Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum.
Leeson's expression [1] for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per hertz) and augmented for flicker noise: [2]
where f0 is the output frequency, Ql is the loaded quality factor, fm is the offset from the output frequency (Hz), fc is the 1/f corner frequency, F is the noise factor of the amplifier, k is the Boltzmann constant, T is absolute temperature, and Ps is the available power at the sustaining amplifier input. [3]
There is often misunderstanding around Leeson's equation, even in text books. In the 1966 paper, Leeson stated correctly that "Ps is the signal level at the oscillator active element input" (often referred to as the power through the resonator now, strictly speaking it is the available power at the amplifier input). F is the device noise factor, however this does need to be measured at the operating power level. The common misunderstanding, that Ps is the oscillator output level, may result from derivations that are not completely general. In 1982, W. P. Robins (IEE Publication "Phase noise in signal sources") correctly showed that the Leeson equation (in the −20 dB/decade region) is not just an empirical rule, but a result that follows from a linear analysis of an oscillator circuit. However, a used constraint in his circuit was that the oscillator output power was approximately equal to the active device input power.
The Leeson equation is presented in various forms. In the above equation, if fc is set to zero the equation represents a linear analysis of a feedback oscillator in the general case (and flicker noise is not included), it is for this that Leeson is most recognised, showing a −20 dB/decade of offset frequency slope. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. If a value for fc is included, the equation also shows a curve fit for the flicker noise. The fc for an amplifier depends on the actual configuration used, because radio-frequency and low-frequency negative feedback can have an effect on fc. So for accurate results, fc must be determined from added noise measurements on the amplifier using R.F., with the actual circuit configuration to be used in the oscillator.
Evidence that Ps is the amplifier input power (often contradicted or very unclear in text books) can be found in the derivation in further reading which also shows experimental results, "Enrico Rubiola, The Leeson Effect" also shows this in a different form.
An electronic oscillator is an electronic circuit that produces a periodic, oscillating or alternating current (AC) signal, usually a sine wave, square wave or a triangle wave, powered by a direct current (DC) source. Oscillators are found in many electronic devices, such as radio receivers, television sets, radio and television broadcast transmitters, computers, computer peripherals, cellphones, radar, and many other devices.
An operational amplifier is a DC-coupled electronic voltage amplifier with a differential input, a (usually) single-ended output, and an extremely high gain. Its name comes from its original use of performing mathematical operations in analog computers.
The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan and expressed mathematically as . The Allan deviation (ADEV), also known as sigma-tau, is the square root of the Allan variance, .
Noise figure (NF) and noise factor (F) are figures of merit that indicate degradation of the signal-to-noise ratio (SNR) that is caused by components in a signal chain. These figures of merit are used to evaluate the performance of an amplifier or a radio receiver, with lower values indicating better performance.
A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is fixed relative to the phase of an input signal. Keeping the input and output phase in lockstep also implies keeping the input and output frequencies the same, thus a phase-locked loop can also track an input frequency. And by incorporating a frequency divider, a PLL can generate a stable frequency that is a multiple of the input frequency.
In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). Generally speaking, radio-frequency engineers speak of the phase noise of an oscillator, whereas digital-system engineers work with the jitter of a clock.
A negative-feedback amplifier is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its performance and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, many amplifiers and control systems use negative feedback.
In electronics, a relaxation oscillator is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. The circuit consists of a feedback loop containing a switching device such as a transistor, comparator, relay, op amp, or a negative resistance device like a tunnel diode, that repetitively charges a capacitor or inductor through a resistance until it reaches a threshold level, then discharges it again. The period of the oscillator depends on the time constant of the capacitor or inductor circuit. The active device switches abruptly between charging and discharging modes, and thus produces a discontinuously changing repetitive waveform. This contrasts with the other type of electronic oscillator, the harmonic or linear oscillator, which uses an amplifier with feedback to excite resonant oscillations in a resonator, producing a sine wave.
A voltage-controlled oscillator (VCO) is an electronic oscillator whose oscillation frequency is controlled by a voltage input. The applied input voltage determines the instantaneous oscillation frequency. Consequently, a VCO can be used for frequency modulation (FM) or phase modulation (PM) by applying a modulating signal to the control input. A VCO is also an integral part of a phase-locked loop. VCOs are used in synthesizers to generate a waveform whose pitch can be adjusted by a voltage determined by a musical keyboard or other input.
A Colpitts oscillator, invented in 1918 by Canadian-American engineer Edwin H. Colpitts using vacuum tubes, is one of a number of designs for LC oscillators, electronic oscillators that use a combination of inductors (L) and capacitors (C) to produce an oscillation at a certain frequency. The distinguishing feature of the Colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the inductor.
A Wien bridge oscillator is a type of electronic oscillator that generates sine waves. It can generate a large range of frequencies. The oscillator is based on a bridge circuit originally developed by Max Wien in 1891 for the measurement of impedances. The bridge comprises four resistors and two capacitors. The oscillator can also be viewed as a positive gain amplifier combined with a bandpass filter that provides positive feedback. Automatic gain control, intentional non-linearity and incidental non-linearity limit the output amplitude in various implementations of the oscillator.
A lock-in amplifier is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment. Depending on the dynamic reserve of the instrument, signals up to a million times smaller than noise components, potentially fairly close by in frequency, can still be reliably detected. It is essentially a homodyne detector followed by low-pass filter that is often adjustable in cut-off frequency and filter order.
A ring oscillator is a device composed of an odd number of NOT gates in a ring, whose output oscillates between two voltage levels, representing true and false. The NOT gates, or inverters, are attached in a chain and the output of the last inverter is fed back into the first.
A low-dropout regulator is a type of a DC linear voltage regulator circuit that can operate even when the supply voltage is very close to the output voltage. The advantages of an LDO regulator over other DC-to-DC voltage regulators include: the absence of switching noise ; smaller device size ; and greater design simplicity. The disadvantage is that linear DC regulators must dissipate heat in order to operate.
This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.
A phase-shift oscillator is a linear electronic oscillator circuit that produces a sine wave output. It consists of an inverting amplifier element such as a transistor or op amp with its output fed back to its input through a phase-shift network consisting of resistors and capacitors in a ladder network. The feedback network 'shifts' the phase of the amplifier output by 180 degrees at the oscillation frequency to give positive feedback. Phase-shift oscillators are often used at audio frequency as audio oscillators.
The Pierce oscillator is a type of electronic oscillator particularly well-suited for use in piezoelectric crystal oscillator circuits. Named for its inventor, George W. Pierce (1872–1956), the Pierce oscillator is a derivative of the Colpitts oscillator. Virtually all digital IC clock oscillators are of Pierce type, as the circuit can be implemented using a minimum of components: a single digital inverter, one resistor, two capacitors, and the quartz crystal, which acts as a highly selective filter element. The low manufacturing cost of this circuit and the frequency stability of quartz crystals make it advantageous for many consumer electronics applications.
In electronics, a transimpedance amplifier (TIA) is a current to voltage converter, almost exclusively implemented with one or more operational amplifiers. The TIA can be used to amplify the current output of Geiger–Müller tubes, photo multiplier tubes, accelerometers, photo detectors and other types of sensors to a usable voltage. Current to voltage converters are used with sensors that have a current response that is more linear than the voltage response. This is the case with photodiodes where it is not uncommon for the current response to have better than 1% nonlinearity over a wide range of light input. The transimpedance amplifier presents a low impedance to the photodiode and isolates it from the output voltage of the operational amplifier. In its simplest form a transimpedance amplifier has just a large valued feedback resistor, Rf. The gain of the amplifier is set by this resistor and because the amplifier is in an inverting configuration, has a value of -Rf. There are several different configurations of transimpedance amplifiers, each suited to a particular application. The one factor they all have in common is the requirement to convert the low-level current of a sensor to a voltage. The gain, bandwidth, as well as current and voltage offsets change with different types of sensors, requiring different configurations of transimpedance amplifiers.
A comparator is an electronic component that compares two input voltages. Comparators are closely related to operational amplifiers, but a comparator is designed to operate with positive feedback and with its output saturated at one power rail or the other. If necessary, an op-amp can be pressed into service as a poorly performing comparator, but its slew Rate will be impaired.
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