Transient Analysis

Dennis Fitzpatrick , in Analog Design and Simulation Using OrCAD Capture and PSpice (Second Edition), 2018

Abstract

Transient analysis calculates a circuit's response over a period of time defined by the user. The accuracy of the transient analysis is dependent on the size of internal time steps, which together make up the complete simulation time known as the Run to time or Stop time. For every time step, the node voltages and currents are calculated and compared to the previous time step DC solution. Only when the difference between two DC solutions falls within a specified tolerance (accuracy) will the analysis move on to the next internal time step. The time step is dynamically adjusted until a solution within tolerance is found. The value for the maximum internal time step can be defined by the user. There are some circuits, where a DC solution cannot be found, as in the case of oscillators. For these circuits, there is an option in the simulation profile to skip over the initial DC bias point analysis. Scheduling allows users to dynamically alter a simulation setting for a transient analysis; for example, users may want to use a smaller step size during periods that require greater accuracy and relax the accuracy for periods of less activity. Check points were introduced in version 16.2 to allow users to effectively mark and save the state of a transient simulation at a check point and to restart transient simulations from defined check points. Input waveforms can also be defined using pairs of time-voltage coordinates, which can be entered in the Property Editor, or read from an external text file.

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Transient Analysis

I.D. Mayergoyz , W. Lawson , in Basic Electric Circuit Theory, 1997

7.8 Summary

In this chapter, we systematically developed various techniques for the analysis of transients in first- and second-order electric circuits. The most important facts and results discussed in the chapter can be summarized as follows:

•

Transients in electric circuits occur due to the presence of energy storage elements (i.e., inductors and capacitors).

•

Transients in electric circuits can be excited by initial conditions, by sources, or by both.

•

Analysis of transients can be broken down into two major steps:

1.

Determination of initial conditions for the energy storage elements by using the continuity of voltage across a capacitor and the continuity of current through an inductor.

2.

Analysis of electric circuits after switching. This step normally involves the solution of initial value problems for ordinary differential equations.

•

Analysis of transients excited by initial conditions requires the solution of homogeneous differential equations subject to nonzero initial conditions.

•

Analysis of transients excited by sources requires the solution of nonhomogeneous differential equations subject to zero initial conditions.

•

Analysis of transients excited by sources rquires the solution of nonhomogeneous differential equations subject to zero initial conditial conditions.

•

Analysis of transients excited by initial conditions and sources requires the solution of nonhomogeneous equations subject to nonzero initial conditions.

•

A complete solution of a nonhomogeneous linear differential equation can be represented as a sum of a particular solution of the nonhomogeneous equation and a general solution of the corresponding homogeneous equation.

•

In the case of excitation by ac sources, the particular solution of the nonhomogeneous equation can be found by using the phasor technique for the calculation of the steady-state response. The particular solution has the physical meaning of forced response.

•

A general solution of the corresponding homogeneous equation has the physical meaning of free (transient) response. Its calculation requires the solution of characteristic equations and determination of unknown constants from initial conditions.

•

The nature of the transient (free) response of second-order circuits is determined by the roots of the characteristic equation. There are four distinct cases:

a)

overdamped response when the two roots are real, negative, and distinct;

b)

critically damped response when the two roots are real, negative, and identical;

c)

underdamped response when the two roots are complex and conjugate;

d)

undamped response when the two roots are imaginary and conjugate.

•

The machinery of transfer functions is a very powerful tool in the analysis of transients in electric circuits. This analysis proceeds as follows. A circuit variable, which we are interested in, is identified as the output. The transfer function is defined as the ratio of the output phasor to the input phasor, i.e., excitation source. By using the phasor technique, the transfer function H ^ ( s ) is found as the function of complex frequency s. The value of this function at the excitation frequency (s = jω) fully determines the ac steady-state (forced) response, while the poles of the transfer function determine the exponents and, consequently, the form of the free response. The described approach is algebraic in nature; it completely avoids the derivation and solution of differential equations and fully exploits the machinery of the phasor technique.

•

To calculate the response of an electric circuit to an arbitrary source, the convolution integral can be used. The convolution integral has the formml:

(7.464) i ( t ) = ∫ 0 t i δ ( t − τ ) v s ( τ ) d τ ,

where i(t) is the response (current) caused by the excitation by the voltage source v s(t), while i δ(t) is the unit impulse response caused by the unit impulse source excitation. The unit impulse response can be found as the time derivative of unit step response, which in turn can be found from the transient analysis of the electric circuit excited by a unit dc source. Thus, the analysis of an electric circuit by using the convolution integral consists of two major steps: a) calculation of unit impulse response; b) evaluation of convolution integral for an arbitrary (but given) voltage source v s(t). Expressions similar to (7.464) hold for the convolution integrals when the desired circuit response is a voltage and/or the circuit excitation is a current source.

•

A diode is a two-terminal element whose resistance depends on the polarity of the applied voltage. Ideal diodes act as short circuits if applied voltages are positive, and they act as open circuits if applied voltages are negative. Diodes are used in rectifier circuits to convert ac voltage sources into dc voltage sources. A single diode can be used to construct a half-wave rectifier. A diode bridge circuit can be used to construct a full-wave rectifier. Energy storage elements are employed in rectifier circuits to reduce the level of ripples. The techniques for transient analysis of electric circuits can be used for the steady-state analysis of rectifiers.

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Introduction to pressure-transient analysis

Sadiq J. Zarrouk , Katie McLean , in Geothermal Well Test Analysis, 2019

Abstract

Pressure-transient analysis (PTA) is a valuable tool for assessing well condition and reservoir parameters. PTA theory is very well established in the petroleum industry but not in the geothermal industry due to various challenges. PTA can be based on analytical models to represent simple systems, or based on numerical models for more complex systems. Geothermal reservoirs are complex systems; however, most PTA is based on analytical models, primarily due to availability and ease of use. In this chapter the definition of PTA is presented along with an overview of typical well test types, and those used for geothermal wells. An historical overview of the development of petroleum industry analytical PTA discusses major developments in both theory and graphical methods, followed by a focus on geothermal PTA developments. The chapter covers the fundamental concepts which underpin PTA and are discussed often, and also details of the major analytical graphical methods.

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The transient response of turbocharger turbines

H. Chen , ... P. Li , in 10th International Conference on Turbochargers and Turbocharging, 2012

4 CONCLUSIONS

The transient response of turbochargers was studied through Newton's equations of motion and a single non-dimensional parameter 2 W ˙ ¯ Δt ω 0 2 I controlling the response was identified. Larger values of this parameter lead to higher turbocharger speed after the transient. Analysis of this parameter showed that:

1.

Reduction of bearing loss has a positive effect on the transient response.

2.

It is better to start the transient at higher turbo speeds.

3.

Smaller turbochargers will have better transient response than larger ones.

4.

Turbine efficiency and expansion ratio have a large effect on the response.

5.

The transient response can be improved by a combination of a smaller turbine housing and a larger turbine wheel exit area (larger trim), and by using a mixed flow turbine.

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Completion and output testing

Sadiq J. Zarrouk , Katie McLean , in Geothermal Well Test Analysis, 2019

6.3.4 Injection pressure transient: build-up/fall-off

PTA is an extensive subject, covered in detail in Chapter 4 , Introduction to Pressure-Transient Analysis, and Chapter 5, Advanced Analytical Pressure-Transient Analysis Relevant to Geothermal Wells and Chapter 8, Numerical Pressure-Transient Analysis Modelling Framework. In this section the analysis will not be discussed, only the manner in which the data collection fits into the completion test design (Section 6.2.1).

It is possible to measure pressure transients at various times throughout the completion test, whenever there is a change in flow rate. When injection rates increase, "build-ups" can be measured (not the same as a build-up after production), and when rates decrease, "fall-offs" can be measured. Ideally, both a build-up and fall-off will be measured, to yield two different types of PTA data set for comparison, and without moving the tool (as discussed in Section 6.2.1).

Unfortunately, pressure fall-offs are commonly measured as an afterthought, to zero flow, when the well is shut in and the tool is still in the well waiting for the first heatup run (often 1-hour heating). From a PTA point of view, a fall-off to zero flow should be avoided due to a range of issues that can arise when injection ceases and the temperature profile in the well changes rapidly as the well heats up. These include expansion of the fluid column (Section 7.10), expansion of the wireline (Section 7.9) and also downflows which were suppressed by injection but start again when injection stops (Section 7.8). A fall-off to a lower flow rate rather than zero flow (these two options are shown in Fig. 6.6) is strongly recommended as it avoids those issues as much as possible, by minimising the change in temperature profile of the well over the duration of the pressure transient.

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Power Supplies

Peter Wilson , in The Circuit Designer's Companion (Fourth Edition), 2017

7.2.13 Transient Response

The transient response of a power supply is a measure of how fast it reacts to a sudden change in load current. This is primarily a function of the bandwidth of the regulator's feedback loop. The regulator has to maintain a constant output in the face of load changes, and the speed at which it can do this is set by its frequency response as with any conventional operational amplifier. The trade-off that the power supply designer has to worry about is against the stability of the regulator under all load conditions; a regulator with a very fast response is likely to be unstable under some conditions of load, and so its bandwidth is "slugged" by a compensation capacitor within the regulator circuit. Too much of this and the transient response suffers. The same effect can be had by siting a large capacitor at the regulator output, but this is a brute-force and inefficient approach because its effect is heavily load-dependent. Note that the 78XX series of three-terminal regulators should have a small, typically 0.1   μF capacitor at the output for good transient response and HF noise decoupling. This is separate from the required 0.33–1   μF capacitor at the input to ensure stability.

Switch-Mode Versus Linear

The transient response of a switch-mode power supply is noticeably worse than that of a linear because the bandwidth of the feedback loop has to be considerably less than the switching frequency. Typically, switch-mode transient recovery time is measured in milliseconds while linear is in the tens of microseconds.

If your circuit only presents slowly varying loads then the power supply's transient response will not interest you. It becomes important when a large proportion of the load can be instantaneously switched—a relay coil or bank of LEDs for example—and the rest of the load is susceptible to short-duration over- or undervoltages.

Although load transient response is usually the most significant, a regulator also exhibits a delayed response to line transients, and this may become important when you are feeding it from a dc input that can change quickly. The line transient response is normally of the same order as the load response.

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Power supplies

Peter Wilson , in The Circuit Designer's Companion (Third Edition), 2012

7.2.13 Transient response

The transient response of a power supply is a measure of how fast it reacts to a sudden change in load current. (See Figure 7.15.) This is primarily a function of the bandwidth of the regulator's feedback loop. The regulator has to maintain a constant output in the face of load changes, and the speed at which it can do this is set by its frequency response as with any conventional operational amplifier. The trade-off that the power supply designer has to worry about is against the stability of the regulator under all load conditions; a regulator with a very fast response is likely to be unstable under some conditions of load, and so its bandwidth is "slugged" by a compensation capacitor within the regulator circuit. Too much of this and the transient response suffers. The same effect can be had by siting a large capacitor at the regulator output, but this is a brute-force and inefficient approach because its effect is heavily load-dependent. Note that the 78XX series of three-terminal regulators should have a small, typically 0.1   μF capacitor at the output for good transient response and HF noise decoupling. This is separate from the required 0.33−1   μF capacitor at the input to ensure stability.

FIGURE 7.15. Load transient behavior

Switch-mode versus linear

The transient response of a switch-mode power supply is noticeably worse than that of a linear because the bandwidth of the feedback loop has to be considerably less than the switching frequency. Typically, switch-mode transient recovery time is measured in milliseconds while linear is in the tens of microseconds.

If your circuit only presents slowly varying loads then the power supply's transient response will not interest you. It becomes important when a large proportion of the load can be instantaneously switched − a relay coil or bank of LEDs for example − and the rest of the load is susceptible to short-duration over- or undervoltages.

Although load transient response is usually the most significant, a regulator also exhibits a delayed response to line transients, and this may become important when you are feeding it from a DC input which can change quickly. The line transient response is normally of the same order as the load response.

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Well Test Analysis: The Use of Advanced Interpretation Models

In Handbook of Petroleum Exploration and Production, 2002

Transient state

Transient responses are observed before constant pressure or closed boundary effects are reached. The pressure variation with time is a function of the well geometry and the reservoir properties, such as permeability and heterogeneity.

Usually, well test interpretation focuses on the transient pressure response. Near wellbore conditions are seen first and later, when the drainage area expands, the pressure response is characteristic of the reservoir properties until boundary effects are seen at late time (then the flow regime changes to pseudo steady or steady state). In the following, several characteristic examples of well behavior are introduced, for illustration of typical well test responses.

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Characterization of Porous Solids VII

V. Fierro , ... C. Mirodatos , in Studies in Surface Science and Catalysis, 2007

2.2. TAP experiments

Transient responses experiments were performed in the TAP-2 reactor under vacuum conditions and at temperatures ranging from 423 to 498 K. An activated carbon loading of 15 mg with a particle size of approximately 0.2-0.3 mm was placed in the center of two layers of 0.2 – 0.3 mm size quartz particles. A microreactor filled with quartz only was employed to determine the Knudsen diffusion coefficients. n-Butane together with argon were introduced in the microreactor (25.4 mm in length and 4 mm in diameter) in a volume ratio of 1:1. The reactor is evacuated continuously and the transient responses of n-butane and argon were monitored by a quadrupole mass spectrometer following the signal of a single atomic mass unit (amu) per pulse as a function of time. The amu's of 43 and 40 were used to monitor butane and argon respectively. Pulse broadening reflects the diffusion between the particles, diffusion inside the particles and adsorption/desorption at the surface. The data-acquisition time to measure the entire pulse response was 1 s for argon and up to 20s for butane. By modeling the pulse responses the adsorption and diffusion parameters can be determined. For details of this technique, the reader is referred to [4].

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Fundamentals, Analysis Methods and Workflow

Christopher R. Clarkson , in Unconventional Reservoir Rate-Transient Analysis, 2021

RTA workflow 92

1.4.1

Step 1: Assess data adequacy (Chapter 2) 93

1.4.2

Step 2: Check for data correlation (Chapter 2) 93

1.4.3

Step 3: Filter and edit data (Chapter 2) 94

1.4.4

Step 4: Identify flow regimes (Chapter 3) 94

1.4.5

Step 5: Perform straight-line analysis (Chapter 4) 95

1.4.6

Step 6: Perform type-curve analysis (Chapter 5) 96

1.4.7

Step 7: Perform history match with model (examples in Chapters 6–9 chapter 6 chapter 7 chapter 8 chapter 9 ) 97

1.4.8

Step 8: Use calibrated model to forecast (examples in Chapter 9) 98

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