Transmission Lines And Networks Johnson Pdf

[Daiana Parthemore]

Thermalnoise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution.[1]

For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors. Thermal noise is distinct from shot noise, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow.

In 1905, in one of Albert Einstein's Annus mirabilis papers the theory of Brownian motion was first solved in terms of thermal fluctuations. The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out the calculation as he considered it to be untestable.[2]

Geertruida de Haas-Lorentz, daughter of Hendrik Lorentz, in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current.[2][3]

Frits Zernike working in electrical metrology, found unusual random deflections while working with high-sensitive galvanometers. He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction.[2]

The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies.[4][5][2] He described his findings to Harry Nyquist, also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928.[6]

Johnson's experiment (Figure 1) found that the thermal noise from a resistance R \displaystyle R at kelvin temperature T \displaystyle T and bandlimited to a frequency band of bandwidth Δ f \displaystyle \Delta f (Figure 3) has a mean square voltage of:[5]

A resistor with thermal noise can be represented by its Thvenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage.

Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 ΩHz for R Δ f \displaystyle R\,\Delta f corresponds to almost one nanovolt of RMS noise.

A resistor with thermal noise can also converted into its Norton equivalent circuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current:

This charge noise is the origin of the term "kTC noise". Although independent of the resistor's value, 100% of the kTC noise arises in the resistor. Therefore, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise,[7] and the temperature of the resistor alone should be used, even if the resistor and the capacitor are at different temperatures. Some values are tabulated below:

An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage must be interpreted not as a time average, but as an average over many such reset events, since the voltage is constant when the bandwidth is zero. In this sense, the Johnson noise of an RC circuit can be seen to be inherent, an effect of the thermodynamic distribution of the number of electrons on the capacitor, even without the involvement of a resistor.

The noise is not caused by the capacitor itself, but by the thermodynamic fluctuations of the amount of charge on the capacitor. Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above. The reset noise of capacitive sensors is often a limiting noise source, for example in image sensors.

For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm. It accomplished this by using Josephson voltage standard and a quantum Hall resistor, held at the triple-point temperature of water. The voltage is measured over a period of 100 days and integrated.[10]

The noise generated at a resistor R S \displaystyle R_\textS can transfer to the remaining circuit. The maximum power transfer happens when the Thvenin equivalent resistance R L \displaystyle R_\rm L of the remaining circuit matches R S \displaystyle R_\textS .[14] In this case, each of the two resistors dissipates noise in both itself and in the other resistor. Since only half of the source voltage drops across any one of these resistors, this maximum noise power transfer is:

Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors"[6] used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell[16] to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors ( R 1 = R 2 \displaystyle R_1=R_2 ). According to the conclusion of Figure 5, the total average power transferred over bandwidth Δ f \displaystyle \Delta f from R 1 \displaystyle R_1 and absorbed by R 2 \displaystyle R_2 was determined to be:

Setting this P 1 \textstyle P_\text1 equal to the earlier average power expression P 1 \textstyle \overline P_1 allows solving for the average of V 1 2 \textstyle V_1^2 over that bandwidth:

Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used k B T \displaystyle k_\rm BT according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant h \displaystyle h (from the new theory of quantum mechanics).[6]

The 4 k B T R \displaystyle 4k_\textBTR voltage noise described above is a special case for a purely resistive component for low to moderate frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissipation theorem. Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely passive and linear.

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors.[6] Such a component can be described by a frequency-dependent complex electrical impedance Z ( f ) \displaystyle Z(f) . The formula for the power spectral density of the series noise voltage is

With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor η ( f ) \displaystyle \eta (f) mentioned earlier is in general given by:[17]

Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.[20] Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of cross-spectral density functions relating the different noise voltages,

where the Z m n \displaystyle Z_mn are the elements of the impedance matrix Z \displaystyle \mathbf Z .Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by

Vermont Electric Power Company (VELCO) President and CEO Tom Dunn announced today that Kerrick Johnson has agreed to serve as VELCO's Vice President of Strategic Innovation. Johnson joins VELCO's senior leadership team, where he will be responsible for all company business development activities and communications work as the company continues to pursue initiatives that deliver grid reliability, cost savings and decarbonization value. This is a return engagement for Johnson having previously served eleven years at VELCO.

"We welcome Kerrick back to VELCO as a catalyst for our strategic growth," said Dunn. "Kerrick's relationships, creativity, and passion are unmatched. We are pleased he is joining our team as we broaden and deepen our strategic partnerships, develop new tools and services, and ultimately deliver more value for Vermonters."

"I am thrilled and grateful to be back," said Johnson, "These former colleagues have remained close friends. They, like me, have continued to evolve and grow and to now rejoin the team and renew those powerful and productive relationships is an exciting opportunity. VELCO serves a unique role in the state and national energy landscape. The opportunities, and the imperative, to capitalize on the company's data, fiber optic network, and expert teams have never been greater." Johnson added, "The time is now and I can't wait to get started."

Johnson comes to VELCO with over 30 years in the energy industry, most recently at Vermont Energy Investment Corporation (VEIC) where he was the Strategy & Corporate Affairs Director responsible for strengthening the firm's alignment in Vermont, as well as developing opportunities to collaboratively deliver energy efficiency and climate services value in New England, Mid-Atlantic and Midwest. Immediately prior to his VEIC role, Johnson was the Co-Founder and Chief Ecosystem Officer for Utopus Insights, a New York-based energy analytics company formed by IBM, Boston Consulting Group, and VELCO that was acquired in 2018 by Vestas Wind Systems A/S.

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