V R V RIRV 1-n n321 ffout 242  MSB LSB
If Rf=R/2        n n321 fout 2842 VVVV IRV  For example, a 4-Bit converter yields        16 1 8 1 4 1 2 1 0123refout bbbbVV Where b3 corresponds to Bit-3, b2 to Bit-2, etc. Advantages Simple Construction/Analysis Fast Conversion Disadvantages Requires large range of resistors (2000:1 for 12-bit DAC) with necessary high precision for low resistors Requires low switch resistances in transistors Can be expensive. Therefore, usually limited to 8-bit resolution.
R-2R Ladder Bit: 0 0 0 0 Each bit corresponds to a switch: If the bit is high, the corresponding switch is connected to the inverting input of the op-amp. If the bit is low, the corresponding switch is connected to ground. 4-Bit Converter
Vref V2V1 V3 R-2R Ladder Ideal Op-amp 2R 2R V3      R RR RR R    22 22 eq
Vref V2V1 V3 Vout R-2R Ladder V2 V3 R R 223 2 1 VV RR R V         Likewise, 12 2 1 VV  ref1 2 1 VV  I IRV out
Vref V2V1 V3 R-2R Ladder Results: ref1ref2ref3 2 1 , 4 1 , 8 1 VVVVVV         R V b R V b R V b R V bRV 16842 ref 0 ref 1 ref 2 ref 3out Where b3 corresponds to bit 3, b2 to bit 2, etc. If bit n is set, bn=1 If bit n is clear, bn=0 Vout
       16 1 8 1 4 1 2 1 0123refout bbbbVV For a 4-Bit R-2R Ladder For general n-Bit R-2R Ladder or Binary Weighted Resister DAC i n i inbVV 2 1 1 refout   • Advantages • Only two resistor values (R and 2R) • Does not require high precision resistors • Disadvantage • Lower conversion speed than binary weighted DAC
Specifications of DACs • Resolution • Speed • Linearity • Settling Time • Reference Voltages • Errors bitsofnumberN V V N ref LSB   where 2 Resolution Resolution Smallest analog increment corresponding to 1 LSB change An N-bit resolution can resolve 2N distinct analog levels Common DAC has a 8-16 bit resolution Speed Rate of conversion of a single digital input to its analog equivalent Conversion rate depends on clock speed of input signal settling time of converter When the input changes rapidly, the DAC conversion speed must be high.
Settling Time • Time required for the output signal to settle within +/- ½ LSB of its final value after a given change in input scale • Limited by slew rate of output amplifier • Ideally, an instantaneous change in analog voltage would occur when a new binary word enters into DAC
Types of Errors Associated with DACs • Gain • Offset • Full Scale • Resolution • Non-Linearity • Settling Time and Overshoot
OSCILLATOR Oscillators are circuits that produce a continuous signal of some type without the need of an input. These signals serve a variety of purposes such as communications systems, digital systems (including computers), and test equipment  An oscillator is a circuit that produces a repetitive signal from a dc voltage. There are two major classificationsfor oscillators: The feedback oscillator relies on a positive feedback of the output to maintain the oscillations.  The relaxation oscillator makes use of an RC timing circuit to generate a non-sinusoidal signal such as square wave.
• A feedback oscillator consists of amplifier for gain and a positive feedback circuit that produces phase shift and provides attenuation. • Instead of feedback, a relaxation oscillator uses an RC timing circuit to generate a waveform that is generally a square wave or other non sinusoidal waveform. Typically, Schmitt trigger or other device are uses for changes states to alternately charge and discharge a capacitor through a resistor. Ve = Vi + Vf (1) Vo = AVe (2) Vf = (AVe)=Vo (3) From (1), (2) and (3) Af ≡ Vo V𝑖 = A 1 − Aβ where A is loop gain As a function of s Af s = Vo Vs s = A s 1−A s β s
 Two conditions are required for a sustainedstate of oscillation: 1. The phase shift around the feedback loop must be 0° or 360°. 2. The voltage gain, Acl , around the closed feedback loop (loop gain) must equal 1 (unity) (Barkhaussen criterion) © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd Feedback oscillators 1.RC Oscillator - Wien Bridge Oscillator - Phase-Shift Oscillator 2. LC Oscillator - Crystal Oscillator Types of oscillator depending on the feedback components, amplifiers and circuit topologies used
1.RC Oscillators RC feedback oscillators are generally limited to frequencies of 1MHz or less The types of RC oscillators that we will discuss are the Wien-Bridge and the Phase Shift Wien-Bridge Oscillator It is a low frequency oscillator which ranges from a few kHz to 1 MHz. sCR R sC RZR ZRZ sC sRC sC RZRZ C CP CS                   1 111 11 11 ZS ZP
Oscillation condition   21 3 1 1 usingelyappropriat andresistorstheselectingby1getcanweThen, 1 frequencynoscillatioat the0termimaginaryThen 1 3 1 1 1 3 1 1 31 1 21 1 )1( 1 Rewriting 1 )1( 1onlyneedThen 0.sinceisalreadyIt.180toequalofPhase 1 2 1 2 21 1 2 1 2 222 1 2 222 1 2 2 1 2 2 1 2 o                                                               R R or R R RRA RC CR CRj R R sCR sCR R R RCssCR sCR R R RCssCRsCR sCR R R sCRsCR sCR R R A sCRsCR sCR R R A AA rf o rf rf rfrf       
• The phase-shift oscillator uses three RC circuits in the feedback path that have a total phase shift of 180° at one frequency – for this reason an inverting amplifier is required for this circuit © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd The Phase-Shift Oscillator • Each of the three RC circuits in the feedback loop provide a maximum phase shift approaching 90°. • Oscillation occurs at the frequency where the total phase shift through the three RC circuits is 180°. • The inversion of the op-amp itself provides the additional 180° to meet the requirement for oscillation of a 360° (or 0°) phase shift around the feedback loop.  The attenuation of the three-section RC feedback circuit is 1/29 To meet the unity loop gain requirement, the closed-loop voltage gain of the inverting op-amp must be 29. This means that Rf /R3 ≥29.
• The frequency of oscillation is given by • Example
LC Oscillators • LC feedback elements are normally used in oscillators that require higher frequencies of oscillation. • Also, because of the frequency limitation (lower unity-gain frequency) of most op-amps, transistors (BJT or FET) are often used as the gain element in LC oscillators. • Several types of resonant LC feedback oscillators like the Colpitts, Clapp, Hartley, and crystal-controlled oscillators. Colpitts oscillators  Colpitts oscillator is one of basic type of resonant circuit feedback oscillator uses an LC circuit in the feedback loop to provide the necessary phase shift and to act as a resonant filter that passes only the desired frequency of oscillation.
The approximate frequency of oscillation is the resonant frequency of the LC circuit and is established by the values of C1 , C2 and L according to the formula: 𝑟𝑟𝑓𝑓 = 1 2𝜋 𝐿𝐶 𝑇𝑇 Where CT is the total capacitance the series capacitors around the tank circuit, given by: 𝑇𝑇 𝐶𝐶 = 𝐶1 𝐶2 𝐶1 + 𝐶2 The output voltage is developed across C1 and the feedback voltage is developed across C2. © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd The Colpitts Oscillator The expression for the attenuation is 𝐵 = 𝐶2 /𝐶1 The condition for oscillation is 𝐴 𝑣 𝐵 = 1 or 𝐴 𝑣= 𝐶1 /𝐶2
The Hartley Oscillator  The Hartley oscillator is similar to the Colpitts oscillator, except the resonant circuit consists of two series inductors (or a single tapped inductor) and a parallel capacitor. © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd  The inductors act in a role similar to C1 and C2 in the Colpitts to determine the attenuation, B, of the feedback circuit B = (L1 / L2). To assure start-up of oscillation, Av must be greater than 1/B
Relaxation Oscillators • Relaxation oscillators make use of an RC timing and a device that changes states to generate a periodic waveform (non-sinusoidal) such as: • Triangular-wave • Square-wave • Sawtooth 1. Triangular-wave oscillator circuit is a combination of a comparator and integrator circuit. 𝑓𝑟 = 1 4𝐶𝑅1 𝑅2 𝑅3 𝑉𝑈𝑇𝑃 = +𝑉𝑚𝑎𝑥 𝑅3 𝑅2 𝑉𝐿𝑇𝑃 = −𝑉max 𝑅3 𝑅2
2. Square-wave Oscillator  A square wave relaxation oscillator is like the Schmitt trigger or Comparator circuit.  The charging and discharging of the capacitor cause the op-amp to switch states rapidly and produce a square wave.  The RC time constant determines the frequency.
3. Sawtooth Voltage-Controlled Oscillator (VCO) When Vout = VP • Vanode > VG , PUT turn ‘ON’ • The capacitor rapidly discharges. • Vout drop until Vout = VF. • Vanode < VG , PUT turn ‘OFF’ Operation VP-maximum peak value VF-minimum peak value Initially, dc input = -VIN • Volt = 0V, Vanode < VG • The circuit is like an integrator. • Capacitor is charging. • Output is increasing positive going ramp. • Sawtooth VCO circuit is a combination of a Programmable Unijunction Transistor (PUT) and integrator circuit. 𝑓 = 𝑉𝐼𝑁 𝑅𝑖 𝐶 1 𝑉𝑃 − 𝑉𝐹
Summary  Sinusoidal oscillators operate with positive feedback.  Two conditions for oscillation are 0º feedback phase shift and feedback loop gain of 1.  The initial startup requires the gain to be momentarily greater than 1.  RC oscillators include the Wien-bridge and phase shift.  LC oscillators include the Crystal Oscillator.  The crystal actually uses a crystal as the LC tank circuit and is very stable and accurate.  A voltage controlled oscillator’s (VCO) frequency is controlled by a dc control voltage.

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Chap 3. signal processing elemnt part three

  • 1. Chapter Three Signal Processing and Conversion Dept. of Electrical and Computer Eng., AASTU Addis Ababa By Biruk T.
  • 2. Analog to Digital Converter (ADC) • An analog signal is a continuous signal that contains time-varying quantities, such as temperature or speed, with infinite possible values in between. which are directly measurable quantities. • An analog signal can be used to measure changes in some physical phenomena such as light, sound, pressure, or temperature. • ADC is an electronic integrated circuit which transforms a signal from analog (continuous) to digital (discrete) form. • ADC Provides a link between the analog world of transducers and the digital world of signal processing and data handling. Examples: • Thermometer – mercury height rises as temperature rises • Car Speedometer – Needle moves farther right as you accelerate
  • 3. There are two step Process :- • Quantizing - breaking down analog value is a set of finite states • Encoding - assigning a digital word or number to each state and matching it to the input signal Quantizing: the number of possible states that the converter can output is: N=2n The number of possible states that the converter can output is: Sampling :- • It is a process of taking a sufficient number of discrete values at point on a waveform that willdefine the shape of waveform. • The more samples you take, the more accurately you will define the waveform. • It converts analog signal into series of impulses, each representing amplitude of the signal atgiven point.
  • 4. ADC SAMPLED AND QUANTIZED WAVEFORM DAC RECONSTRUCTED WAVEFORM ADC DAC DSP MemoryChannel Analog Digital timetime Analog Digital Amplitude Value “Real World” Sampled Data Systems Consist of ADCs and DACs Nyquist Rule:  Use a sampling frequency at least twice as high as the maximum frequency in the signal to avoid aliasing. maxsamples 2 ff 
  • 5. Example: Quantizing You have 0-10V signals. Separate them into a set of discrete states with 1.25V increments. (How did we get 1.25V?) The number of possible states that the converter can output is: N=2n where n is the number of bits in the AD converter Example: For a 3 bit A/D converter, N=23=8. Analog quantization size: Q=(Vmax-Vmin)/N = (10V – 0V)/8 = 1.25V Output States Discrete Voltage Ranges (V) 0 0.00-1.25 1 1.25-2.50 2 2.50-3.75 3 3.75-5.00 4 5.00-6.25 5 6.25-7.50 6 7.50-8.75 7 8.75-10.0
  • 6. Encoding • Here we assign the digital value (binary number) to each state for the computer to read. There are two ways to best improve accuracy of A/D conversion: • increasing the resolution which improves the accuracy in measuring the amplitude of the analog signal. • increasing the sampling rate which increases the maximum frequency that can be measured. Output States Output Binary Equivalent 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111  Resolution (number of discrete values the converter can produce) = Analog Quantization size (Q) = Vrange / 2^n, where Vrange is the range of analog voltages which can be represented limited by signal-to-noise ratio (should be around 6dB) In example: Q = 1.25V, this is a high resolution. A lower resolution would be if we used a 2-bit converter, then the resolution would be 10/2^2 = 2.50V.
  • 7. Types of A/D convertor:- • FlashADC • Dual slope/Counter slopeADC • Successive ApproximationADC Flash ADC: series of comparators, each one compares input to a unique reference voltage. • comparator outputs connect to a priority encoder circuit  produces binary output Fast – but more expensive : Single cycle - Uses many Comparators in parallel with different reference voltages How Flash Works  As the analog input voltage exceeds the reference voltage at each comparator, the comparator outputs will sequentially saturate to a high state.  The priority encoder generates a binary number based on the highest-order active input, ignoring all other active inputs.
  • 8. Figure: Parallel, simultaneous flash A/D conversion. Analog Digital• 2N-1 comparators for N-bits • Each reference voltage equivalent to a quantization level • Encoding logic produces word Advantages • Simplest in terms of operational theory, Most efficient in terms of speed, very fast, limited only in terms of comparator and gate propagation delays Disadvantages • Lower resolution, Expensive, For each additional output bit, the number of comparators is doubled, i.e. for 8 bits, 256 comparators needed
  • 9. 9 Integrating or Single & Dual Slope • Accumulate the input current on a capacitor for a fixed time and then measure the time (T) to discharge the capacitor at a fixed discharge rate. 1) S1->V1:Integrate the input on the cap. For N clock ticks 2) S1-> -Vref: restart clock (S2->counter) discharge C at know rate(governed by -Vref and R) 3) When the cap. is discharged to 0 voltage, the comparator will stop the counter. problem --very slow 0 1 arg _ / constant T fixed ch e held Q Idt I V R       Discharge time for stopping counter by S2 depends on RC and Q
  • 10. Successive Approximation: continuously convert analog signal to discreet digital signal. • Assume that • Vin= 0.6 volts • Vref=1volts • Find the digital value of Vin Divided Vref by 2 Compare Vref /2 with Vin If Vin is greater than Vref /2 , turn MSB on (1) If Vin is less than Vref /2 , turn MSB off (0) Vin =0.6V and V=0.5 Since Vin>V, MSB = 1 (on)
  • 11. • Next Calculate MSB-1 (bit 8) • Compare Vin=0.6 V to V=Vref/2 + Vref/4= 0.5+0.25 =0.75V • Since 0.6<0.75, MSB is turned off • Calculate MSB-2 (bit 7) • Go back to the last voltage that caused it to be turned on (Bit 9) and add it to Vref/8, and compare with Vin • Compare Vin with (0.5+Vref/8)=0.625 • Since 0.6<0.625, MSB is turned off • Calculate the state of MSB-3 (bit 6) • Go to the last bit that caused it to be turned on (In this case MSB-1) and add it to Vref/16, and compare it to Vin • Compare Vin to V= 0.5 + Vref/16= 0.5625 • Since 0.6>0.5625, MSB-3=1 (turned on)
  • 12. • This process continues for all the remaining bits.
  • 13. ADC example 2. using successive approximation • Given an analog input signal whose voltage should range from 0 to 15 volts, and an 8-bit digital encoding, calculate the correct encoding for 5 volts. Then trace the successive- approximation approach to find the correct encoding. • Assume M = 2n – 1, a / Vmax = d / M, 5 / 15 = d / (256 - 1), d = 85 or binary 01010101 Step 1-4: determine bits 0-3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 ½(Vmax – Vmin) = 7.5 volts Vmax = 7.5 volts. ½(7.5 + 0) = 3.75 volts, Vmin = 3.75 volts. ½(7.5 + 3.75) = 5.63 volts, Vmax = 5.63 volts ½(5.63 + 3.75) = 4.69 volts, Vmin = 4.69 volts. 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 ½(5.63 + 4.69) = 5.16 volts, Vmax = 5.16 volts. ½(5.16 + 4.69) = 4.93 volts, Vmin = 4.93 volts. ½(5.16 + 4.93) = 5.05 volts, Vmax = 5.05 volts. ½(5.05 + 4.93) = 4.99 volts
  • 14. Constructing SA ADC SARComparator DAC Vmax Vmin Analog input SAR BUF Digital output State machine SAR: Successive approximation register Timing control
  • 15. Digital to Analog Converter (DAC) • Signals are easily stored and transmitted in digital form, but a DAC is needed for the signal to be recognized by human senses or other non-digital systems. • A digital-to-analog converter (DAC or D-to-A) is a device that converts a digital (usually binary) code signal to an analog signal (current, voltage, or electric charge). • Digital signals is a type of signal that can take on a set of discrete values (a quantized signal) • It can represent a discrete set of values using any discrete set of waveforms .. And we can represent it like (0 or 1) ,( on or off )….. etc Examples:  Light switch can be either on or off
  • 16. ANALOG OUTPUT DIGITAL INPUT REFERENCE INPUT Digital Input Analog Output = x Reference Input (2N - 1) What is a Digital-Analog Converter? • Produces a Quantized (Discrete Step) Analog Output (Voltage or Current) in Response to Binary Digital Input Code • A reference quantity (either voltage or current) is accurately divided into binary and/or linear segments. • The digital input drives switches that connect an appropriate number of segments to the output. • Finite Number of Discrete Values : 2N Resulting in Quantization Uncertainty • Sampling and Quantization Impose Fundamental yet Predictable Limitations RESOLUTION = N BITS
  • 17. 10111001 10100111 10000110010101000011001000010000 AnalogOutputSignal Digital Input Signal Two Types of DAC: • Binary Weighted Resistor • R-2R Ladder
  • 18. Terms & Equations Offset: minimum analog value Span (or Range): difference between maximum and minimum analog values Max - Min n: number of bits in digital code (sometimes referred to as n-bit resolution) Bit Weight: analog value corresponding to a bit in the digital number Step Size (or Resolution): smallest analog change resulting from changing one bit in the digital number, or the analog difference between two consecutive digital numbers; also the bit weight of the Span / 2n (Assume M = 2n) Let AV be Analog Value; DN be Digital Number: AV = DN * Step Size + Offset = (DN / 2n )* Span + Offset DN = (AV - Offset) / Step Size = (AV - Offset) * 2n / Span
  • 19. Bit Weight Each bit is weighted with an analog value, such that a 1 in that bit position adds its analog value to the total analog value represented by the digital encoding. Digital Bit Bit Weight (V) 7 10/2 = 5 6 10/4 = 2.5 5 10/8 = 1.25 4 10/16 = 0.625 3 10/32 = 0.313 2 10/64 = 0.157 1 10/128 = 0.078 0 10/256 = 0.039 Binary Weighted Resistor • Utilizes a summing op-amp circuit • Weighted resistors are used to distinguish each bit from the most significant to the least significant • Transistors are used to switch between Vref and ground (bit high or low) Example: -5 V to +5 V analog range, n=8 Solution Vref(pp)=10v
  • 20. Binary Weighted Resistor • Assume Ideal Op-amp • No current into op-amp • Virtual ground at inverting input • Vout= -IRf - + R 2R 4R 2nR Rf Vout I Vref Voltages V1 through Vn are either Vref if corresponding bit is high or ground if corresponding bit is low V1 is most significant bit, Vn is least significant bit I - + R 2R 4R 2n-1R Rf Vout Vref V1 V2 V3 Vn        R V R V R V R V RIRV 1-n n321 ffout 242  MSB LSB
  • 21. If Rf=R/2        n n321 fout 2842 VVVV IRV  For example, a 4-Bit converter yields        16 1 8 1 4 1 2 1 0123refout bbbbVV Where b3 corresponds to Bit-3, b2 to Bit-2, etc. Advantages Simple Construction/Analysis Fast Conversion Disadvantages Requires large range of resistors (2000:1 for 12-bit DAC) with necessary high precision for low resistors Requires low switch resistances in transistors Can be expensive. Therefore, usually limited to 8-bit resolution.
  • 22. R-2R Ladder Bit: 0 0 0 0 Each bit corresponds to a switch: If the bit is high, the corresponding switch is connected to the inverting input of the op-amp. If the bit is low, the corresponding switch is connected to ground. 4-Bit Converter
  • 23. Vref V2V1 V3 R-2R Ladder Ideal Op-amp 2R 2R V3      R RR RR R    22 22 eq
  • 24. Vref V2V1 V3 Vout R-2R Ladder V2 V3 R R 223 2 1 VV RR R V         Likewise, 12 2 1 VV  ref1 2 1 VV  I IRV out
  • 25. Vref V2V1 V3 R-2R Ladder Results: ref1ref2ref3 2 1 , 4 1 , 8 1 VVVVVV         R V b R V b R V b R V bRV 16842 ref 0 ref 1 ref 2 ref 3out Where b3 corresponds to bit 3, b2 to bit 2, etc. If bit n is set, bn=1 If bit n is clear, bn=0 Vout
  • 26.        16 1 8 1 4 1 2 1 0123refout bbbbVV For a 4-Bit R-2R Ladder For general n-Bit R-2R Ladder or Binary Weighted Resister DAC i n i inbVV 2 1 1 refout   • Advantages • Only two resistor values (R and 2R) • Does not require high precision resistors • Disadvantage • Lower conversion speed than binary weighted DAC
  • 27. Specifications of DACs • Resolution • Speed • Linearity • Settling Time • Reference Voltages • Errors bitsofnumberN V V N ref LSB   where 2 Resolution Resolution Smallest analog increment corresponding to 1 LSB change An N-bit resolution can resolve 2N distinct analog levels Common DAC has a 8-16 bit resolution Speed Rate of conversion of a single digital input to its analog equivalent Conversion rate depends on clock speed of input signal settling time of converter When the input changes rapidly, the DAC conversion speed must be high.
  • 28. Settling Time • Time required for the output signal to settle within +/- ½ LSB of its final value after a given change in input scale • Limited by slew rate of output amplifier • Ideally, an instantaneous change in analog voltage would occur when a new binary word enters into DAC
  • 29. Types of Errors Associated with DACs • Gain • Offset • Full Scale • Resolution • Non-Linearity • Settling Time and Overshoot
  • 30. OSCILLATOR Oscillators are circuits that produce a continuous signal of some type without the need of an input. These signals serve a variety of purposes such as communications systems, digital systems (including computers), and test equipment  An oscillator is a circuit that produces a repetitive signal from a dc voltage. There are two major classificationsfor oscillators: The feedback oscillator relies on a positive feedback of the output to maintain the oscillations.  The relaxation oscillator makes use of an RC timing circuit to generate a non-sinusoidal signal such as square wave.
  • 31. • A feedback oscillator consists of amplifier for gain and a positive feedback circuit that produces phase shift and provides attenuation. • Instead of feedback, a relaxation oscillator uses an RC timing circuit to generate a waveform that is generally a square wave or other non sinusoidal waveform. Typically, Schmitt trigger or other device are uses for changes states to alternately charge and discharge a capacitor through a resistor. Ve = Vi + Vf (1) Vo = AVe (2) Vf = (AVe)=Vo (3) From (1), (2) and (3) Af ≡ Vo V𝑖 = A 1 − Aβ where A is loop gain As a function of s Af s = Vo Vs s = A s 1−A s β s
  • 32.  Two conditions are required for a sustainedstate of oscillation: 1. The phase shift around the feedback loop must be 0° or 360°. 2. The voltage gain, Acl , around the closed feedback loop (loop gain) must equal 1 (unity) (Barkhaussen criterion) © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd Feedback oscillators 1.RC Oscillator - Wien Bridge Oscillator - Phase-Shift Oscillator 2. LC Oscillator - Crystal Oscillator Types of oscillator depending on the feedback components, amplifiers and circuit topologies used
  • 33. 1.RC Oscillators RC feedback oscillators are generally limited to frequencies of 1MHz or less The types of RC oscillators that we will discuss are the Wien-Bridge and the Phase Shift Wien-Bridge Oscillator It is a low frequency oscillator which ranges from a few kHz to 1 MHz. sCR R sC RZR ZRZ sC sRC sC RZRZ C CP CS                   1 111 11 11 ZS ZP
  • 34. Oscillation condition   21 3 1 1 usingelyappropriat andresistorstheselectingby1getcanweThen, 1 frequencynoscillatioat the0termimaginaryThen 1 3 1 1 1 3 1 1 31 1 21 1 )1( 1 Rewriting 1 )1( 1onlyneedThen 0.sinceisalreadyIt.180toequalofPhase 1 2 1 2 21 1 2 1 2 222 1 2 222 1 2 2 1 2 2 1 2 o                                                               R R or R R RRA RC CR CRj R R sCR sCR R R RCssCR sCR R R RCssCRsCR sCR R R sCRsCR sCR R R A sCRsCR sCR R R A AA rf o rf rf rfrf       
  • 35. • The phase-shift oscillator uses three RC circuits in the feedback path that have a total phase shift of 180° at one frequency – for this reason an inverting amplifier is required for this circuit © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd The Phase-Shift Oscillator • Each of the three RC circuits in the feedback loop provide a maximum phase shift approaching 90°. • Oscillation occurs at the frequency where the total phase shift through the three RC circuits is 180°. • The inversion of the op-amp itself provides the additional 180° to meet the requirement for oscillation of a 360° (or 0°) phase shift around the feedback loop.  The attenuation of the three-section RC feedback circuit is 1/29 To meet the unity loop gain requirement, the closed-loop voltage gain of the inverting op-amp must be 29. This means that Rf /R3 ≥29.
  • 36. • The frequency of oscillation is given by • Example
  • 37. LC Oscillators • LC feedback elements are normally used in oscillators that require higher frequencies of oscillation. • Also, because of the frequency limitation (lower unity-gain frequency) of most op-amps, transistors (BJT or FET) are often used as the gain element in LC oscillators. • Several types of resonant LC feedback oscillators like the Colpitts, Clapp, Hartley, and crystal-controlled oscillators. Colpitts oscillators  Colpitts oscillator is one of basic type of resonant circuit feedback oscillator uses an LC circuit in the feedback loop to provide the necessary phase shift and to act as a resonant filter that passes only the desired frequency of oscillation.
  • 38. The approximate frequency of oscillation is the resonant frequency of the LC circuit and is established by the values of C1 , C2 and L according to the formula: 𝑟𝑟𝑓𝑓 = 1 2𝜋 𝐿𝐶 𝑇𝑇 Where CT is the total capacitance the series capacitors around the tank circuit, given by: 𝑇𝑇 𝐶𝐶 = 𝐶1 𝐶2 𝐶1 + 𝐶2 The output voltage is developed across C1 and the feedback voltage is developed across C2. © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd The Colpitts Oscillator The expression for the attenuation is 𝐵 = 𝐶2 /𝐶1 The condition for oscillation is 𝐴 𝑣 𝐵 = 1 or 𝐴 𝑣= 𝐶1 /𝐶2
  • 39. The Hartley Oscillator  The Hartley oscillator is similar to the Colpitts oscillator, except the resonant circuit consists of two series inductors (or a single tapped inductor) and a parallel capacitor. © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Electronic Devices, 9th edition Thomas L. Floyd  The inductors act in a role similar to C1 and C2 in the Colpitts to determine the attenuation, B, of the feedback circuit B = (L1 / L2). To assure start-up of oscillation, Av must be greater than 1/B
  • 40. Relaxation Oscillators • Relaxation oscillators make use of an RC timing and a device that changes states to generate a periodic waveform (non-sinusoidal) such as: • Triangular-wave • Square-wave • Sawtooth 1. Triangular-wave oscillator circuit is a combination of a comparator and integrator circuit. 𝑓𝑟 = 1 4𝐶𝑅1 𝑅2 𝑅3 𝑉𝑈𝑇𝑃 = +𝑉𝑚𝑎𝑥 𝑅3 𝑅2 𝑉𝐿𝑇𝑃 = −𝑉max 𝑅3 𝑅2
  • 41. 2. Square-wave Oscillator  A square wave relaxation oscillator is like the Schmitt trigger or Comparator circuit.  The charging and discharging of the capacitor cause the op-amp to switch states rapidly and produce a square wave.  The RC time constant determines the frequency.
  • 42. 3. Sawtooth Voltage-Controlled Oscillator (VCO) When Vout = VP • Vanode > VG , PUT turn ‘ON’ • The capacitor rapidly discharges. • Vout drop until Vout = VF. • Vanode < VG , PUT turn ‘OFF’ Operation VP-maximum peak value VF-minimum peak value Initially, dc input = -VIN • Volt = 0V, Vanode < VG • The circuit is like an integrator. • Capacitor is charging. • Output is increasing positive going ramp. • Sawtooth VCO circuit is a combination of a Programmable Unijunction Transistor (PUT) and integrator circuit. 𝑓 = 𝑉𝐼𝑁 𝑅𝑖 𝐶 1 𝑉𝑃 − 𝑉𝐹
  • 43. Summary  Sinusoidal oscillators operate with positive feedback.  Two conditions for oscillation are 0º feedback phase shift and feedback loop gain of 1.  The initial startup requires the gain to be momentarily greater than 1.  RC oscillators include the Wien-bridge and phase shift.  LC oscillators include the Crystal Oscillator.  The crystal actually uses a crystal as the LC tank circuit and is very stable and accurate.  A voltage controlled oscillator’s (VCO) frequency is controlled by a dc control voltage.