Linear circuit

Network with linear circuit parameters and constant voltage sources... 

It was observed that with a linear circuit and in the absence of any source of energy (except probably the residual charges in condensers) the circuit becomes self-excited and builds up the voltage indefinitely until the insulation is punctured, which is in accordance with (6-138). In the second experiment these physicists inserted a nonlinear resistor in series with the circuit and obtained a stable oscillation with fixed amplitude and phase, as follows from the analysis of the differential equation (6-127). 

The literature refers the term as a first order filter. It only makes sense if you recall your linear circuit analysis or if you wait until the chapter on frequency response analysis. 

B.J. Ley, S.G. Lutz, C.F. Rehberg Linear circuit analysis, McGraw-Hill, New York (1959)... 

This circuit is difficult because it contains a nonlinear element (the diode) and a complex linear circuit. If we could replace VI, Rl, R2, and R3 by a simpler circuit, the analysis of the nonlinear element would be much easier. To simplify the analysis of the diode, we will find the Thevenin and Norton equivalent circuits of the circuit connected to the diode that is, we will find the Thevenin and Norton equivalents of the circuit below ... 

Texas Instruments. 1992. Linear Circuits Data Book (Vol. 3). 

Thanks to Ron Rohrer, Larry Nagel, and all the students at the University of California, Berkeley, who worked hard in 1969 and 1970 to develop the first computer simulation software, Cancer (Computer Analysis of Non-Linear Circuits Excluding Radiation). This effort would result in the release of SPICE into the public domain in 1971. 

Consider a general linear circuit whose instantaneous electrical state is described by a set of currents and voltages. If we denote all these variables by vv the network equations have the general form... 

Table 1 Linear Circuit Elements Commonly Used in Electrochemical Impedance...

One of the most important strategies to simplify or reduce a linear circuit is superposition. The superposition theorem states that the response of a linear network to a number of simultaneously applied sources is equal to the sum of the individual responses due to each source acting alone. 

By analyzing separately a single-input circuit, superposition allows us to analyze linear circuits with more than one independent source. For example, Figure... 

Other methods to simplify the circuit are Thevenin s and Norton s theorems. These two theorems can be used to replace the entire circuit by employing equivalent circuits. For example, Figure 2.34 shows a circuit separated into two parts. Circuit A is linear. Circuit B contains non-linear elements. The essence of Thevenin s and Norton s theorems is that no dependent source in circuit A can be controlled by a voltage or current associated with an element in circuit B, and vice versa. 

Thevenin s theorem states that a section of a linear circuit containing one or more sources and impedances can be replaced with an equivalent circuit model containing only one voltage source and one series-connected impedance, as shown in Figure 2.35. 

An electronically stabilized power source may show an idealized voltage-current diagram according to the right side curve. The output voltage remains constant U = U l at all load conditions I < Isc, and in the short-circuit-condition I = Isc, U drops down to zero. At point A, the output power is P UNL Isc, four times the maximum output power of a linear circuit at identical values for UNL and Isc. 

Figure 1.2.5 Left Two-electrode cell with an ideal polarized mercury drop electrode and an SCE. Right Representation of the cell in terms of linear circuit elements.

Logically, there could be another kind of non-linear circuit with the opposite characteristic i.e., a good insulator for large voltages and a good conductor for small voltages. Such a unit could be put in parallel with the output of the transmitter to short out only the low level tail of the transmitter pulse without attenuating the main part of the pulse. Unfortunately, its implementation is limited to cross coil setups... 

Rupp CJ, Dunn ML, Maute K (2010) Analysis of piezoelectric energy harvesting systems with non-linear circuits using the harmonic balance method. J Intell Mater Syst Str 2010 21 1383. originally published online 10 Sept 2010... 

The models discussed in this chapter involve the time behavior of electrochemical activity in excitable cells. These models are systems of ordinary differential equations where the independent variable is time. While a good understanding of linear circuit theory is useful in order to understand the models presented in this chapter, most of the phenomena of interest involve nonlinear circuits with time-varying components. 

The DHP to CPD interconversion also perturbs conjugation pathways around the perimeter of the molecule. This fact was capitalized upon to yield an optoelectronic redox switch (Scheme 3c). Specifically, the 16-7i linear circuit extending from thiophene A through DHP to thiophene B is interrupted upon conversion to Th-CPD-Th. This process is reversible, and only the DHP form exhibits anodic activity within a window of 0.0-0.75 V. Thus, the state of the photochrornic switch can be written photochemically and read electrochemically (the CPD form is redox silent, and the DHP form is redox active). 

In the field of frequencies, spectral transmittance H(co) is usually used for linear circuits to describe electrical answers. It is defined as a relation of complex responses Y and excitation X of investigated circuit ... 

GRAPH 10.30 Three dipoles with common flow inductive CPE, resistor, and capacitive CPE and their representation as an equivalent electrical circuit in series (a) and as a single unit of Eormal Graph (b). Each CPE is built by complementing its own paths with its own evolution mode before being added to the resistance. As the linear circuit is not powered, the common flow is zero. 

It is possible to replace the Kramers-Kronig integration by approximation. If the system can be well approximated by a linear circuit, then it must be Kramers-Kronig transformable. Orazem and coworkers [572, 573] proposed using the Voigt circuit displayed in Fig. 13.2. 

Experimental impedance data should be validated before further analysis. Raw data might be verified using Kramers-Kronig or Z-Hit transforms. It should be kept in mind that these transforms are not very sensitive to system nonlinearities, and an additional test with different amplitudes could be carried out. An alternative to the aforementioned transforms is the approximation to linear circuits (Figs. 13.2 and 13.4). 

A disadvantage common to both classes of spectrum analyzer discussed so far is that they do not measure absolute amplitudes accurately, and they do not measure phase at all. Although this last limitation can be circumvented by the use of KK transformations (see Section 3.1.2.9), these instruments generally are poor choices for linear circuit (ac impedance) analysis. 

When a sinusoidal source is applied to a stable linear circuit all of the steady-state node voltages and branch currents in the circuit will be sinusoids having the same frequency as the input. A sinusoidal input signal produces a sinusoidal steady-state output (response). The steady-state response of a given node voltage or branch current, however, may have different ampU-... 

FIGURE 1.2 Input-output relationship for linear circuits in sinusoidal steady state. 

Time-domain methods are often used to characterize linear circuits, and can also be used to describe resonance. When an electrical circuit exhibits an undamped oscillatory or slightly damped behavior it is said to be in resonance, and the waveforms of the voltages and currents in the circuit can oscillate indefinitely. 

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