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'''鋰離子電池等效電路模型'''，常簡稱'''等效電路模型'''（equivalent circuit model）或'''ECM'''，是[[锂离子电池]]的[[集總電路]]模型<ref name=":4">{{Cite journal |last1=Hu |first1=Xiaosong |last2=Li |first2=Shengbo |last3=Peng |first3=Huei |date=January 2012 |title=A comparative study of equivalent circuit models for Li-ion batteries |url=https://doi.org/10.1016/j.jpowsour.2011.10.013 |journal=Journal of Power Sources |volume=198 |pages=359–367 |doi=10.1016/j.jpowsour.2011.10.013 |issn=0378-7753}}</ref><ref name=":0">{{Cite journal |last1=Lin |first1=Xinfan |last2=Kim |first2=Youngki |last3=Mohan |first3=Shankar |last4=Siegel |first4=Jason B. |last5=Stefanopoulou |first5=Anna G. |date=2019-05-03 |title=Modeling and Estimation for Advanced Battery Management |url=https://www.annualreviews.org/doi/10.1146/annurev-control-053018-023643 |journal=Annual Review of Control, Robotics, and Autonomous Systems |language=en |volume=2 |issue=1 |pages=393–426 |doi=10.1146/annurev-control-053018-023643 |issn=2573-5144 |access-date=2025-01-14 |archive-date=2022-02-25 |archive-url=https://web.archive.org/web/20220225044855/https://www.annualreviews.org/doi/10.1146/annurev-control-053018-023643 |dead-url=no }}</ref><ref name=":5">{{Cite journal |last1=Liaw |first1=Bor Yann |last2=Nagasubramanian |first2=Ganesan |last3=Jungst |first3=Rudolph G. |last4=Doughty |first4=Daniel H. |date=2004-11-30 |title=Modeling of lithium ion cells—A simple equivalent-circuit model approach |url=https://www.sciencedirect.com/science/article/pii/S0167273804006678 |journal=Solid State Ionics |series=Fourteenth International Conference on Solid State Ionics |volume=175 |issue=1 |pages=835–839 |doi=10.1016/j.ssi.2004.09.049 |issn=0167-2738}}</ref>。等效電路模型用電阻及电容等被動元件以及电压源組成的等效電路，來模擬鋰離子電池的端電壓特性。等效電路模型可以用在許多的領域，因為結構簡單、計算需求低、容易表現其電池特性，以及在結構上的靈活度，可以用在[[计算机模拟]]上<ref name=":0" /><ref name=":1">{{Cite journal |last1=Zhang |first1=Lijun |last2=Peng |first2=Hui |last3=Ning |first3=Zhansheng |last4=Mu |first4=Zhongqiang |last5=Sun |first5=Changyan |date=October 2017 |title=Comparative Research on RC Equivalent Circuit Models for Lithium-Ion Batteries of Electric Vehicles |journal=Applied Sciences |language=en |volume=7 |issue=10 |pages=1002 |doi=10.3390/app7101002 |doi-access=free |issn=2076-3417}}</ref><ref>{{Cite journal |last1=Nejad |first1=S. |last2=Gladwin |first2=D. T. |last3=Stone |first3=D. A. |date=2016-06-01 |title=A systematic review of lumped-parameter equivalent circuit models for real-time estimation of lithium-ion battery states |url=https://www.sciencedirect.com/science/article/pii/S0378775316302427 |journal=Journal of Power Sources |volume=316 |pages=183–196 |doi=10.1016/j.jpowsour.2016.03.042 |bibcode=2016JPS...316..183N |issn=0378-7753}}</ref><ref>{{Cite journal |last1=Tekin |first1=Merve |last2=Karamangil |first2=M. İhsan |date=2024-05-10 |title=Comparative analysis of equivalent circuit battery models for electric vehicle battery management systems |url=https://www.sciencedirect.com/science/article/pii/S2352152X24009125 |journal=Journal of Energy Storage |volume=86 |pages=111327 |doi=10.1016/j.est.2024.111327 |bibcode=2024JEnSt..8611327T |issn=2352-152X}}</ref>。等效電路模型的特點也適合用在實時的[[电池管理系统]]（BMS）上，用來檢測[[電量狀態]]（SoC）<ref>{{Cite journal |last1=Xiong |first1=Rui |last2=Cao |first2=Jiayi |last3=Yu |first3=Quanqing |last4=He |first4=Hongwen |last5=Sun |first5=Fengchun |date=2018 |title=Critical Review on the Battery State of Charge Estimation Methods for Electric Vehicles |url=https://ieeexplore.ieee.org/document/8168251 |journal=IEEE Access |volume=6 |pages=1832–1843 |doi=10.1109/ACCESS.2017.2780258 |bibcode=2018IEEEA...6.1832X |issn=2169-3536 |doi-access=free |access-date=2025-01-14 |archive-date=2024-08-12 |archive-url=https://web.archive.org/web/20240812033333/https://ieeexplore.ieee.org/document/8168251/ |dead-url=no }}</ref>、[[電池健康狀態]]（SoH）等資訊<ref>{{Cite journal |last1=Berecibar |first1=M. |last2=Gandiaga |first2=I. |last3=Villarreal |first3=I. |last4=Omar |first4=N. |last5=Van Mierlo |first5=J. |last6=Van den Bossche |first6=P. |date=2016-04-01 |title=Critical review of state of health estimation methods of Li-ion batteries for real applications |url=https://www.sciencedirect.com/science/article/pii/S1364032115013076 |journal=Renewable and Sustainable Energy Reviews |volume=56 |pages=572–587 |doi=10.1016/j.rser.2015.11.042 |bibcode=2016RSERv..56..572B |issn=1364-0321}}</ref>，也可以用在電池熱管理上<ref>{{Cite journal |last1=Liu |first1=Jie |last2=Yadav |first2=Saurabh |last3=Salman |first3=Mohammad |last4=Chavan |first4=Santosh |last5=Kim |first5=Sung Chul |date=2024-01-01 |title=Review of thermal coupled battery models and parameter identification for lithium-ion battery heat generation in EV battery thermal management system |url=https://www.sciencedirect.com/science/article/pii/S0017931023008931 |journal=International Journal of Heat and Mass Transfer |volume=218 |pages=124748 |doi=10.1016/j.ijheatmasstransfer.2023.124748 |bibcode=2024IJHMT.21824748L |issn=0017-9310 |access-date=2025-01-14 |archive-date=2024-07-09 |archive-url=https://web.archive.org/web/20240709140910/https://www.sciencedirect.com/science/article/pii/S0017931023008931 |dead-url=no }}</ref>。
[[File:First-order equivalent circuit model for Li-ion cell.png|thumb|鋰離子電池的一階等效電路模型|294x294px]]

== 模型結構 ==
等效電路模型是用來模擬電池充電或是放電時，電池上對應的電壓。最常見的電路是由三個電件串聯組成：表示電池[[開路電壓]]（OCV）的可變電壓源、表示電池[[欧姆定律|欧姆]][[内阻]]的電阻、表示電池動態[[電壓降]]的[[RC電路]]<ref name=":4" /><ref name=":0" /><ref name=":5" />。

=== 開路電壓 ===
[[File:Open circuit voltage of li-ion cells with different chemical composition.png|thumb|鋰離子電池不同陽極材料下的開路電壓<ref>{{Cite journal |last1=Al-Shammari |first1=Hammad |last2=Farhad |first2=Siamak |date=January 2022 |title=Performance of Cathodes Fabricated from Mixture of Active Materials Obtained from Recycled Lithium-Ion Batteries |journal=Energies |language=en |volume=15 |issue=2 |pages=410 |doi=10.3390/en15020410 |doi-access=free |issn=1996-1073}}</ref>|292x292px]]
鋰離子電池的開路電壓是指在平衡條件（沒有負載電流，靜置很長一段時間之後）下量測的電壓。開路電壓是其電量狀態的遞減[[非線性系統|非線性函數]]，和電池[[陽極]]（多半是[[石墨]]）和[[陰極]]（[[磷酸铁锂电池|LFP]]、[[镍钴锰酸锂|NMC]]、[[鎳鈷鋁酸鋰|NCA]]、[[氧化鈷鋰|LCO]]等）有關<ref name=":7">{{Cite journal |last1=Somakettarin |first1=Natthawuth |last2=Funaki |first2=Tsuyoshi |date=March 2017 |title=Study on Factors for Accurate Open Circuit Voltage Characterizations in Mn-Type Li-Ion Batteries |journal=Batteries |language=en |volume=3 |issue=1 |pages=8 |doi=10.3390/batteries3010008 |doi-access=free |issn=2313-0105}}</ref>。開路電壓在等效電路中是由電荷產生電壓的元件，是電路中電壓的主要提供者，也是電池電量狀態最有效的指標<ref>{{Cite journal |last1=Zhang |first1=Ruifeng |last2=Xia |first2=Bizhong |last3=Li |first3=Baohua |last4=Cao |first4=Libo |last5=Lai |first5=Yongzhi |last6=Zheng |first6=Weiwei |last7=Wang |first7=Huawen |last8=Wang |first8=Wei |last9=Wang |first9=Mingwang |date=September 2018 |title=A Study on the Open Circuit Voltage and State of Charge Characterization of High Capacity Lithium-Ion Battery Under Different Temperature |journal=Energies |language=en |volume=11 |issue=9 |pages=2408 |doi=10.3390/en11092408 |doi-access=free |issn=1996-1073}}</ref><ref>{{Cite journal |last1=Yang |first1=Jie |last2=Du |first2=Chunyu |last3=Wang |first3=Ting |last4=Gao |first4=Yunzhi |last5=Cheng |first5=Xinqun |last6=Zuo |first6=Pengjian |last7=Ma |first7=Yulin |last8=Wang |first8=Jiajun |last9=Yin |first9=Geping |last10=Xie |first10=Jingying |last11=Lei |first11=Bo |date=December 2018 |title=Rapid Prediction of the Open-Circuit-Voltage of Lithium Ion Batteries Based on an Effective Voltage Relaxation Model |journal=Energies |language=en |volume=11 |issue=12 |pages=3444 |doi=10.3390/en11123444 |doi-access=free |issn=1996-1073}}</ref>。

===內阻 ===
內阻在電路中是以一個電阻來表示，用來表示電池因為歐姆效應產生的電壓降，其中包括[[电极]]的電阻率<ref name=":1" /><ref name=":3" />、[[电解质]]的電導率<ref name=":1" /><ref name=":3" /><ref name=":2">{{Cite journal |last1=Dong |first1=T. K. |last2=Kirchev |first2=A. |last3=Mattera |first3=F. |last4=Kowal |first4=J. |last5=Bultel |first5=Y. |date=2011 |title=Dynamic Modeling of Li-Ion Batteries Using an Equivalent Electrical Circuit |url=https://iopscience.iop.org/article/10.1149/1.3543710 |journal=Journal of the Electrochemical Society |language=en |volume=158 |issue=3 |pages=A326 |doi=10.1149/1.3543710}}</ref>，以及{{link-en|接觸電阻|contact resistance}}<ref name=":3" /><ref name=":2" />（例如固態電解質介面 solid-electrolyte interface以及集電體的接觸電阻）。

內阻會受到許多因素所影響，例如：
* 溫度：在低溫時內阻會明顯上昇<ref>{{Cite journal |last1=Hossain Ahmed |first1=Sazzad |last2=Kang |first2=Xiaosong |last3=Bade Shrestha |first3=S. O. |date=2015-05-01 |title=Effects of Temperature on Internal Resistances of Lithium-Ion Batteries |url=https://asmedigitalcollection.asme.org/energyresources/article/doi/10.1115/1.4028698/373140/Effects-of-Temperature-on-Internal-Resistances-of |journal=Journal of Energy Resources Technology |language=en |volume=137 |issue=3 |doi=10.1115/1.4028698 |issn=0195-0738 |access-date=2025-01-15 |archive-date=2023-11-21 |archive-url=https://web.archive.org/web/20231121223309/https://asmedigitalcollection.asme.org/energyresources/article/doi/10.1115/1.4028698/373140/Effects-of-Temperature-on-Internal-Resistances-of |dead-url=no }}</ref><ref name=":3">{{Cite journal |last1=Barcellona |first1=Simone |last2=Colnago |first2=Silvia |last3=Dotelli |first3=Giovanni |last4=Latorrata |first4=Saverio |last5=Piegari |first5=Luigi |date=June 2022 |title=Aging effect on the variation of Li-ion battery resistance as function of temperature and state of charge |url=https://doi.org/10.1016/j.est.2022.104658 |journal=Journal of Energy Storage |volume=50 |pages=104658 |doi=10.1016/j.est.2022.104658 |bibcode=2022JEnSt..5004658B |issn=2352-152X}}</ref>，這也是鋰電池在低溫時效能較差的原因<ref>{{Cite news |last1=Schmall |first1=Emily |last2=Gross |first2=Jenny |date=2024-01-17 |title=Electric Car Owners Confront a Harsh Foe: Cold Weather |url=https://www.nytimes.com/2024/01/17/business/tesla-charging-chicago-cold-weather.html |access-date=2024-07-10 |work=The New York Times |language=en-US |issn=0362-4331 |archive-date=2025-01-16 |archive-url=https://web.archive.org/web/20250116212728/https://www.nytimes.com/2024/01/17/business/tesla-charging-chicago-cold-weather.html |dead-url=no }}</ref><ref>{{Cite web |date=2024-01-18 |title=Why Teslas and other electric vehicles have problems in cold weather — and how EV owners can prevent issues - CBS News |url=https://www.cbsnews.com/news/teslas-electric-vehicles-cold-weather/ |access-date=2024-07-10 |website=www.cbsnews.com |language=en-US |archive-date=2025-01-15 |archive-url=https://web.archive.org/web/20250115022606/https://www.cbsnews.com/news/teslas-electric-vehicles-cold-weather/ |dead-url=no }}</ref>。
* 電量狀態：內阻會受到電量狀態的影響，造成其值顯著變化<ref name=":6">{{Cite journal |last1=Kim |first1=Daehyun |last2=Koo |first2=Keunhwi |last3=Jeong |first3=Jae Jin |last4=Goh |first4=Taedong |last5=Kim |first5=Sang Woo |date=October 2013 |title=Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model |journal=Energies |language=en |volume=6 |issue=10 |pages=5538–5551 |doi=10.3390/en6105538 |doi-access=free |issn=1996-1073}}</ref>。特別是在低電量狀態（接近完全放電）以及滿電量狀態（接近完全充電）時，內阻會增加<ref name=":6" />。
* 電池老化：隨著鋰電池老化，其內阻會漸漸增加<ref name=":3" />。電阻增加的主因是因為固態電解質介面（solid-electrolyte interface，簡稱SEI）的產生，這是在陰極表面自然生成，有保護作用的固態物質，其成份是電解液成份衍生的化合物<ref>{{Cite journal |last1=Han |first1=Xuebing |last2=Lu |first2=Languang |last3=Zheng |first3=Yuejiu |last4=Feng |first4=Xuning |last5=Li |first5=Zhe |last6=Li |first6=Jianqiu |last7=Ouyang |first7=Minggao |date=August 2018 |title=A review on the key issues of the lithium ion battery degradation among the whole life cycle |url=https://doi.org/10.1016/j.etran.2019.100005 |journal=ETransportation |volume=1 |pages=100005 |doi=10.1016/j.etran.2019.100005 |issn=2590-1168}}</ref><ref>{{Cite journal |last1=Vetter |first1=J. |last2=Novák |first2=P. |last3=Wagner |first3=M.R. |last4=Veit |first4=C. |last5=Möller |first5=K.-C. |last6=Besenhard |first6=J.O. |last7=Winter |first7=M. |last8=Wohlfahrt-Mehrens |first8=M. |last9=Vogler |first9=C. |last10=Hammouche |first10=A. |date=September 2005 |title=Ageing mechanisms in lithium-ion batteries |url=https://doi.org/10.1016/j.jpowsour.2005.01.006 |journal=Journal of Power Sources |volume=147 |issue=1–2 |pages=269–281 |doi=10.1016/j.jpowsour.2005.01.006 |bibcode=2005JPS...147..269V |issn=0378-7753}}</ref>。

===RC並聯電路===
在模型中會加入一個或是多個RC並聯電路，模擬其動態的電壓變化。並聯電路的數量是建模時可以決定的：一般而言，RC電路越多，模型會越精準，但其識別流程的複雜度會增加，而且會增加運算上的負擔，而RC電路少，在計算上的負擔較少，較容易找到特微，但其動態電壓估測會較不精準。一般來說，最佳作法是使用一個或二個RC並聯電路<ref name=":4" />。
== 模型方程式 ==
等效電路模型可以用[[状态空间]]表示，以電流（<math display="inline">i</math>）為輸入，電池電壓（<math display="inline">V</math>）為輸出。考慮有數個RC並聯電路的通用等效電路模型<math display="inline">N</math>。模型狀態（[[微分方程]]中隨時間變化的變數）是[[電量狀態]]（<math display="inline">SoC</math>）以及各RC並聯電路上的電壓（<math display="inline">V_{c,1}, V_{c,2} \dots  V_{c,N}</math>）<ref name=":0" />。
[[File:Equivalent circuit model of generic order with SoC computation for Li-ion cell.png|thumb|鋰離子電池任意階數的等效電路模型。左側：透過庫侖計數積分公式的電量狀態電路表示法。右側：電池電壓模擬|550x550px]]
電量狀態一般會用對電池充電電流以及電池放電電流[[積分]]而得，此方法稱為庫侖計數（Coulomb Counting）<ref>{{Cite journal |last1=Zhang |first1=Shuzhi |last2=Guo |first2=Xu |last3=Dou |first3=Xiaoxin |last4=Zhang |first4=Xiongwen |date=August 2020 |title=A data-driven coulomb counting method for state of charge calibration and estimation of lithium-ion battery |url=https://doi.org/10.1016/j.seta.2020.100752 |journal=Sustainable Energy Technologies and Assessments |volume=40 |pages=100752 |doi=10.1016/j.seta.2020.100752 |bibcode=2020SETA...4000752Z |issn=2213-1388}}</ref>：

<math>SoC(t)= SoC(t_0) + \int_{t_0}^t\dfrac{1}{3600Q}i(t) dt</math>

其中<math display="inline">Q</math>是電池額定容量（以[[安培小時]]表示）。每一個RC並聯電路上的電壓可以用以下方式模擬<ref name=":0" />：

<math>\dfrac{dV_{c,i}}{dt}(t)=-\dfrac{1}{R_iC_i}V_{c,i}(t) + \dfrac{1}{C_i}i(t)</math>

其中<math display="inline">R_i</math>和<math display="inline">C_i</math>是極化電阻和電容。在知道開路電壓和電量狀態關係<math>V_{OC}(SoC)</math>，以及電池內阻<math>R_0</math>後，可以用以下方式計算電池端電壓<ref name=":0" />：

<math>V(t) = V_{OC}(SoC(t)) + R_0i(t) + \sum_{i=1}^NV_{c,i}(t)</math>
== 應用 ==
以下是一些可以用到等效電池模型的應用：
* [[電池管理系統]]的線上狀態估測：等效電池模型常用在預測電池內不可量測狀態（例如電量狀態、[[電池健康狀態]]），以模型為基礎的估測器。例如可以使用各種階數的等效電池模型，配合{{le|扩展卡尔曼滤波器|Extended Kalman filter}}（EKF）來進行電量狀態的線上估測<ref>{{Cite journal |last=Sepasi |first=Saeed |last2=Ghorbani |first2=Reza |last3=Liaw |first3=Bor Yann |date=2014-01-01 |title=A novel on-board state-of-charge estimation method for aged Li-ion batteries based on model adaptive extended Kalman filter |url=https://www.sciencedirect.com/science/article/pii/S0378775313011154 |journal=Journal of Power Sources |volume=245 |pages=337–344 |doi=10.1016/j.jpowsour.2013.06.108 |issn=0378-7753}}</ref>。
* 模擬以及系統設計：等效電池模型常用在[[電池組]]的設計階段<ref>{{Cite journal |last=Pham |first=Cong-Toan |last2=Månsson |first2=Daniel |date=2018-08-01 |title=Optimal energy storage sizing using equivalent circuit modelling for prosumer applications (Part II) |url=https://www.sciencedirect.com/science/article/pii/S2352152X17305868 |journal=Journal of Energy Storage |volume=18 |pages=1–15 |doi=10.1016/j.est.2018.04.015 |issn=2352-152X}}</ref>。模擬電池芯的電子負載分佈可以依容量和電壓來定義系統尺寸。而且等效電池模型也可以用來模擬電池產生的熱，以此設計電池冷卻系統<ref>{{Cite journal |last=Hou |first=Guiqi |last2=Liu |first2=Xianqing |last3=He |first3=Wenxuan |last4=Wang |first4=Changhong |last5=Zhang |first5=Jiangyun |last6=Zeng |first6=Xiaoxing |last7=Li |first7=Zhuoming |last8=Shao |first8=Dan |date=2022-11-30 |title=An equivalent circuit model for battery thermal management system using phase change material and liquid cooling coupling |url=https://www.sciencedirect.com/science/article/pii/S2352152X22018229 |journal=Journal of Energy Storage |volume=55 |pages=105834 |doi=10.1016/j.est.2022.105834 |issn=2352-152X}}</ref>。

== 實驗識別的簡介 ==
等效電路模型的實驗識別是另外對電池進行實驗，來識別未知的參數，特別是電池容量<math display="inline">Q</math>，開路電壓曲線<math>V_{OC}(SoC)</math>、被動元件<math>R_0</math>、<math display="inline">R_i</math>和<math display="inline">C_i</math>。一般來說，識別會有幾個不同的步驟<ref name=":8">{{Cite journal |last1=Wang |first1=Jianfeng |last2=Jia |first2=Yongkai |last3=Yang |first3=Na |last4=Lu |first4=Yanbing |last5=Shi |first5=Mengyu |last6=Ren |first6=Xutong |last7=Lu |first7=Dongchen |date=2022-08-25 |title=Precise equivalent circuit model for Li-ion battery by experimental improvement and parameter optimization |url=https://www.sciencedirect.com/science/article/pii/S2352152X22009859 |journal=Journal of Energy Storage |volume=52 |pages=104980 |doi=10.1016/j.est.2022.104980 |bibcode=2022JEnSt..5204980W |issn=2352-152X}}</ref>。

=== 電池容量評估 ===
電池容量<math display="inline">Q</math>多半會透過定電流的完全放電來偵測<ref name=":12">{{Cite web |title=Battery capacity test for Lithium-ion battery – Neware battery testers |url=https://newarebattery.com/battery-capacity-test-for-lithium-ion-battery/ |access-date=2024-07-12 |language=en-US}}</ref>。電池容量測試一般會從電池電壓上限<math>V_{max}</math>放到電池電壓下限<math>V_{min}</math>，以[[放電速率]]0.5C或1C的電流量進行放電（0.5C／1C是指在額定電量下，二小時/一小時可以從電池滿電將電完全放完），再將其完全充電（會依照定電流-定電壓的策略）<ref name=":12" />。電池容量可以用下式計算：<math display="inline">Q=  \int_{t\mid_{V(t)=V_{max}}}^{t\mid_{{V(t)=V_{min}}}}\dfrac{1}{3600}i(t) dt</math>。

=== 開路電壓特徵化 ===
要找到開路電壓的特徵，主要會用以下兩種實驗方式：

#脈衝測試<ref name=":7" />：用連續電流脈衝，將電池完全放電/充電。每一次脈衝放電時會放掉電池一部份的電量，因此讓電池到新的<math display="inline">SoC</math>點。在每次電流脈衝之後，電池會靜置數小時不充電也不放電，以便量測其開路電壓。最後會將記錄的[<math display="inline">SoC</math>, <math>V_{OC}</math>]資料，用任意選定的函數（一般是多項式）進行曲線擬合，以找到<math>V_{OC}=f(SoC)</math>曲線。一般認為此方法快而且有效，但其結果準確性取決於實驗設計，以及要進行多久的實驗<ref name=":7" />。
# 慢恆電流放電（Slow galvanostatic discharge）<ref name=":7" />：另一種評估電池開路電壓的方式，就是小的恆電流條件下充電或是放電。在小電流下，以下的近似成立：<math display="inline">V = V_{OC}(SoC) + R_0i + \sum_{i=1}^NV_{c,i} \; \underset{i \rightarrow0}{\simeq}\; V_{OC}(SoC)</math>。在此作法中，估測的精準度取決於所充電/放電電流有多小，而估測結果的好壞也取決於要花多久進行實驗<ref name=":7" />。
[[File:Pulse discharge test performed on a li-ion battery.svg|thumb|3.2Ah磷酸鋰鐵電池脈衝放電測試的實驗結果。從上到下分別是：脈衝電流波形；所得電壓；電壓響應的一些細節（開路電壓、歐姆壓降、RC暫態等）|512x512px]]
=== 動態響應特徵化 ===
動態響應的參數，像是歐姆電阻<math>R_0</math>、RC電路相關參數<math display="inline">R_i</math>,<math display="inline">C_i</math>多半可以用以下兩種方式來識別：

# [[時域]]識別<ref name=":8" /><ref name=":9">{{Cite journal |last1=Madani |first1=Seyed Saeed |last2=Schaltz |first2=Erik |last3=Kær |first3=Søren Knudsen |date=2018-11-26 |title=A Review of Different Electric Equivalent Circuit Models and Parameter Identification Methods of Lithium-Ion Batteries |url=https://iopscience.iop.org/article/10.1149/08701.0023ecst |journal=ECS Transactions |language=en |volume=87 |issue=1 |pages=23–37 |doi=10.1149/08701.0023ecst |bibcode=2018ECSTr..87a..23S |issn=1938-6737 |access-date=2025-01-17 |archive-date=2024-07-11 |archive-url=https://web.archive.org/web/20240711142002/https://iopscience.iop.org/article/10.1149/08701.0023ecst |dead-url=no }}</ref>：提供指定的電流波形，分析電池電壓的波形，以此找到最合適的參數。例如，脈波測試可以提供以下的功能：在施加或移取電流的瞬間，可以量測其電壓變化，求得不同電量狀態下的<math>R_0</math>，而<math display="inline">R_i</math> 和<math display="inline">C_i</math>可以用電池電壓動態變化的資料進行[[最佳化]]來求得<ref name=":8" /><ref name=":9" /> 
# [[頻域]]識別<ref name=":10">{{Cite journal |last1=Zhao |first1=Zhaoyang |last2=Zou |first2=Yang |last3=Liu |first3=Peng |last4=Lai |first4=Zhaogui |last5=Wen |first5=Lei |last6=Jin |first6=Ying |date=June 2022 |title=EIS equivalent circuit model prediction using interpretable machine learning and parameter identification using global optimization algorithms |url=https://doi.org/10.1016/j.electacta.2022.140350 |journal=Electrochimica Acta |volume=418 |pages=140350 |doi=10.1016/j.electacta.2022.140350 |issn=0013-4686}}</ref><ref name=":11">{{Cite web |last=Shepard |first=Jeff |date=2023-01-04 |title=What does electrochemical impedance spectroscopy have to do with Li-ion health? |url=https://www.batterypowertips.com/what-does-electrochemical-impedance-spectroscopy-have-to-do-with-li-ion-health-faq/ |access-date=2024-07-11 |website=Battery Power Tips |language=en-US}}</ref>：分析電池的[[频率响应]]來找到參數。為此會在電池中注入不同[[頻率 (物理學)|頻率]]的[[交流電|交流]]電壓（或電流）信號，再計算所得電流（或電壓）的[[振幅]]和[[相位]]。此分析稱為{{le|電化學交流阻抗法|Electrochemical Impedance Spectroscopy}}（EIS），需要專門的實驗室儀器，可以產生高可靠度的結果。EIS結果一般會用奈奎斯特圖來評估，可以識別出電池不同的阻抗成份（<math>R_0</math>、<math display="inline">R_i</math>和<math display="inline">C_i</math>）<ref name=":10" /><ref name=":11" />。

== 相關條目 ==
* [[内阻]]

== 參考資料 ==
{{Reflist|2}}

== 外部連結 ==
* [https://www.powerelectronicsnews.com/modeling-li-ion-batteries-with-equivalent-circuit-technology/#:~:text=Equivalent%20circuit%20modeling%20provides%20a,characteristics%20of%20the%20battery's%20behavior Li-ion battery modeling through equivalent circuit models]
* [https://www.powersystemsdesign.com/articles/equivalent-circuit-modelling-for-li-ion-batteries/22/19993 Equivalent circuit models for Li-ion cells]
* [https://it.mathworks.com/help/simscape-battery/ref/batteryequivalentcircuit.html Matlab tool for equivalent circuit models development]
* [https://www.batterypowertips.com/what-does-electrochemical-impedance-spectroscopy-have-to-do-with-li-ion-health-faq/ Introduction to EIS methodology]

[[Category:数学模型]]
[[Category:鋰離子電池]]