Lecture 8: Oscillators Noise in electronic systems

Relaxation Oscillator Noise Oscillators Wien-Bridge Relaxation Oscillator Noise Type of Noise Noise Sources Noise Analysis

Oscillators An oscillator is a circuit that produces a periodically oscillating waveform on its output with dc input. Two major classifications: Feedback oscillators Relaxation oscillators

Feedback Oscillators Feedback oscillator operation is based on the principle of positive feedback. A fraction of output signal is returned to input with no net phase shift resulting in a re-inforcement of the output signal.

Feedback Oscillators Conditions of Oscillations: The phase shift around the feedback loop must be 0 degree. Closed feedback loop gain Acl must be 1.

Feedback Oscillators Vf is amplified to produce the output voltage , which in turn produces the feedback voltage. A loop is created and signal sustain itself and produces continuous oscillations. In some types of oscillators feedback shifts the phase by 180. inverting amplifier are used there to produce another 180 degree

Start-up conditions Feedback oscillators require a small disturbance such as that generated by thermal noise to start oscillations. This initial voltage starts the feedback process and oscillations. The feedback circuit permits only a voltage with a frequency equal to selected frequency to appear in phase on the amplifier’s input.

Wien-Bridge oscillators RC feedback is used in various lower frequency (up to 1 MHz) sine-wave oscillators. At resonant frequency fr the attenuation of the circuit is 1/3. The lead-lag circuit is used in the feedback of Wien-Bridge oscillator. It gives 0 phase shift and 1/3 attenuation at resonant frequency.

Wien-Bridge oscillators The basic Wien-bridge uses the lead-lag network to select a specific frequency that is amplified. The voltage-divider sets the gain to make up for the attenuation of the feedback network. The non-inverting amplifier must have a gain of exactly 3.0 as set by R1 and R2 to make up for the attenuation. If it is too little, oscillations will not occur; if it is too much the sine wave will be clipped. Voltage-divider Lead-lag network Basic Circuit Wien Bridge Oscillator

Wien-Bridge oscillation conditions The phase shift around the positive feedback loop must be 0o and the gain around the loop must be 1. The 0o phase-shift condition is met when the frequency is fr.

Wien-Bridge oscillator Startup The loop gain should be greater than 1 at startup to build up output.

Wien-Bridge oscillator Startup

Relaxation oscillator A simple relaxation oscillator that uses a Schmitt trigger is the basic square-wave oscillator. The two trigger points, UTP and LTP are set by R2 and R3. The capacitor charges and discharges between these levels: The period of the waveform is given by:

Noise Noise is a random fluctuation in an electrical signal. Noise in electronic devices varies greatly, as it can be produced by several different effects. Noise is a fundamental parameter to be considered in an electronic design as it typically limits the overall performance of the system.

Noise is a purely random signal, the instantaneous value and/or phase of the waveform cannot be predicted at any time. The amplitude of the signal has very nearly a Gaussian probability density function.

External and internal Noise Noise can either be generated internally in the op amp, from its associated passive components, or superimposed on the circuit by external sources. “External” refers to noise present in the signal being applied to the circuit or to noise introduced into the circuit by another means, such as conducted on a system ground or received on one of the many antennas formed by the traces and components in the system.

Types of internal Noise Thermal Noise Shot Noise Flicker Noise Burst Noise Avalanche Noise Some or all of these noises may be present in a design, presenting a noise spectrum unique to the system. It is not possible in most cases to separate the effects, but knowing general causes may help the designer optimize the design, minimizing noise in a particular bandwidth of interest.

Thermal Noise Generated by the random thermal motion of charge carriers (usually electrons), inside an electrical conductor. It happens regardless of any applied voltage.  Power Spectral Density is nearly equal throughout the frequency spectrum, approximately white noise.

Thermal Noise The RMS voltage due to thermal noise , generated in a resistance R (ohms) over bandwidth Δf (hertz), is given by: The noise from a resistor is proportional to its resistance and temperature. Lowering resistance values also reduces thermal noise. See example in section 10.3.2 ‘Op-amp for every one’

Shot Noise The name ‘Shot Noise’ is short of Schottky noise, also called quantum noise. It is caused by random fluctuations in the motion of charge carriers in a conductor.

Shot Noise Some characteristics of shot noise: Shot noise is always associated with current flow. It stops when the current flow stops. Shot noise is independent of temperature. Shot noise is spectrally flat or has a uniform power density, meaning that when plotted versus frequency it has a constant value. Shot noise is present in any conductor

Flicker Noise Flicker noise is also called 1/f noise. Its origin is one of the oldest unsolved problems in physics. It is present in all active and many passive devices. It may be related to imperfections in crystalline structure of semiconductors, as better processing can reduce it.

Flicker Noise Some characteristics of flicker noise: It increases as the frequency decreases, hence the name 1/f It is associated with a dc current in electronic devices It has the same power content in each octave (or decade)

Burst Noise Burst noise consists of sudden step-like transitions between two or more levels. is related to imperfections in semiconductor material and heavy ion implants. As high as several hundred microvolts. Lasts for several milli-seconds. Burst noise makes a popping sound at rates below 100 Hz when played through a speaker — it sounds like popcorn popping, hence also called popcorn noise. Low burst noise is achieved by using clean device processing, and therefore is beyond the control of the designer.

Avalanche Noise Avalanche noise is created when a PN junction is operated in the reverse breakdown mode. Under the influence of a strong reverse electric field within the junction’s depletion region, electrons have enough kinetic energy. They collide with the atoms of the crystal lattice, to form additional electron-hole pair. These collisions are purely random and produce random current pulses similar to shot noise, but much more intense.

Avalanche Noise When electrons and holes in the depletion region of a reversed-biased junction acquire enough energy to cause the avalanche effect, a random series of large noise spikes will be generated. The magnitude of the noise is difficult to predict due to its dependence on the materials.

Measuring Noise RMS, P-P or PDF Instantaneous noise voltage amplitudes are as likely to be positive as negative. Noise values form a random pattern centered on zero. Since amplitudes vary randomly with time, they can only be specified by a probability density function, most commonly by Gaussian density function. σ is the standard deviation of the Gaussian distribution and the rms value of the noise voltage and current. The instantaneous noise amplitude is within ±1σ 68% of the time, is within ±3σ of the mean 99.7% of the time and within ±3.4σ 99.94% of the time.

Signal to Noise ratio The noisiness of a signal is defined as: In other words, it is a ratio of signal voltage to noise voltage (hence the name signal-to-noise ratio).

Multiple Noise sources When there are multiple noise sources in a circuit, the total root-mean-square (rms) noise signal is the square root of the sum of the average mean-square values of the individual sources: If there are two noise sources of equal amplitude in the circuit, the total noise is not doubled (increased by 6 dB). It only increases by 3 dB. Consider a very simple case, two noise sources with amplitudes of 2 Vrms:

Noise Unit Internal noise is normally specified as a noise spectral density in rms volts or amps per root Hertz, V/√Hz or A /√ Hz. In datasheet it is often expressed with a plot: Example: An op-omp TLE2027 has noise specification of 2.5 nV/ √ Hz Noise characteristic for TLE2027 http://www.ti.com/lit/ds/symlink/tle2027.pdf

Equivalent noise EIN TLE2027 is used in a system that operates over an audio frequency range of 20 Hz to 20 kHz with a gain of 40db (100). Equivalent noise over the whole bandwidth is : 2.5nV * 20000−20 2.5nV * 141.35 EIN = 353.38nV If the gain of the system is 100 Eout= 353.38nV x 100 = 35.3 microV Noise characteristic for TLE2027

Calculating SNR If the output signal is of 1V SNR = 1V/ 35.3 uV = 28328 SNRdB= 20log(28328) = 89 dB Noise characteristic for TLE2027

Corner Frequency Noise in Spectral Density Usually a plot for Noise Spectral Density is given in op-amp datasheets. These graphs usually show two distinct regions: Lower frequencies where pink noise is the dominant effect Higher frequencies where white noise is the dominant effect The point in the frequency spectrum where 1/f noise and white noise are equal is referred to as the noise corner frequency, fnc

Corner Frequency in Noise Spectral Density The point in the frequency spectrum where 1/f noise and white noise are equal is referred to as the noise corner frequency, fnc The fnc can be determined visually from the graph: Take the white noise portion of the curve, and extrapolate it down to 10 Hz as a horizontal line. Pink noise White noise Take the portion of the pink noise from 10 Hz to 100 Hz, and extrapolate it as a straight line. The point where the two intercept is fnc, the point where the white noise and pink noise are equal in amplitude.

Corner Frequency Once the corner frequency is known, the individual noise components can be added together (if the bandwidth includes corner frequency): If fnc is not included in bandwidth, all of the contribution will be from either the 1/f noise or the white noise. Similarly, if the bandwidth is very large, and extends to three decades or so above fnc, the contribution of the 1/f noise can be ignored.

Op-amp circuit noise model Noise in op-amp circuits can be modeled as voltage noise source and current noise source. Input voltage noise is always represented by a voltage source in series with the non-inverting input. Input current noise is always represented by current sources from both inputs to ground.

Inverting op-amp circuit noise Model Sources e1, e2 and e3 represent the thermal noise contribution from the resistors. Note: Noise current sources are missing here. e1 R1 e2 R2 E0 e3

Noninverting Op Amp Circuit Noise Model Sources e1, e2 and e3 represent the thermal noise contribution from the resistors. Note: Noise current sources are missing here.

General Noise model eni= (𝑒𝑛 2 +4𝑘𝑇(𝑅𝑔||𝑅𝑓)+(𝑖𝑛− 𝑅𝑓 𝑅𝑔 2 Figure describes the noise model for the non-inverting amplifier configuration showing all noise sources. In addition to the intrinsic input voltage noise (en) and current noise (in=in+=in-) sources, there also exists thermal voltage noise (et 4 TR = k ) associated with each of the external resistors. Input Noise expression: eni= (𝑒𝑛 2 +4𝑘𝑇(𝑅𝑔||𝑅𝑓)+(𝑖𝑛− 𝑅𝑓 𝑅𝑔 2

Non-inverting Noise model Adding input noise from signal source at non-inverting input: Output Noise expression: Eo= 𝑒𝑛 1+ 𝑅𝑓 𝑅𝑔 2+ 𝐸𝑖𝑛 1+ 𝑅𝑓 𝑅𝑔 2+ 4𝑘𝑇𝑅𝑓 2+ 4𝑘𝑇𝑅𝑔 ( 𝑅𝑓 𝑅𝑔 ) 2+(𝑖𝑛− 𝑅𝑓 𝑅𝑔 ( 𝑅𝑓 𝑅𝑔 ) 2

inverting Noise model Adding input noise from signal source at inverting input: Output Noise expression: Eo= 𝑒𝑛 1+ 𝑅𝑓 𝑅𝑔 2+ 𝐸𝑖𝑛 𝑅𝑓 𝑅𝑔 2+ 4𝑘𝑇𝑅𝑓 2+ 4𝑘𝑇𝑅𝑔 ( 𝑅𝑓 𝑅𝑔 ) 2+(𝑖𝑛− 𝑅𝑓 𝑅𝑔 ( 𝑅𝑓 𝑅𝑔 ) 2

Reducing resistance values Reducing resistance value can help in reducing thermal noise.

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