Academic literature on the topic 'Discrete-time domain modelling'

Abstract The paper proposes an approach to constructing a mathematical model of lattice functions, which are mainly used in the study of discrete control systems in the time and domain of the Laplace transform. The proposed approach is based on the assumption of the physical absence of an impulse element. An alternative to the classical approach to the description of discrete data acquisition - the process of quantization in time, is considered. As a result, models of the lattice function in the time domain and the domain of the discrete Laplace transform are obtained. Based on the obtained mathematical models of lattice functions, a mathematical model of the time quantization element of the system is obtained. This will allow in the future to proceed to the construction of mathematical models of various discrete control systems, incl. expanding the proposed approaches to the construction of mathematical models of multi-cycle continuous-discrete automatic control systems

The equivalent model of offshore DC power collection network for the harmonic susceptibility study is proposed based on the discrete time-domain modelling technique and frequency scan approach in the frequency domain. The proposed methodology for modelling a power converter and a DC collection system in the frequency domain can satisfy harmonic studies of any configuration of wind farm network and thereby find suitable design of power components and array network. The methodology is intended to allow studies on any configuration of the wind power collection, regardless of choice of converter topology, array cable configuration, and control design. To facilitate harmonic susceptibility study, modelling DC collection network includes creating the harmonic model of the DC turbine converter and modelling the array network. The current harmonics within the DC collection network are obtained in the frequency domain to identify the resonance frequency of the array network and potential voltage amplification issues, where the harmonic model of the turbine converter is verified by the comparison of the converter switching model in the PLECS™ circuit simulation tool and laboratory test bench, and show a good agreement.

In this paper we present pddl+, a planning domain description language for modelling mixed discrete-continuous planning domains. We describe the syntax and modelling style of pddl+, showing that the language makes convenient the modelling of complex time-dependent effects. We provide a formal semantics for pddl+ by mapping planning instances into constructs of hybrid automata. Using the syntax of HAs as our semantic model we construct a semantic mapping to labelled transition systems to complete the formal interpretation of pddl+ planning instances. An advantage of building a mapping from pddl+ to HA theory is that it forms a bridge between the Planning and Real Time Systems research communities. One consequence is that we can expect to make use of some of the theoretical properties of HAs. For example, for a restricted class of HAs the Reachability problem (which is equivalent to Plan Existence) is decidable. pddl+ provides an alternative to the continuous durative action model of pddl2.1, adding a more flexible and robust model of time-dependent behaviour.

Abstract In this paper, a modified time-domain model of millimeter-wave all-digital phase-locked loop (ADPLL) is implemented. In order to reflect the true behaviour of ADPLL, a quantified output digitally controlled-oscillator (DCO) with time domain jitter is proposed. In this ADPLL time-domain model, the DCO model can only output discrete frequency points to imitate the quantization effect of true DCO, and the overlap of different level tuning band is added into this model to imitate the true situation. In addition, the DCO time domain jitter and wander are also added into this model by using the Box-Muller method. Finally, a period estimation method is used to calculate the phase power spectrum density of the output signal, and then the phase noise is obtained through subsequent processing.

This article presents a general six-step discrete-time Zhang neural network (ZNN) for time-varying tensor absolute value equations. Firstly, based on the Taylor expansion theory, we derive a general Zhang et al. discretization (ZeaD) formula, i.e., a general Taylor-type 1-step-ahead numerical differentiation rule for the first-order derivative approximation, which contains two free parameters. Based on the bilinear transform and the Routh–Hurwitz stability criterion, the effective domain of the two free parameters is analyzed, which can ensure the convergence of the general ZeaD formula. Secondly, based on the general ZeaD formula, we design a general six-step discrete-time ZNN (DTZNN) for time-varying tensor absolute value equations (TVTAVEs), whose steady-state residual error changes in a higher order manner than those presented in the literature. Meanwhile, the feasible region of its step size, which determines its convergence, is also studied. Finally, experiment results corroborate that the general six-step DTZNN model is quite efficient for TVTAVE solving.

Soil-structure-interaction (SSI) analyses are essential to evaluate the seismic performance of important structures before finalizing their structural design. SSI under seismic condition involves much more complex interaction with soil compared to the dynamic loads having source on the structure. Seismic SSI analysis requires due consideration of site-specific and structure-specific properties to estimate the actual ground motion (scattered motion) experienced at the base of the structure, and subsequently the effects of the scattered motion on the structure. Most challenging aspect of seismic SSI analysis is to implement transmitting boundaries that absorb the artificial reflections of stress waves at the truncated interface of the finite and infinite domains, while allowing the seismic waves to enter the finite domain. In this paper, the time domain implementation of seismic analysis of a soil-structure system is presented using classical discrete models of structure and interactive force boundary conditions for soil. These models represent typical SSI systems- a single Degree of Freedom (DOF) of a spherical cavity with mass attached to its wall, a two DOF system consisting of a mass attached by a nonlinear spring to a semi-infinite rod on elastic foundation, and a three DOF system with additional DOFs for modelling the structural stiffness and damping. The convolution integral representing the force boundary condition on the truncated interface, is evaluated interactively using UAMP user-subroutine in ABAQUS and applied as concentrated forces at the interface (truncated interface) nodes of the bounded domain or generalized-structure domain. The verification problems presented in the paper show the satisfactory performance of the developed MATLAB code and ABAQUS implementation with FORTRAN user-subroutines. The classical phenomena associated with the dynamic soil-structure systems are discussed through the present work.

This paper focuses on the modelling of the series resonant converter proposed as a DC/DC converter for DC wind turbines. The closed-loop control design based on the discrete time domain modelling technique for the converter (named SRC#) operated in continuous-conduction mode (CCM) is investigated. To facilitate dynamic analysis and design of control structure, the design process includes derivation of linearized state-space equations, design of closed-loop control structure, and design of gain scheduling controller. The analytical results of system are verified in z-domain by comparison of circuit simulator response (in PLECS™) to changes in pulse frequency and disturbances in input and output voltages and show a good agreement. Furthermore, the test results also give enough supporting arguments to proposed control design.

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh. In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.

Thesis (MScEng)--Stellenbosch University, 2012
ENGLISH ABSTRACT: The class-D audio power amplifier has found widespread use in both the consumer and professional audio industry for one reason: efficiency. A higher efficiency leads to a smaller and cheaper design, and in the case of mobile devices, a longer battery life. Unfortunately, the basic class-D amplifier has some serious drawbacks. These include high distortion levels, a load dependent frequency response and the potential to radiate EMI. Except for EMI, the aforementioned issues can be mitigated by the proper implementation of global negative feedback. Negative feedback also has the potential to indirectly reduce EMI, since the timing requirements of the output devices can be relaxed. This thesis discusses the design of a clocked analogue controlled pulse-width modulated class-D audio amplifier with global negative feedback. The analogue control loop is converted to the z-domain by modelling the PWM comparator as a sampling operation. A method is implemented that improves clip recovery and ensures stability during over-modulation. Loop gain is shaped to provide a high gain across the audio band, and ripple compensation is implemented to minimize the negative effect of ripple feedback. Experimental results are presented.
AFRIKAANSE OPSOMMING: Die klas-D klankversterker geniet wydverspreide gebruik in beide die verbruiker en professionele oudio industrie vir een rede: benuttingsgraad. ’n Hoër benuttingsgraad lei tot ’n kleiner en goedkoper ontwerp, en in die geval van draagbare toestelle, tot langer batterylewe. Ongelukkig het die basiese klas-D klankversterker ernstige tekortkominge, naamlik hoë distorsievlakke, ’n lasafhanklike frekwensierespons en die vermoë om EMI te genereer. Behalwe vir EMI kan hierdie kwessies deur die korrekte toepassing van globale negatiewe terugvoer aangespreek word. Negatiewe terugvoer het ook die potensiaal om EMI indirek te verminder, aangesien die tydvereistes van die skakel stadium verlaag kan word. Hierdie tesis bespreek die ontwerp van ’n geklokte analoog-beheerde pulswydte-modulerende klas-D klankversterker met globale negatiewe terugvoer. Die analoogbeheerlus word omgeskakel na die z-vlak deur die PWM vlakvergelyker as ’n monster operasie te modelleer. ’n Metode word geïmplementeer wat die stabiliteit van die lus verseker tydens oormodulasie. Die lusaanwins word gevorm om ’n hoë aanwins in die oudioband te verseker en riffelkompensasie word geïmplementeer om die negatiewe effek van terugvoerriffel teen te werk. Eksperimentele resultate word voorgelê.

Accurate testing of a device under development, in near to real-life environment, is essential to the rapid growth of emerging technologies such as distributed generation, renewable energy sources, electrical storage, microgrid and electrical vehicles. Power hardware in loop (PHIL) simulation is an emerging methodology which allows for the testing of a physical hardware, i.e., a device under test (DUT), in a safe and controlled environment, without the rest of the system being available. The rest of the system for the DUT is emulated through its mathematical models computed in a real-time simulator (RTS). The DUT and the RTS interact with each other through a power amplifier (PA), which scales up the signals provided by the RTS for the DUT, and sensors. For accurate testing of a DUT, its response in a PHIL simulation should be similar to that in a real-life environment. The two responses are often different due to several factors which are present in a PHIL simulation but not in an actual system. An RTS is a discrete-time domain system while the DUT is a continuous-time domain system. Finite computation time requirement of the RTS and conversion of the continuous-time domain signals to the discretetime domain signals, and vice versa, results in an inaccurate response at high frequencies. Even worse, the PA employed usually introduces additional dynamics into a PHIL simulation and results in inaccuracies at much lower frequencies. The inaccuracy can be so significant that the PHIL simulation of a system can be unstable even though the actual system is stable. This thesis deals with the power amplification required for the accurate replication of the fast transients of a system in a PHIL simulation, accurate modelling of high frequency phenomena of the discrete-time domain characteristics of a PHIL simulation and associated stability analysis, and stabilizing methods for PHIL simulation. Conventional linear PAs employed in a PHIL simulation are too bulky, expensive and lossy to be used at medium to high power applications. Whereas, a conventional filter-based switched-mode PA has a limited dynamic response to emulate fast transients of a system in a PHIL simulation. An output filter-less voltage source inverter is proposed as a high bandwidth PA for a PHIL simulation with inductive DUT. The proposed PA, realized through an IGBT-based PWM converter stack, is utilized to emulate the transients of synchronous generator, including fast transient corresponding to the field excitation controller, while feeding a balanced linear load. Along with a proposed modification to the conventionally used synchronous generator model, in order to include the effects of stator transients, improved accuracy is obtained for unbalanced and non-linear loads also. The applicability of the proposed PA is extended for it to be interfaced to PWM converters by proposing an in-phase synchronization of PWM carriers of the PA and the converter under test. The PA is utilized for testing the control of a three-phase 415 V, 3 kW PWM rectifier For various applications, a PA is required to have power sinking capability, which can be achieved by supplying it from a grid-connected PWM rectifier. A simple input voltage sensor-less vector control of PWM rectifier is proposed in the thesis. While the performance of the proposed method, in terms of THD and power factor, is comparable to the sensorbased method and existing sensor-less methods, its computation time requirement is much lower than those for these methods. The proposed control is validated through simulations and experiments on a three-phase 415 V, 3 kW grid-connected PWM rectifier, generating 800 V dc supply. Conventional continuous-time domain transfer functions of current control loop of a PWM rectifier, and PWM converters in general, do not represent accurately the closed-loop system when the bandwidth of the loop is comparable to the switching frequency of the converter. Third-order reference and disturbance transfer functions of discrete-PI controlled current loop of PWM converters are proposed in the thesis which are utilized to derive closed-form expressions for the current response of the converter for step changes in current reference and voltage disturbance. Consequently, optimized PI-controller parameters are obtained for the fastest disturbance rejection settling time. Further, a pre-filter to the current control loop is proposed to achieve dead-beat reference tracking response. The pre-filter, along with the optimized PI-controller, results in reference tracking and disturbance rejection settling times of two and eight switching cycles, respectively. A PHIL simulation, being a combinational of continuous-time domain and discrete-time domain systems, is conventionally represented using continuous-time models. A discretetime domain model is proposed in the thesis which represents a PHIL simulation much more accurately than the conventional model. The proposed model is used to conduct stability analysis of PHIL simulations. The stability limits in terms of the parameters of the simulated and physical quantities are more accurately estimated through the proposed method as compared to the conventional methods. The stability limits of PHIL simulation are verified through simulations and experiments. Low-pass filter (LPF) based feedback current filtering (FCF) method is widely used for stabilizing an unstable PHIL simulation. The proposed discrete-time domain modelling method is utilized to show that the low-pass filter-based FCF method is ineffective in stabilizing PHIL simulations having highly inductive physical impedances. Phase-lag compensator (PLC) is proposed to be a superior alternative to low-pass filter in such cases. Further, a novel cross-coupled compensator (CCC) is proposed in this thesis. The same is utilized as a filter in FCF method for stabilizing those PHIL simulations where both LPF and PLC are ineffective. CCC-based FCF method is employed for the PHIL simulation of a single machine infinite bus system. The proposed CCC is also utilized for realizing a fault-tolerant synchronous inverter, i.e., a renewable energy source-fed grid tied-inverter which is controlled to act as a synchronous generator. The proposed synchronous inverter draws significantly less current in case of a grid fault as compared to a conventional synchronous inverter and hence avoids damage to the inverter without additional current limiting methods.
DST, MHRD

AbstractIn time series analysis auto regressive (AR) modelling of zero mean data is widely used for system identification, signal decorrelation, detection of outliers and forecasting. An AR process of order p is uniquely defined by r coefficients and the variance of the noise. The roots of the characteristic polynomial can be used as an alternative parametrization of the coefficients, which is used to construct a continuous covariance function of the AR processes or to verify that the AR processes are stationary. In this contribution we propose an approach to estimate an AR process of time varying coefficients (TVAR process). In the literature, roots are evaluated at discrete times, rather than a continuous function like we have for time varying systems. By introducing the assumption that the movement of the roots are linear functions in time, stationarity for all possible epochs in the time domain is easy to accomplish. We will illustrate how this assumption leads to TVAR coefficients where the k-th coefficient is a polynomial of order k with further restrictions on the parameters of the coefficients. At first we study how to estimate TVAR process parameters by using a Least Squares approach in general. As any AR process can be rewritten as a combination of AR processes of order two with two complex conjugated roots and AR processes of order one, we limit our investigations to these orders. Higher order TVAR processes are computed by successively estimating TVAR processes of orders one or two. Based on a simulation, we will demonstrate the advantages of a time varying model and compare them to the stationary time stable model. In addition, we will give a method to identify time series, for which the model of the TVAR processes with linear roots is suitable.

Mathematical morphology, used extensively in image processing, tracks the support domains for the operation of convolution and deconvolution. Morphology is also important in the modelling of signals on time scales. Time scale theory provides a generalization tent under which the operations of discrete and continuous time signal and Fourier analysis rest as special cases. The time scale paradigm provides modelling under which a rich class of hybrid signals and systems can be analyzed. We begin with introductory material on mathematical morphology which is foundational to the development of time scale theory. The support of convolution is related to the operation of dilation in mathematical morphology. Mathematical morphology is most commonly associated with image processing. Applications of morphology was initially applied to binary black and white images by Matheron [966]. The field is richly developed [506, 578]. Here, we outline the fundamentals. In N dimensions, let X and H denote a set of vectors or, in the degenerate case of one dimension, a set of real numbers.

Abstract Over the past years, nonlinear frequency-domain methods have become a state-of-the-art technique for the numerical simulation of unsteady flow fields within multistage turbomachinery as they are capable of fully exploiting the given spatial and temporal periodicities, as well as modelling flow nonlinearities in a computationally efficient manner. Despite this success, it still remains a significant challenge to capture nonlinear interaction effects within the context of configurations with multiple fundamental frequencies. If all frequencies are integer multiples of a common fundamental frequency, the interval spanned by the sampling points typically resolves the period of the common base frequency. For configurations in which the common frequency is very low in relation to the frequencies of primary interest, many sampling points are required to resolve the highest harmonic of the common fundamental frequency and the method becomes inefficient. In addition when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations or when simulating the nonlinear interaction in the context of forced response or flutter, then the standard discrete Fourier transform is no longer suitable and the basic harmonic balance method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or non-equidistant time sampling. However, the definition of suitable sampling points that lead to well-conditioned Fourier transform matrices and small aliasing errors is an intricate issue and far from straightforward. To overcome the issues regarding multi-frequency problems described above, a new harmonic balance approach based on multidimensional Fourier transforms in time is presented. The basic idea of the approach is that, instead of defining common sampling points in a common time period, separate time domains, one for each base frequency, are spanned and the sampling points are computed equidistantly within each base frequency’s period. Since the sampling domain is now extended to a multidimensional time-domain, all time instant combinations covering the whole multidimensional domain are computed as the Cartesian product of the sampling points on the axes. In a similar fashion the frequency-domain is extended to a multidimensional frequency-domain by the Cartesian product of the harmonics of each base frequency, so that every point defined by the Cartesian product is an integer linear combination of the occurring frequencies. In this way the proposed method is capable of fully integrating the nonlinear coupling effects between higher harmonics of different fundamental frequencies by using multidimensional discrete Fourier transforms within the harmonic balance solution procedure. The aim of this paper is to introduce the multidimensional harmonic balance method in detail and demonstrate the capability of the approach to simultaneously capture unsteady disturbances with arbitrary excitation frequencies. Therefore the well established aeroelasticity testcase standard configuration 10 in the presence of an artificial inflow disturbance, that mimics an upstream blade wake, is investigated. The crucial aspect of the proposed testcase is that a small ratio of the frequency of the inflow disturbance and the blades vibration frequency is chosen. To demonstrate the advantages of the newly proposed multidimensional harmonic balance approach, the results are compared to unsteady simulations in the time-domain and to state-of-the-art frequency-domain methods based on one-dimensional discrete Fourier transforms.

Abstract The significant difference in length scales between the flow around a moving fish net and the flow around each twine of the net prevents the resolution of the complete structure within a discrete fluid domain. In this paper, this issue is overcome by calculating the net and fluid dynamics separately and incorporate their interaction implicitly. The forces on the net are approximated using a screen force model, and the motion of the net is computed with a lumped mass method. Here, a linear system of equations is derived from the dynamic equilibria and kinematic relations. The net model is coupled to the CFD solver REEF3D which solves the incompressible Navier-Stokes equations using high-order finite differences in space and time. Several numerical calculations are provided, and the comparison of loads and velocity reduction with available measurements indicates the good performance of the proposed model.

Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are usually dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows, (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as modelling flow nonlinearities. A nontrivial issue in the application of frequency-domain methods to turbomachinery flows is to simultaneously capture disturbances with multiple fundamental frequencies in one relative system. In case of harmonically related frequencies, the interval spanned by the sampling points typically resolves the common fundamental frequency. To avoid signal aliasing the highest harmonic of the common frequency should be sampled with an appropriate number of sampling points. However, when the common fundamental frequency is very low in relation to the frequencies of primary interest, equidistant time sampling leads to a high number of sampling points, hence frequency-domain methods can become computationally inefficient. Furthermore, when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations, the standard discrete Fourier transform is no longer suitable and the basic harmonic balance method requires extension. In this article two nonlinear frequency-domain approaches for dealing with the accounted issues are demonstrated and compared. The first approach is a generalized harmonic balance method based on almost periodic Fourier transforms with non-equidistant time sampling. Then the so-called harmonic set approach, developed by the authors, is evaluated. Based on the neglection of the nonlinear, quadratic cross-coupling terms between higher harmonics of different fundamental frequencies, the harmonic set approach allows the superposition of periodic disturbances with different fundamental frequencies as well as the separated, equidistant sampling of the highest harmonic of each base frequency. The aim of this paper is to compare the computational efficiency and accuracy of the two methods and assess the impact of neglecting the quadratic cross-coupling terms.

The technique of pre-swirling cooling air to reduce its relative total temperature, as felt by rotating components, is well established. It is important to optimise the design of such systems in order to achieve maximum cooling effectiveness and to minimise the impact on cycle efficiency. Traditionally, these cooling systems have been developed by a combination of experimental investigation and careful evolution. However, more recently it has become practical to apply CFD to such problems. The nature of gas turbine cooling systems generally mandates the presence of discrete features on both static and rotating components, so that a fully rigorous analysis would need to be both 3D and unsteady, with the sub-domains adjacent to static and rotating surfaces solved in an appropriate frame of reference, together with a suitable interfacing procedure to communicate the evolving solution between each sub-domain. Such analyses are challenging for current CFD codes, both in terms of computation time and numerical stability. The present work explores the various options that are available to make such computations more practical and hence more accessible to the secondary systems modelling community. Significant reductions in set-up time can be achieved by adopting unstructured calculational meshes, although this may be at the expense of some loss of accuracy and increase in computational time relative to structured meshes. In the present work, an attempt has been made to quantify the effect of these choices. Depending on the configuration of the system under investigation, it may be permissible to ignore the unsteady interactions and to model the system using the more computationally efficient multiple reference frame (MRF) approach. Guidelines are proposed for assessing the likely impact of these simplifications on the results obtained.

Abstract The 3-D unsteady turbulent flow inside a centrifugal fan and its downstream pipe is investigated at the best efficiency point (BEP) flow rate using the computational fluid dynamics (CFD) package ANSYS FLUENT. The impeller with an outlet diameter of 400 mm has 12 forward curved blades. The computational domain comprises four parts: the inlet part, the impeller, the volute, and the downstream pipe. The flow domain was meshed in ANSYS ICEM-CFD with structured hexahedron cells, and nearly 9 million cells were used. The Detached Eddy Simulation (DES) turbulence modelling approach was employed with this fine enough mesh scheme. The impeller was set as the rotating domain at a speed of 2900 rpm. A sliding mesh technique was applied to the interfaces in order to allow unsteady interactions between the rotating impeller and the stationary parts; the unsteady interactions generate pressure fluctuations inside the centrifugal fan. One impeller revolution is divided into 2048 time steps, in order to capture the transient flow phenomena with high resolution. Monitoring points were set along the volute casing profile, and along the downstream pipe centerline. When the numerical simulation became stable after several impeller revolutions, the statistics of the unsteady flow was initiated with a total of 16384 time steps (8 impeller revolutions) data. The time history data of the pressure and velocity magnitude at the monitoring points were saved and with Fourier transform applied to obtain the frequency spectra. The time-averaged flow fields show clearly the static pressure rises gradually through the impeller, and further recovers from the velocity in the volute, and decreases gradually along the downstream pipe due to the friction. The mean pressure at the pressure side of the impeller blade is larger than it at the suction side, forming the circumferential nonuniform flow pattern. Owing to the forward-curved blades, large velocity region exists around the impellor exit, and the maximum velocity near the trailing edge can reach 1.5u2, where u2 is the circumferential velocity at the impeller outlet. The root mean square (rms) value distribution of pressure fluctuations show that most parts inside the centrifugal fan undergo large pressure fluctuation with the magnitude about 10% of the reference dynamic pressure pref = 0.5ρu22; the maximum value locating at the tongue tip can reach 30% of pref. The pressure fluctuation magnitude decreases quickly along the outlet pipe: after 5D (D is the outlet pipe diameter) the magnitude is 0.5% of pref. The pressure and velocity fluctuation spectra at the monitoring points in the volute show striking discrete components at the blade-passing frequency (BPF) and its 2nd, 3rd harmonics. The BPF component has the maximum value of 15% of pref in the tongue region, and it decreases dramatically along the downstream pipe with the amplitude less than 0.2% of pref after 5D distance.