A review of vibration-based MEMS piezoelectric energy harvesters
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A review of vibration-based MEMS piezoelectric energy harvesters

Course Number: EECS 101, Fall 2012

College/University: Beacon FL

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Energy Conversion and Management 52 (2011) 500504 Contents lists available at ScienceDirect Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman A review of vibration-based MEMS piezoelectric energy harvesters Salem Saadon *, Othman Sidek Collaborative Microelectronic Design Excellence Center (CEDEC), School of Electrical and Electronic Engineering, Universiti Sains Malaysia,...

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Conversion Energy and Management 52 (2011) 500504 Contents lists available at ScienceDirect Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman A review of vibration-based MEMS piezoelectric energy harvesters Salem Saadon *, Othman Sidek Collaborative Microelectronic Design Excellence Center (CEDEC), School of Electrical and Electronic Engineering, Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia article info Article history: Received 28 November 2009 Accepted 14 July 2010 Available online 5 August 2010 Keywords: PZT Vibration Energy harvesting MEMS abstract The simplicity associated with the piezoelectric micro-generators makes it very attractive for MEMS applications, especially for remote systems. In this paper we reviewed the work carried out by researchers during the last three years. The improvements in experimental results obtained in the vibration-based MEMS piezoelectric energy harvesters show very good scope for MEMS piezoelectric harvesters in the eld of power MEMS in the near future. 2010 Elsevier Ltd. All rights reserved. 1. Introduction The exibility associated with piezoelectric materials is very attractive for power harvesting. They possess more mechanical energy for conversion into electrical energy and can also withstand large amounts of strain. Many methods have been reported to improve the harvested power of MEMS micro-generators. One of the methods is by selecting a proper coupling mode of operation. This involves two modes of operation. The rst mode called 31 mode, involves the excited vibration force being applied perpendicular to the poling direction (pending beam). And the other is called 33 mode, in which the force is applied on the same side as the poling direction. The two modes 33 mode and 31 mode are as shown in Fig. 1. Between the two modes, 31 mode is most commonly used. It produces a lower coupling coefcient k, when compared to the 33 mode. The second method for harvested power improvement is by changing the device conguration. This is accomplished by adding multiple pieces of piezoelectric materials to the harvester. The uni-morph cantilever beam conguration is as shown in Fig. 2c. Johnson et al. [1] demonstrated using this conguration that highest power can be generated under lower excitation frequencies and load resistances. Two combinations of bimorph structures are possible: (a) Series type. (b) Parallel type. * Corresponding author. Tel.: +60 4 5995800. E-mail address: saadonsalem@yahoo.com (S. Saadon). 0196-8904/$ - see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.07.024 Series and parallel triple layer bimorph structures represented by Ng and Liao [2,3] are shown in Fig. 2a and b respectively. The series triple layer bimorph is constructed out of a metallic layer, sandwiched between two piezoelectrics and the piezoelectric patches are electrically connected in series. In the case of the parallel triple, which is also sandwiched between two piezoelectric layers bimorph, the piezoelectric materials are connected in parallel. The parallel triple layer bimorph has the highest power under medium excited frequencies and load resistances, whereas the series triple layer bimorph produces highest power when excited under higher frequencies and load resistances. A series connection will increase the device impedance as well as improve the output delivered power at higher loads. Several researchers have carried out studies to improve the bimorph efciency. Jiang et al. [4] studied a bimorph cantilever with a proof mass attached to its end. Their results showed that, by reducing the bimorph thickness and increasing the attached proof mass decreases the harvester resonant frequency and produces a maximum harvested power. Similarly, Anderson and Sexton [5], found that, by varying the length and width of the proof mass affects the output of the harvested power. Cantilever geometrical structure also plays an important role in improving the harvesters efciency. The rectangular shaped cantilever structures are most commonly used in MEMS-based piezoelectric harvesters. They are easy to implement and effective in harvesting energy from ambient vibrations. However, the study proposed by Mateu and Moll [6], showed that the triangular shaped cantilever beam with a small free end can withstand higher strains and allows maximum deections, resulting in higher power output when compared to a rectangular S. Saadon, O. Sidek / Energy Conversion and Management 52 (2011) 500504 501 Fig. 1. Piezoelectric coupling modes. beam having width and length equal to the base and height of the proposed triangular cantilever beam. Roundy et al. [7], found that the strain on a trapezoidal shaped cantilever beam can be distributed more throughout its structure, and also observed that, for the same PZT volume a trapezoidal cantilever beam can deliver more than twice the energy than a rectangular shaped beam. Similarly, Baker et al. [8], experimentally tested a nearly triangular trapezoidal shaped cantilever beam along with a rectangular shaped beam of the same volume, and found that 30% more power can be achieved by using the trapezoidal beam than the rectangular one. A circular shaped structure called cymbal was developed by Kim et al. [9]. This structure consists of two dome-shaped metal bonded on a piezoelectric circular plate, as illustrated in Fig. 3. Another method of improving the efciency of the power harvester is by tuning the device so that its resonant frequency matches the ambient vibrations resonant frequency. Shahruz [10,11], designed a power harvester which resonates at various frequency range and without the need of any adjustments. This device consists of different cantilever beams with different lengths and tip masses attached to its common base frame, such that each cantilever has its own resonant frequency, and this resulted in mechanical band-pass lter, which led to the increase of size and cost of the device. Rastegar et al. [12] designed a passive tuning system, which has a two-stage system in which a very low frequency in the range of 0.20.5 Hz can be converted into potential energy and then transferred to the system with a higher natural frequency. The schematic diagram of the harvester is as shown in Fig. 4. Similar works on the modeling, design and fabrication of MEMS-based piezoelectric power harvesters is being referred in the literature survey [1323]. 2. Literature survey Most of the previous studies on piezoelectric energy harvesting micro-generators, concentrated on bulk prototypes. However, only a few of them have demonstrated MEMS micro-generators capability to deliver useful power. Marzencki et al. [13], successfully designed and fabricated a thin lm AlN cantilever micro-generator, that can generate a power of 0.038 lW from a 0.5 g (g = 9.81 m/s2) acceleration at Fig. 3. Piezoelectric cymbal circular shaped cantilever. Fig. 4. Schematic of a typical energy harvesting power source using the two-stage design [12]. 204 Hz resonant frequency. However, the output power is limited to low power levels due to the properties of AlN material. In his later works Marzencki et al. [14], improved the power generated by increasing the vibration amplitude and frequency of their device to 4 g at 1368 Hz resonant frequency to generate a power of 1.97 lW. Shen et al. [15], designed and fabricated a PZT cantilever MEMSbased micro-generator with an integrated Si proof mass, that can generate 2.15 lW from 2 g (g = 9.81 m/s2) acceleration at its resonant frequency of 461.15 Hz. Renaud et al. [16], fabricated a MEMS-based PZT cantilever micro-generator with an integrated proof mass that can generate 40 lW at 1.8 kHz vibration frequency. This device known as unimorph cantilever is shown in Fig. 5. Fang et al. [17] also designed a composite cantilever structure with nickel metal mass as shown in Fig. 6. Fig. 2. (a) A series triple layer type cantilever. (b) A parallel triple layer type cantilever. (c) A uni-morph cantilever. 502 S. Saadon, O. Sidek / Energy Conversion and Management 52 (2011) 500504 Fig. 8. Schematic conguration of a single cantilever beam [18]. Fig. 5. Schematic of a PZT uni-morph cantilever [16]. Fig. 6. Cross-sectional sketch and the fabrication process of micro piezoelectric power generator [17]. improve power output and frequency exibility. The cantilevers are designed for working under low frequency range. The schematic conguration of a single cantilever is shown in Fig. 8. It was found that the output voltage drops off when the excited frequency deviates from the resonant frequency for the available bandwidth of only just 23 Hz as described in Fig. 9. Normally, the derived frequency should be determined rst before any design or fabrication of those devices, and this taken as advantageous to design a device that can perfectly operate over a range of frequencies. Cantilevers having closer resonant frequency are designed by considering appropriate structural parameters. The center level of resonance frequency is determined by the target vibration frequency level. The structure parameters are selected from the nodal simulation using ANSYS software. The frequency bandwidth ranged from 226 to 234 Hz. This shows that the cantilever array has a wider range of bandwidth than that of a single cantilever. When the cantilevers are excited under frequency of 229 Hz, the AC output voltage after direct serial connection was found to Fig. 7. Fabrication process of micro piezoelectric power generator: (1) functional lms preparation: SiO2/Ti/Pt/PZT/Ti/Pt, (2) functional lms pattern, (3) silicon slot etching by RIE, (4) back silicon deep etching by KOH solution, (5) cantilever release by RIE, and (6) metal mass micro-fabrication and assemblage [17]. The metal mass on the tip of the cantilever is used to decrease p the structures natural frequency (w = k/m, where k is the material stiffness and m is the mass), for the application under low-frequency vibration. As described in Fig. 7, the composite cantilever is made up of an upper PZT thick lm sandwiched between a pair of (Pt/Ti) metal electrodes and a lower non-piezoelectric element. When the base of the system is vibrated by the environmental groundwork, input force is fed to the mechanical parts. Some parts move relatively to the base frame causing the PZT material to be tensed or compressed. This in turn induces charge shift as a result of the piezoelectric effect. The magnitude of this electric charge is proportional to the mechanical stress induced by the displacement. The micro-generator should be designed that so it can mechanically resonate at frequency tuned to the ambient vibration in order to generate maximum electrical power, and the structural natural frequency can be regulated by varying the dimensions of the moving parts. The device that was micro-fabricated involved functional lms preparation, pattern and bulk silicon micromachining processes as shown in Fig. 7. The prototype fabricated by MEMS technology under resonant operation with about 609 Hz resulted in the output power in the range of 898 mV and 2.16 lW. Similar to the previous work, Liu et al. [18] used the previous cantilever structure [ref] to construct a power generator array to Fig. 9. Output voltage as a function of excited frequency [18]. Fig. 10. AC output voltage of three cantilevers and their overall output after serial connection [18]. S. Saadon, O. Sidek / Energy Conversion and Management 52 (2011) 500504 be only about 3.06 V, which is less than the actual total value of the cantilevers (5.256 V). Fig. 10 shows the array performance excited under 229 Hz. As shown in Fig. 10, phase difference of about 120 between C1 and C2 is observed. This phase difference impairs the electrical accumulation of the cantilevers and the dc voltage obtained across the capacitor after rectication is only 2.51 V, and the maximum power obtained is about 3.15 lW. The experimental results showed that the arrayed device is very promising and it improved the operation bandwidth and the output power of the generator. This shows the potential use of the arrayed device in the development of the power generator especially in the wireless/embedded sensor network applications. The MEMS vibration-based harvesting device has AC output that needs to be rectied. Almost all the rectifying semiconductor devices consume at least 500 mV as dropped voltage. Hence to overcome this high voltage requirement it is proposed to use the inter-digitated electrodes instead of the proposed PZT parallel electrodes [1921]. Jeon et al. [19] developed a {3-3} mode thin lm PZT cantilever device with inter-digitated electrodes that can generate 1.0 lW from 10.8 g vibration at 13.9 kHz resonant frequency. Lee et al. [20], designed and fabricated piezoelectric MEMS micro-generator with laminated {3-3} mode PZT cantilever and interdigitated electrodes that can generate 0.123 lW under 2 g (g = 9.81 m/s2) acceleration amplitude. Similarly Lee et al. [21] developed two piezoelectric MEMS generators with {3-1} mode and {3-3} mode, having a cantilever made by a silicon micromachining process. The experimental results showed that {3-1} mode micro-generator could generate output power of 2.765 lW excited at 2.5 g amplitude and 255.9 Hz resonant frequency, while the {3-3} mode generator could generate an output power of 1.288 lW under 2 g amplitude and 214 Hz. The schematic diagram of {3-1} mode and {3-3} mode conguration is illustrated in Fig. 11. In the case of {3-3} mode, the inter-digitated electrodes were fabricated with a width of 30 lm and gap of 30 lm. The proof masses for both MEMS generators were fabricated under the beam structure with dimensions of 500 1500 500 lm3 and 750 1500 500 lm3 for the {3-1} and {3-3} modes respectively. A different proof mass dimensions were used to demonstrate the ability of the structure to adjust the resonant frequency. The fabrication process for {3-1} and {3-3} modes conguration are represented in Fig. 12a and b. Fig. 11. Schematic diagram of the piezoelectric MEMS generators: (a) {3-1} mode conguration, (b) {3-3} mode conguration [21]. 503 Fig. 12. Fabrication process of the generators in: (a) {3-1} mode and (b) {3-3} mode [21]. Muralt et al. [22], designed and fabricated a micro power generator of thin lm PZT laminated cantilever with proof mass and inter-digitated electrodes which could generate about 1.6 V and 1.4 lW when excited under 2 g at 870 Hz resonant frequency. Elfrink et al. [23], designed and fabricated a MEMS-based AlN piezoelectric cantilever micro-generator, that can generate an output power of 60 lW under 2 g (g = 9.81 m/s2) acceleration at 572 Hz resonant frequency. Devices with different cantilever beams and mass geometries were produced. Glass wafers were used for the top and bottom covers. Fig. 13 (top) shows the generator design and its package conguration at the rest position, while the Fig. 13 (bottom), shows the movement of the mass when the generator is at resonance [23]. Fig. 13. (Top) Vibration energy harvester packaged in between glass substrates at the rest position. (Bottom) The mass movement [23]. 504 S. Saadon, O. Sidek / Energy Conversion and Management 52 (2011) 500504 The devices were packaged with top and bottom glass substrates within the cavities to allow mass displacement up to about 400 lm. 3. Conclusion Various designs of harvesters and their experimentally obtained results in the last 3 years have been summarized in this review. The following two observations were derived from this study. The rst observation was that, the maximum output power harvested was about 60 lW without any interface power conversion circuit. This produced power losses in the form of consumption power and also resulted in the reduction of the delivered power. Secondly, the absence of the vibration source control affected the delivered output power. According to the literature reviewed thus far, it appears that the delivered power output of those MEMS-based devices is still inadequate to be used as dc power supply to power mobile electronic devices as well as remote sensors and other medical monitoring devices. However, the prospect for improvement looks positively inclined. 4. Future work Our future work is to design and fabricate a novel vibration-based MEMS micro power harvesting device consisting of piezoelectric cantilever type together with interface power conversion circuitry. This is expected to provide the optimal desired dc output power characteristics with high efciency satisfying all the desired parameters and also maintain an output power that can be used to power wireless sensor networks instead of the conventional batteries. Acknowledgments The authors would like to thank our colleague Nazer for his help on the redrawing the gures for clarity. The authors also would like to knowledge the use of the following Grant for this research work: USM/FRGS 6071173. References [1] Johnson TJ, Charnegie D, Clark WW, Buric M, Kusic G. Energy harvesting from mechanical vibrations using piezoelectric cantilever beams. Smart structures and materials: damping and isolation, vol. 6169; 2006. p. D1690 [art. no. 61690D]. [2] Ng TH, Liao WH. Feasibility study of a self-powered piezoelectric sensor. Smart structures and materials: smart electronics, MEMS bio-mems and nanotechnology, vol. 5389; 2004. p. 37788. . [3] Ng TH, Liao WH. Sensitivity analysis and energy harvesting for a self-powered piezoelectric sensor. J Intell Mater Syst Struct 2005;16:78597. [4] Jiang SN, Li XF, Guo SH, Hu YT, Yang JS, Jiang Q. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart Mater Struct 2005;14:76974. [5] Anderson TA, Sexton DW. A vibration energy harvesting sensor platform for increased industrial efciency. In: Smart structures and materials 2006: sensors and smart structures technologies for civil, mechanical, and aerospace systems, proceedings of the SPIE 6174; 2006. p. 6219. [6] Mateu L, Moll F. Optimum piezoelectric bending beam structures for energy harvesting using shoe inserts. J Intell Mater Syst Struct 2005;16: 83545. [7] Roundy S, Leland ES, Baker J, Carleton E, Reilly E, Lai E, et al. Improving power output for vibration-based energy scavengers. IEEE Pervasive Comput 2005;4:2836. [8] Baker J, Roundy S, Wright P. Alternative geometries for increasing power density in vibration energy scavenging for wireless sensor networks. In: Proceeding 3rd international energy conversion engineering conference, San Francisco, California; August 1518, 2005. [9] Kim HW, Batra A, Priya S, Uchino K, Markley D, Newnham RE, et al. Energy harvesting using a piezoelectric cymbal transducer in dynamic environment. Jpn J Appl Phys 1, Regul Pap Short Notes Rev Pap 2004;43:617883. [10] Shahruz SM. Design of mechanical band-pass lters for energy scavenging. J Sound Vib 2006;292:98798. [11] Shahruz SM. Limits of performance of mechanical band-pass lters used in energy scavenging. J Sound Vib 2006;293:44961. [12] Rastegar J, Pereira C, Nguyen HL. Piezoelectric-based power sources for harvesting energy from platforms with low frequency vibration. Smart structures and materials: industrial and commercial applications of smart structures technologies, vol. 6171; 2006. p. 17101 [art. no. 617101]. [13] Marzencki M, Basrour S, Charlot B, Grasso A, Colin M, Valbin L. Design and fabrication of piezoelectric micro power generators for autonomous microsystems, DTIP05, Montreux, Switzerland; June 0103, 2005. [14] Marzencki M, Ammar Y, Basrour S. Integrated power harvesting system including a MEMS generator and a power management circuit. Sensors Actuat A Phys 2008;145:36370. [15] Shen D, Park JH, Ajitsaria J, Choe SY, Wikle HC, Kim DJ. The design, fabrication and evaluation of a MEMS PZT cantilever with an integrated Si proof mass for vibration energy harvesting. J Micromech Microeng 2008;18. [16] Renaud M, Karakaya K, Sterken T, Fiorini P, Van Hoof C, Puers R. Fabrication, modeling and characterization of MEMS piezoelectric vibration harvesters. Sensors Actuat A Phys 2008;145:3806. [17] Fang HB, Liu JQ, Xu ZY, Dong L, Wang L, Chen D, et al. Fabrication and performance of MEMS-based piezoelectric power generator for vibration energy harvesting. Microelectron J 2006;37:12804. [18] Liu JQ, Fang HB, Xu ZY, Mao XH, Shen XC, Chen D, et al. A MEMS-based piezoelectric power generator array for vibration energy harvesting. Microelectron J 2008;39:8026. [19] Jeon YB, Sood R, Jeong JH, Kim SG. MEMS power generator with transverse mode thin lm PZT. Sensors Actuat A Phys 2005;122:1622. [20] Lee BS, Wu WJ, Shih WP, Vasic D, Costa F. Power harvesting using piezoelectric MEMS generator with inter-digital electrodes. In: 2007 IEEE ultrasonics symposium proceedings, vol. 16; 2007. p. 1598601. [21] Lee BS, Lin SC, Wu WJ, Wang XY, Chang PZ, Lee CK. Piezoelectric MEMS generators fabricated with an aerosol deposition PZT thin lm. J Micromech Microeng 2009;19. [22] Muralt P, Marzencki M, Belgacem B, Calame F, Basrour S. Vibration energy harvesting with PZT micro device. Procedia Chem 2009;1:11914. [23] Elfrink R, Kamel TM, Goedbloed M, Matova S, Hohlfeld D, van Andel Y, et al. Vibration energy harvesting with aluminum nitride-based piezoelectric devices. J Micromech Microeng 2009;19.

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01/20/1200:31:54CS 61B: Lecture 2Friday, January 20, 2012Todays reading:102Sierra & Bates, Chapter 2; pp. 54-58, 154-160, 661, 669.OBJECTS AND CONSTRUCTORS=String s;/ Step 1:s = new String();/ Steps 2, 3:METHODS=Lets look at some methods t
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02/27/1202:42:42CS 61B: Lecture 3Monday, January 23, 2012Todays reading:103Sierra & Bates, pp. 71-74, 76, 85, 240-249, 273-281, 308-309.DEFINING CLASSES=An object is a repository of data. _Fields_ are variables that hold the datastored in objec
Berkeley - CS - 61b
01/23/1219:46:10CS 61B: Lecture 4Wednesday, January 25, 2012Todays reading:S&B pp. 10-14, 49-53, 75, 78-79, 86, 117, 286-287, 292, 660.PRIMITIVE TYPES=Not all variables are references to objects. Some variables are primitivetypes, which store val
Berkeley - CS - 61b
01/28/1200:16:25CS 61B: Lecture 5Friday, January 27, 2012Todays reading:105Sierra & Bates pp. 59-62, 83, 114-116, 293-300, 670.LOOPS="while" Loops-A "while" statement is like an "if" statement, but the body of the statement isexecuted repeated
Berkeley - CS - 61b
01/28/1203:31:57CS 61B: Lecture 6Monday, January 30, 2012Todays reading:106main()s Parameter-What is the array of Strings that the main() method takes as a parameter?Its a list of command-line arguments sent to your Java program. Consider thefol
Berkeley - CS - 61b
02/01/1216:52:45CS 61B: Lecture 7Wednesday, February 1, 2012Todays reading:Goodrich & Tamassia, Section 3.2.LISTS=We can store a list of ints as an array, but there are disadvantages to thisrepresentation. First, arrays have a fixed length that c
Berkeley - CS - 61b
02/02/1220:57:1208CS 61B: Lecture 8Friday, February 3, 2012Todays reading:Goodrich & Tamassia, Section 3.3.THE "public" AND "private" KEYWORDS=Thus far, weve usually declared fields and methods using the "public" keyword.However, we can also dec
Berkeley - CS - 61b
02/03/1223:04:53CS 61B: Lecture 9Monday, February 6, 2012Todays reading:109Sierra & Bates pp. 77, 235-239, 258-265, 663.THE STACK AND THE HEAP=Java stores stuff in two separate pools of memory:Parameter Passing-As in Scheme, Java passes all pa
Berkeley - CS - 61b
02/08/1219:16:33CS 61B: Lecture 10Wednesday, February 8, 2012Todays reading:110Sierra & Bates, pp. 95-109, 662.equals()=Every class has an equals() method. If you dont define one explictly, then"r1.equals(r2)" returns the same boolean value as
Berkeley - CS - 61b
02/08/1215:23:57CS 61B: Lecture 11Friday, February 10, 2012Todays reading:111All of Chapter 7, plus pp. 28-33, 250-257.The zero-parameter SList() constructor is always called by default, regardlessof the parameters passed to the TailList construc
Berkeley - CS - 61b
02/13/1213:28:22CS 61B: Lecture 12Monday, February 13, 2012Todays reading:112What are abstract classes good for?Sierra & Bates, Chapter 8.ABSTRACT CLASSES=An abstract class is a class whose sole purpose is to be extended.public abstract class
Berkeley - CS - 61b
02/15/1203:44:30CS 61B: Lecture 13Wednesday, February 15, 2012Todays reading:113Sierra & Bates, pp. 154-160, 587-591, 667-668.JAVA PACKAGES=In Java, a _package_ is a collection of classes and Java interfaces, andpossibly subpackages, that trust
Berkeley - CS - 61b
02/17/1201:20:14CS 61B: Lecture 14Friday, February 15, 2012Todays reading:114Sierra & Bates, pp. 315-338.EXCEPTIONS=When a run-time error occurs in Java, the JVM "throws an exception," printsan error message, and quits. Oddly, an exception is a
Berkeley - CS - 61b
02/23/1203:36:10CS 61B: Lecture 16Friday, February 24, 2012Todays reading:116Sierra & Bates, pp. 189, 283.EXCEPTIONS (continued)=The "finally" keyword-A finally clause can also be added to a "try."try cfw_statementX;return 1; catch (SomeExc
Berkeley - CS - 61b
CS 61B: Lecture 17Monday, February 27, 2012one, the computer is guaranteed a win, which is obtained by choosing the second of the three possible moves.The following pseudocode computes a grids score and the bestmove from that grid (which determines th
Berkeley - CS - 61b
02/27/1223:25:0118CS 61B: Lecture 18Wednesday, February 29, 2012Todays reading:Sierra & Bates, pp. 80-84.ENCAPSULATION=A _module_ is a set of methods that work together as a whole to perform sometask or set of related tasks. A module is _encapsu
Berkeley - CS - 61b
02/29/1220:39:38CS 61B: Lecture 19Friday, March 2, 2012Todays reading:119Sierra & Bates, p. 664.ENCAPSULATED LISTS (a case study in encapsulation)=Homeworks 3, 4, and 5 introduced you to three different implementations oflinked lists, each fund
Berkeley - CS - 61b
03/04/1201:54:30CS 61B: Lecture 20Monday, March 5, 2012Todays reading:120Goodrich & Tamassia, Chapter 4 (especially 4.2 and 4.3).Next, you must learn how to express this idea rigorously. Here is thecentral lesson of todays lecture, which will bea
Berkeley - CS - 61b
03/09/1221:57:3421CS 61B: Lecture 21Wednesday, March 7, 2012Theta is symmetric: if f(n) is in Theta(g(n), then g(n) is in Theta(f(n).For instance, n^3 is in Theta(3 n^3 - n^2), and 3 n^3 - n^2 is in Theta(n^3).n^3 is not in Theta(n), and n is not i
Berkeley - CS - 61b
03/09/1213:21:47CS 61B: Lecture 22Friday, March 9, 2012Todays reading:122Goodrich & Tamassia, Sections 9.1, 9.2, 9.5-9.5.1.DICTIONARIES=Suppose you have a set of two-letter words and their definitions. You want tobe able to look up the definiti
Berkeley - CS - 61b
03/11/1221:40:37CS 61B: Lecture 23Monday, March 12, 2012Todays reading:Goodrich & Tamassia, Chapter 5.DICTIONARIES (continued)=Hash Codes-Since hash codes often need to be designed specially for each new object,youre left to your own wits. Here
Berkeley - CS - 61b
03/14/1205:35:44CS 61B: Lecture 24Wednesday, March 14, 2012Todays reading:124Goodrich & Tamassia, Chapter 7.ROOTED TREES=A _tree_ consists of a set of nodes and a set of edges that connect pairs ofnodes. A tree has the property that there is ex
Berkeley - CS - 61b
03/16/1204:54:15CS 61B: Lecture 25Friday, March 16, 2012Todays reading:125A _binary_heap_ is a complete binary tree whose entries satisfy the_heap-order_property_: no child has a key less than its parents key.Observe that every subtree of a binar
Berkeley - CS - 61b
03/21/1209:24:10126CS 61B: Lecture 26Monday, March 19, 2012Todays reading:Goodrich & Tamassia, Section 10.1.Representing Binary Trees-Recall that a binary tree is a rooted tree wherein no node has more thantwo children. Additionally, every child