Carbon nanocages bridged with graphene enable fast kinetics for dual-carbon lithium-ion capacitors

Lithium-ion capacitors(LICs) combining the advantages of lithium-ion batteries and supercapacitors are considered a promising nextgeneration energy storage device. However, the sluggish kinetics of battery-type anode cannot match the capacitor-type cathode, restricting the development of LICs. Herein, hierarchical carbon framework(HCF) anode material composed of 0D carbon nanocage bridged with 2D graphene network are developed via a template-confined synthesis process. The HCF with nanocage structure reduces the Li^(+) transport path and benefits the rapid Li^(+) migration, while 2D graphene network can promote the electron interconnecting of carbon nanocages. In addition, the doped N atoms in HCF facilitate to the adsorption of ions and enhance the pseudo contribution, thus accelerate the kinetics of the anode. The HCF anode delivers high specific capacity, remarkable rate capability. The LIC pouch-cell based on HCF anode and active HCF(a-HCF) cathode can provide a high energy density of 162 Wh kg^(-1) and a superior power density of 15.8 kW kg^(-1), as well as a long cycling life exceeding 15,000cycles. This study demonstrates that the well-defined design of hierarchical carbon framework by incorporating 0D carbon nanocages and 2D graphene network is an effective strategy to promote LIC anode kinetics and hence boost the LIC electrochemical performance.

Coupling of Adhesion and Anti‑Freezing Properties in Hydrogel Electrolytes for Low‑Temperature Aqueous‑Based Hybrid Capacitors

Solid-state zinc-ion capacitors are emerging as promising candidates for large-scale energy storage owing to improved safety,mechanical and thermal stability and easy-to-direct stacking.Hydrogel electrolytes are appealing solid-state electrolytes because of eco-friendliness,high conductivity and intrinsic flexibility.However,the electrolyte/electrode interfacial contact and anti-freezing properties of current hydrogel electrolytes are still challenging for practical applications of zinc-ion capacitors.Here,we report a class of hydrogel electrolytes that couple high interfacial adhesion and anti-freezing performance.The synergy of tough hydrogel matrix and chemical anchorage enables a well-adhered interface between hydrogel electrolyte and electrode.Meanwhile,the cooperative solvation of ZnCl2 and LiCl hybrid salts renders the hydrogel electrolyte high ionic conductivity and mechanical elasticity simultaneously at low temperatures.More significantly,the Zn||carbon nanotubes hybrid capacitor based on this hydrogel electrolyte exhibits low-temperature capacitive performance,delivering high-energy density of 39 Wh kg^(-1)at-60°C with capacity retention of 98.7%over 10,000 cycles.With the benefits of the well-adhered electrolyte/electrode interface and the anti-freezing hydrogel electrolyte,the Zn/Li hybrid capacitor is able to accommodate dynamic deformations and function well under 1000 tension cycles even at-60°C.This work provides a powerful strategy for enabling stable operation of low-temperature zinc-ion capacitors.

Defect engineering on BiFeO_(3) through Na and V codoping for aqueous Na-ion capacitors

Sodium with low cost and high abundance is considered as a substitute element of lithium for batteries and supercapacitors,which need the appropriate host materials to accommodate the relatively large Na^(+) ions.Compared to Li^(+) storage,Na^(+) storage makes higher demands on the structural optimization of perovskite bismuth ferrite(BiFeO_(3)).We propose a novel strategy of defect engineering on BiFeO_(3) through Na and V codoping for high-efficiency Na^(+) storage,to reveal the roles of oxygen vacancies and V ions played in the enhanced electrochemical energy storage performances of Na-ion capacitors.The formation of the oxygen vacancies in the Na and V codoped BiFeO_(3)(denoted as NV-BFO),is promoted by Na doping and suppressed by V doping,which can be demonstrated by XPS and EPR spectra.By the first-principles calculations,the oxygen vacancies and V ions in NV-BFO are confirmed to substantially lower the Na^(+)migration energy barriers through the space and electric field effects,to effectively promote the Na^(+) transport in the crystals.Electrochemical kinetic analysis of the NV-BFO//NV-BFO capacitors indicates the dominant capacitive-controlled capacity,which depends on fast Na^(+) deintercalation-intercalation process in the NV-BFO electrode.The NV-BFO//NV-BFO capacitors open up a new avenue for developing highperformance Na-ion capacitors.

MOF-derived porous graphitic carbon with optimized plateau capacity and rate capability for high performance lithium-ion capacitors

The development of anode materials with high rate capability and long charge-discharge plateau is the key to improve per-formance of lithium-ion capacitors(LICs).Herein,the porous graphitic carbon(PGC-1300)derived from a new triply interpenetrated co-balt metal-organic framework(Co-MOF)was prepared through the facile and robust carbonization at 1300°C and washing by HCl solu-tion.The as-prepared PGC-1300 featured an optimized graphitization degree and porous framework,which not only contributes to high plateau capacity(105.0 mAh·g^(−1)below 0.2 V at 0.05 A·g^(−1)),but also supplies more convenient pathways for ions and increases the rate capability(128.5 mAh·g^(−1)at 3.2 A·g^(−1)).According to the kinetics analyses,it can be found that diffusion regulated surface induced capa-citive process and Li-ions intercalation process are coexisted for lithium-ion storage.Additionally,LIC PGC-1300//AC constructed with pre-lithiated PGC-1300 anode and activated carbon(AC)cathode exhibited an increased energy density of 102.8 Wh·kg^(−1),a power dens-ity of 6017.1 W·kg^(−1),together with the excellent cyclic stability(91.6%retention after 10000 cycles at 1.0 A·g^(−1)).

Vacancy defect MoSeTe embedded in N and F co-doped carbon skeleton for high performance sodium ion batteries and hybrid capacitors

Sodium-ion batteries(SIBs) and hybrid capacitors(SIHCs) have garnered significant attention in energy storage due to their inherent advantages,including high energy density,cost-effectiveness,and enhanced safety.However,developing high-performance anode materials to improve sodium storage performa nce still remains a major challenge.Here,a facile one-pot method has been developed to fabricate a hybrid of MoSeTe nanosheets implanted within the N,F co-doped honeycomb carbon skeleton(MoSeTe/N,F@C).Experimental results demonstrate that the incorporation of large-sized Te atoms into MoSeTe nanosheets enlarges the layer spacing and creates abundant anion vacancies,which effectively facilitate the insertion/extraction of Na^(+) and provide numerous ion adsorption sites for rapid surface capacitive behavior.Additionally,the heteroatoms N,F co-doped honeycomb carbon skeleton with a highly conductive network can restrain the volume expansion and boost reaction kinetics within the electrode.As anticipated,the MoSeTe/N,F@C anode exhibits high reversible capacities along with exceptional cycle stability.When coupled with Na_(3)V_(2)(PO_(4))_(3)@C(NVPF@C) to form SIB full cells,the anode delivers a reversible specific capacity of 126 mA h g^(-1) after 100 cycles at 0.1 A g^(-1).Furthermore,when combined with AC to form SIHC full cells,the anode demonstrates excellent cycling stability with a reversible specific capacity of50 mA h g^(-1) keeping over 3700 cycles at 1.0 A g^(-1).In situ XRD,ex situ TEM characterization,and theoretical calculations(DFT) further confirm the reversibility of sodium storage in MoSeTe/N,F@C anode materials during electrochemical reactions,highlighting their potential for widespread practical application.This work provides new insights into the promising utilization of advanced transition metal dichalcogenides as anode materials for Na^(+)-based energy storage devices.

Status and Opportunities of Zinc Ion Hybrid Capacitors: Focus on Carbon Materials, Current Collectors, and Separators

Zinc ion hybrid capacitors(ZIHCs), which integrate the features of the high power of supercapacitors and the high energy of zinc ion batteries, are promising competitors in future electrochemical energy storage applications. Carbon-based materials are deemed the competitive candidates for cathodes of ZIHC due to their cost-effectiveness, high electronic conductivity, chemical inertness, controllable surface states, and tunable pore architectures. In recent years, great research efforts have been devoted to further improving the energy density and cycling stability of ZIHCs. Reasonable modification and optimization of carbon-based materials offer a remedy for these challenges. In this review, the structural design, and electrochemical properties of carbon-based cathode materials with different dimensions, as well as the selection of compatible, robust current collectors and separators for ZIHCs are discussed. The challenges and prospects of ZIHCs are showcased to guide the innovative development of carbon-based cathode materials and the development of novel ZIHCs.

Boosting High-Voltage Dynamics Towards High-Energy-Density Lithium-Ion Capacitors

Lithium-ion capacitors(LICs)are becoming important electrochemical energy storage systems due to their great potential to bridge the gap between supercapacitors and lithium-ion batteries.However,capacity lopsidedness and low output voltage greatly hinder the realization of high-energy-density LICs.Herein,a strategy of balancing capacity towards fastest dynamics is proposed to enable high-voltage LICs.Through electrochemical prelithiation of Nb_(2)C to be 1.1 V with 165 mAh g^(-1),Nb_(2)C//LiFePO_(4) LICs show a broadened potential window from 3.0 to 4.2 V and an according high energy density of 420 Wh kg^(-1).Moreover,the underlying mechanism between prelithiation and high voltage is disclosed by electrochemical dynamic analysis.Prelithiation declines the Nb_(2)C anode potential that facilitates electron transmission in the interlayer of two-dimensional Nb_(2)C MXene.This effect induces small drive force for Li^(+)ions deposition and hence weakens the repulsive force from adsorbed ions on the electrode surface.Benefiting from even more Li^(+)ions deposition,a higher voltage is eventually delivered.In addition,prelithiation significantly increases Coulomb efficiency of the 1st cycle from 74%to 90%,which is crucial to commercial application of LICs.

Nitrogen-doped porous carbon nanosheets as both anode and cathode for advanced potassium-ion hybrid capacitors

Potassium-ion hybrid capacitors(PIHCs)as a burgeoning research hotspot are an ideal replacement for lithium-ion hybrid capacitors(LIHCs).Here,we report nitrogen-doped porous carbon nanosheets(NPCNs)with enlarged interlayer spacing,abundant defects,and favorable mesoporous structures.The structural changes of NPCNs in potassiation and depotassiation processes are analyzed by using Raman spectroscopy and transmission electron microscopy.Due to the unique structure of NPCNs,the PIHC device assembled using NPCNs as both the anode and cathode material(double-functional self-matching material)exhibits a superior energy density of 128 Wh kg^(-1)with a capacity retention of 90.8%after 9000 cycles.This research can promote the development of double-functional self-matching materials for hybrid energy storage devices with ultra-high performance.

Low-Temperature Carbonized Nitrogen-Doped Hard Carbon Nanofiber Toward High-Performance Sodium-Ion Capacitors

Carbon nanofiber(CNF)was widely utilized in the field of electrochemical energy storage due to its superiority of conductivity and mechanics.However,CNF was generally prepared at relatively high temperature.Herein,nitrogen-doped hard carbon nanofibers(NHCNFs)were prepared by a lowtemperature carbonization treatment assisted with electrospinning technology.Density functional theory analysis elucidates the incorporation of nitrogen heteroatoms with various chemical states into carbon matrix would significantly alter the total electronic configurations,leading to the robust adsorption and efficient diffusion of Na atoms on electrode interface.The obtained material carbonized at 600°C(NHCNF-600)presented a reversible specific capacity of 191.0 mAh g^(−1)and no capacity decay after 200 cycles at 1 A g^(−1).It was found that the sodium-intercalated degree had a correlation with the electrochemical impedance.A sodium-intercalated potential of 0.2 V was adopted to lower the electrochemical impedance.The constructed sodium-ion capacitor with activated carbon cathode and presodiated NHCNF-600 anode can present an energy power density of 82.1 Wh kg^(−1)and a power density of 7.0 kW kg^(−1).

The key challenges and future opportunities of electrochemical capacitors

Electrochemical capacitors(ECs)with unique merits of fast charge/discharge rate and long cyclability are one of the representative electrochemical energy storage systems,possessing wide applications in power electronics and automotive transportation,etc.[1,2].Furthermore.

Defect Engineering of Disordered Carbon Anodes with Ultra-High Heteroatom Doping Through a Supermolecule-Mediated Strategy for Potassium-Ion Hybrid Capacitors

Amorphous carbons are promising anodes for high-rate potassium-ion batteries.Most low-temperature annealed amorphous carbons display unsatisfactory capacities.Heteroatom-induced defect engineering of amorphous carbons could enhance their reversible capacities.Nevertheless,most lignocellulose biomasses lack heteroatoms,making it a challenge to design highly heteroatom-doped carbons(>10 at%).Herein,we report a new preparation strategy for amorphous carbon anodes.Nitrogen/sulfur co-doped lignin-derived porous carbons(NSLPC)with ultra-high nitrogen doping levels(21.6 at%of N and 0.8 at%of S)from renewable lignin biomacromolecule precursors were prepared through a supramolecule-mediated pyrolysis strategy.This supermolecule/lignin composite decomposes forming a covalently bonded graphitic carbon/amorphous carbon intermediate product,which induces the formation of high heteroatom doping in the obtained NSLPC.This unique pyrolysis chemistry and high heteroatom doping of NSLPC enable abundant defective active sites for the adsorption of K+and improved kinetics.The NSLPC anode delivered a high reversible capacity of 419 mAh g^(-1)and superior cycling stability(capacity retention of 96.6%at 1 A g^(-1)for 1000 cycles).Potassiumion hybrid capacitors assembled by NSLPC anode exhibited excellent cycling stability(91%capacity retention for 2000 cycles)and a high energy density of 71 Wh kg^(-1)at a power density of 92 W kg^(-1).

Freestanding fibers assembled by CoPSe@N-doped carbon heterostructures as an anode for fast potassium storage in hybrid capacitors

Although the fast development of potassium-ion hybrid capacitors(PIHC)recently,the issues such as the slow kinetics and poor durability of potassium ion hosts greatly restric their applications.Herein,a freestanding fiber(NHF fiber)with necklace-like configuration and CoPSe@N-doped carbon(CoPSe@NCNT)heterostructured units is introduced as the anode in PIHC.The highly porous network of NHF fiber facilitates the fast ion transports and promises the good high-rate property.Additionally,the nanoscle crystallites inside in-situ grown NCNT favor the high adaption to volume expansion/shrinkage and endow good structure stability during ion insertion/deinsertion.Density function theoretical(DFT)calculations disclose the CoPSe@NCNT heterostructure has improved intrinsic conductivity,fast potassium migration,and decreased energy barrier.Meanwhile,the finite element simulation analysis(FEA)reveals the decreased stress inside the NHF architecture during charge/discharge processes.Moreover,the electrochemical tests confirm the fast and durable properties of the CoPSe@NCNT NHF fibers for potassium storage.Furthermore,the PIHC full cell with the anode of CoPSe@NCNT NHF fiber is assembled,which obtains the superior energy/power densities and high capacity retention(89%)after 2000 cycles at 2 A g^(-1).When the polymer electrolyte is incooperated,the flexible PIHC device achieves the good pliability and good adaptation during wide temperature changes from-20 to 25℃.Therefore,this work introduces a novel anode for fast potassium ion storage,and opens a new approach to assemble the power sources for flexible electronics in diverse conditions.

Ordered Macroporous MoS_(2)-Carbon Composite with Fast and Robust Sodium Storage Properties to Solve the Issue of Kinetics Mismatch of Sodium-Ion Capacitors

Metal-ion capacitors(including Li^(+),Na^(+),and K^(+))effectively combine a battery negative electrode capable of reversibly intercalating metal cations,together with an electrical double-layer positive electrode.However,such novel cell design has a birth defect,namely kinetics mismatch between sluggish negative electrode and fast positive electrode,thus limiting the energy-power performance.Herein,we design a MoS_(2)-carbon composite anode with the ordered macroporous architecture and interlayer-expanded feature,exhibiting the fast and reversible Na^(+)redox processes.This kinetically favored anode is coupled with a homemade activated carbon cathode that allows for the excellent electrochemical performance of sodiumion capacitor with respect to large specific capacity,high-rate capability,and robust cycling.Through quantification of the potential swings of anode and cathode via a three-electrode Swagelok cell,we for the first time observe the abnormal variation law of potential swings and thus directly providing the evidence that the kinetics gap has been filled up by this kinetically favored anode.Our results represent a crucial step toward understanding the key issues of kinetics mismatch for hybrid cell,thus propelling the development of design of kinetically favored anode materials for high-performance metalion capacitors.

A survey of hybrid energy devices based on supercapacitors

Developing multifunctional energy storage systems with high specific energy, high specific power and long cycling life has been the one of the most important research directions. Compared to batteries and traditional capacitors, supercapacitors possess more balanced performance with both high specific power and long cycle-life. Nevertheless, regular supercapacitors can only achieve energy storage without harvesting energy and the energy density is still not very high compared to batteries. Therefore, combining high specific energy and high specific power,long cycle-life and even fast self-charging into one cell has been a promising direction for future energy storage devices. The multifunctional hybrid supercapacitors like asymmetric supercapacitors, batteries/supercapacitors hybrid devices and self-charging hybrid supercapacitors have been widely studied recently. Carbon based electrodes are common materials used in all kinds of energy storage devices due to their fabulous electrical and mechanical properties. In this survey, the research progress of all kinds of hybrid supercapacitors using multiple effects and their working mechanisms are briefly reviewed. And their advantages and disadvantages are discussed. The hybrid supercapacitors have great application potential for portable electronics, wearable devices and implantable devices in the future.

Mechanically flexible reduced graphene oxide/carbon composite films for high-performance quasi-solid-state lithium-ion capacitors

Practical applications of diverse flexible wearable electronics require electrochemical energy storage(EES)devices with multiple configurations.Moreover,to fabricate flexible EES devices with high energy density and stability,organic integration from electrode design to device assembly is required.To address these challenges,a free-standing reduced graphene oxide(rGO)/carbon film with a unique sandwich structure has been designed via the assistance of vacuum-assistant filtration for lithium-ion capacitors(LICs).The graphene acts as not only a binder to construct a three-dimensional conductive network but also an active material to provide additional capacitive lithium storage sites,thus enabling fast ion/electron transport and improving the capacity.The designed rGO/hard carbon(rGO/HC)and rGO/activated carbon(rGO/AC)free-standing films exhibit enhanced specific capacities(513.7 mA h g^(-1)for rGO/HC and 102.8 mA h g^(-1)for rGO/AC)and excellent stability.Moreover,the integrated flexible quasi-solid-state rGO/AC//rGO/HC LIC devices possess a maximum energy density of 138.3 Wh kg^(-1),a high power density of 11 kW kg^(-1),and improved cycling performance(84.4%capacitance maintained after 10,000 cycles),superior to the AC//HC LIC(43.5%retention).Such a strategy enlightens the development of portable flexible LICs.

Interlayer and intralayer co-modified flexible V_(2)CT_(X) MXene@SWCNT films for high-power Li-ion capacitors

As an emerging member of the two-dimensional(2D)material family,V_(2)CT_(X)MXene shows great potential in the application of lithium-ion capacitors(LICs)due to its unique structure and excellent electrical conductivity.However,severe nanosheets stacking and intra-layer transport barriers have limited the further development of V_(2)CT_(X)MXene-based materials.Herein,we prepared Kions and–O functional group co-modified V_(2)CT_(X)MXene(VCT-K)and further incorporated it with single-walled carbon nanotube(SWCNT),obtaining freestanding V_(2)CT_(X)composite films(VCT-K@C)with the 3D conductive network.Significantly,K+ions were introduced into V_(2)CT_(X)MXene to stabilize the interlayer structure and prevent the aggregation of nanosheets,the terminal group of–O was controllably modified on the surface of MXene to improve the Li+ions storage reversible capacities and the SWCNT acted as the bridge between MXene nanosheets to opens up the channels for ion/electron transportation in the longitudinal direction.Benefited from the synergistic effect of VCT-K and SWCNT,the VCT-K@C exhibits superior reversible specific capacities of 671.8 mA h g^(-1)at 0.1 A g^(-1)and 318 mA h g^(-1)at 1.0 A g^(-1).Furthermore,the assembled LICs with VCT-K@C anode coupling activated carbon(AC)cathode deliver an outstanding power density of 19.0 kW kg^(-1)at 67.4 Wh kg^(-1),a high energy density of 140.5 Wh kg^(-1)at 94.8 W kg^(-1)and a stable capacitance retention of 86%after 6000 cycles at 10 A g^(-1).Such unique structures and excellent electrochemical properties are expected to pave the way for the large-scale application in LICs of MXene-based materials.

Sustainable Lignin-Derived Carbon as Capacity-Kinetics Matched Cathode and Anode towards 4.5 V High-Performance Lithium-Ion Capacitors

The Li-ion capacitors(LICs)develop rapidly due to their double-high features of high-energy density and high-power density.However,the relative low capacity of cathode and sluggish kinetics of anode seriously impede the development of LICs.Herein,the precisely pore-engineered and heteroatomtailored defective hierarchical porous carbons(DHPCs)as large-capacity cathode and high-rate anode to construct high-performance dual-carbon LICs have been developed.The DHPCs are prepared based on triple-activation mechanisms by direct pyrolysis of sustainable lignin with urea to generate the interconnected hierarchical porous structure and plentiful heteroatominduced defects.Benefiting from these advanced merits,DHPCs show the well-matched high capacity and fast kinetics of both cathode and anode,exhibiting large capacities,superior rate capability and long-term lifespan.Both experimental and computational results demonstrate the strong synergistic effect of pore and dopants for Li storage.Consequently,the assembled dual-carbon LIC exhibits high voltage of 4.5 V,high-energy density of 208 Wh kg^(−1),ultrahigh power density of 53.4 kW kg^(−1)and almost zerodecrement cycling lifetime.Impressively,the full device with high mass loading of 9.4 mg cm^(−2)on cathode still outputs high-energy density of 187 Wh kg^(−1),demonstrative of their potential as electrode materials for high-performance electrochemical devices.

Alternatingly stacked thin film electrodes-based compact aqueous hybrid electrochemical capacitors for hundred-volts AC line filtering

Filtering capacitor with compact configuration and a wide range of operating voltage has been attracting increasing attention for the smooth conversion of the electric signal in modern circuits.Lossless integration of capacitor units can be regarded as one of the efficient ways to achieve a wider voltage range,which has not yet been fully conquered due to the lack of rational designs of the electrode structure and integration technology.This study presents an alternatingly stacked assemble technology to conveniently fabricate compact aqueous hybrid integrated filtering capacitors on a large scale,in which a unit consists of rGO/MXene composite film as a negative electrode and PEDOT:PSS based film as a positive electrode.Benefiting from the synergistic effect of rGO and MXene components,and morphological characteristics of PEDOT:PSS,the capacitor unit exhibits outstanding AC line filtering with a large areal specific energy density of 1,015 μF V^(2)cm^(-2)(0.28 μW h cm^(-2)) at 120 Hz.After rational integration,the assembled capacitors present compact/lightweight configuration and lossless frequency response,as reflected by almost constant resistor-capacitor time constant of 0.2 ms and dissipation factor of 15% at120 Hz,identical to those of the single capacitor unit.Apart from standing alone steadily on a flower,a small volume(only 8.1 cm^(3)) of the integrated capacitor with 70 units connected in series achieves hundred-volts alternating current line filtering,which is superior to most reported filtering capacitors with sandwich configuration.This study provides insight into the fabrication and application of compact/ultralight filtering capacitors with lossless frequency response,and a wide range of operating voltage.

Novel Bilayer-Shelled N,O-Doped Hollow Porous Carbon Microspheres as High Performance Anode for Potassium-Ion Hybrid Capacitors

With the advantages of high energy/power density,long cycling life and low cost,dual-carbon potassium ion hybrid capacitors(PIHCs)have great potential in the field of energy storage.Here,a novel bilayer-shelled N,O-doped hollow porous carbon microspheres(NOHPC)anode has been prepared by a self-template method,which is consisted of a dense thin shell and a hollow porous spherical core.Excitingly,the NOHPC anode possesses a high K-storage capacity of 325.9 mA h g^(−1)at 0.1 A g^(−1)and a capacity of 201.1 mAh g^(−1)at 5 A g^(−1)after 6000 cycles.In combination with ex situ characterizations and density functional theory calculations,the high reversible capacity has been demonstrated to be attributed to the co-doping of N/O heteroatoms and porous structure improved K+adsorption and intercalation capabilities,and the stable long-cycling performance originating from the bilayer-shelled hollow porous carbon sphere structure.Meanwhile,the hollow porous activated carbon microspheres(HPAC)cathode with a high specific surface area(1472.65 m^(2)g^(−1))deriving from etching NOHPC with KOH,contributing to a high electrochemical adsorption capacity of 71.2 mAh g^(−1)at 1 A g^(−1).Notably,the NOHPC//HPAC PIHC delivers a high energy density of 90.1 Wh kg^(−1)at a power density of 939.6 W kg^(−1)after 6000 consecutive charge-discharge cycles.

Three Dimensional Electric Circuits with Multiple Capacitors and Resistors

Two cubical 3D electric circuits with single and double capacitors and twelve ohmic resistors are considered. The resistors are the sides of the cube. The circuit is fed with a single internal emf. The charge on the capacitor(s) and the current distributions of all twelve sides of the circuit(s) vs. time are evaluated. The analysis requires solving twelve differential-algebraic intertwined symbolic equations. This is accomplished by applying a Computer Algebra System (CAS), specifically Mathematica. The needed codes are included. For a set of values assigned to the elements, the numeric results are depicted.