Cylindrical Battery Electrode
Dimensions Design
According to different packaging methods and shapes, lithium batteries can be classified into square, soft pack and cylindrical batteries.
Among them, cylindrical batteries have core advantages such as good consistency, high production efficiency, and low manufacturing cost.
Since their birth in 1991, they have a development history of more than 30 years. In recent years, with the release of Tesla’s alltab technology, the application of large cylindrical batteries in the field of power batteries and energy storage has accelerated. And that has become a research hotspot for major lithium battery companies.
Figure 1: Lithium battery cells of different shapes and performance comparison at the system level
Winding Technology for Cylindrical Batteries
Cylindrical battery casings can be steel casings, aluminum casings or soft packs. Their commonality is that the manufacturing process adopts winding technology. That is, the winding needle is used as the core, and the isolation film and the pole piece are wound together in layers by driving the winding needle to rotate. And finally a relatively uniform cylindrical winding core is formed.
As shown in the figure below, a typical winding process is:
 First, the needle clamps the diaphragm for prewinding.
 Then the negative electrode sheet is inserted between two layers of separator film for negative electrode prewinding.
 Then insert the positive electrode sheet for highspeed winding.
 After the winding is completed, the cutter mechanism cuts off the pole piece and the diaphragm.
 Finally, stick a layer of adhesive tape at the end to fix the shape.
Figure 2: Schematic diagram of the winding process
The control of the winding core diameter after winding is very critical.
If the diameter is too large, it cannot be assembled. But if too small, there will be a waste of space. Therefore, it is very important to accurately design the diameter of the winding core.
Fortunately, the cylindrical battery is a relatively regular geometry. The circumference of each pole piece and separator can be calculated by an approximate circle method. Besides, the total length of the pole pieces can be obtained by adding up at last, so as to design the capacity. The cumulative value of layers amount, pole pieces amount, and separator layers amount is the diameter of the winding core after winding. It should be noted that the core elements of lithiumion battery design are capacity design and size design.
In addition, through theoretical calculations, we can also design tabs at any position of the winding core, not limited to the head, tail or middle, and also cover the multitab and fulltab design methods for cylindrical batteries.
In order to explore the length of the pole piece and the diameter of the winding core, we first need to study the three processes of infinite prerolling of the separator, infinite prerolling of the negative pole piece, and infinite winding of the positive pole piece.
Assume that:
 the diameter of the needle is p.
 the thickness of the separator is s.
 the thickness of the negative plate is a.
 the thickness of the positive plate is c.
 the unit is mm.
Infinite preroll process of separator film
During the diaphragm prerolling process, two layers of diaphragms are wound at the same time.
Therefore, the diameter of the outer diaphragm during winding is always 1 more than the thickness of the inner diaphragm (+1s)
And the initial diameter of the inner diaphragm is the previous one.
The end diameter of the coil winding, and the diameter of the core increases by 4 layers of diaphragm thickness (+4s) for each prerolled diaphragm.
Table 1: Variation of Diameter of Isolation Film During Infinite Prerolling
Diaphragm preroll  Initial Diameter of Inner Diaphragm Winding  Initial Diameter of Outer Diaphragm Winding  Diaphragm circumference of the m+1th circle  Circumference of the outer diaphragm of the m+1 circle  Core diameter after prerolled diaphragm 
Diaphragm preroll 1st turn  P  p+s  πp  π (p+s)  p+4s 
Diaphragm preroll 2nd turn  p+4s  p+5s  π(p+4s)  π (p+5s)  p+8s 
Diaphragm preroll 3rd turn  p+8s  p+9s  π(p+8s)  π (p+9s)  p+12s 
…  ……  ……  ……  …… 

Diaphragm preroll m+1 turn  p+4m ms  p+(4m+1)s  π(p+4ms)  [p+(4m+1)s]  p+4(m+1)s 
Infinite preroll process of negative electrode sheet
In the prerolling process, due to the addition of a layer of negative electrode sheet, the diameter of the outer separator is always 1 layer more than the thickness of the inner separator and the thickness of the negative electrode sheet (+1s+1a).
And the initial diameter of the inner diaphragm winding is always equal to the ending diameter of the previous coil.
At this time, every time the negative electrode sheet is prerolled one turn, the core diameter increases by 4 layers of separator + 2 layers of negative electrode sheet thickness (+4s+2a).
Table 2: Variation of Diameter of negative electrode sheet during infinite prerolling
Negative preroll  Initial Diameter of Inner Diaphragm Winding  Initial Diameter of Outer Diaphragm Winding  Negative electrode sheet winding initial diameter  Negative sheet prerolled reference core diameter 
Negative preroll 1st lap  p+4(m+1)s  p+4(m+1)s+s+a  p+4(m+1)s+s  p+4(m+1)s+4s+2a 
Negative Tax Preroll Round 2  p+4(m+1)s+4s+2a  p+4(m+1)s+5s+3a  p+4(m+1)s+5s+2a  p+4(m+1)s+8s+4a 
Negative Tax Preroll Round 3  p+4(m+1)s+8s+4a  p+4(m+1)s+9s+5a  p+4(m+1)s+9s+4a  p+4(m+1)s+12s+6a 
···  ····· 

 ···· 
Negative plate preroll n+1 turn  p+4(m+1)s+4ns+2na  p+4(m+1)s+(4n+1)s+(2n+1)a  p+4(m+1)s+(4n+1)s+2na  p+4(m+1)s+4(n+1)s+2(n+1)a 
Merger of similar items  p+4(m+n+1)s+2na  p+(4m+4n+5)s+(2n+1)a  p+(4m+4n+5)s+2na  p+4(m+n+2)s+2(n+1)a 
Infinite winding process of positive electrode sheet
During the winding process, since a new layer of positive electrode sheet is added, the initial diameter of the positive electrode sheet is always equal to the end diameter of the previous circle.
The initial diameter of the inner separator coil becomes the end diameter of the previous coil plus the thickness of 1 positive electrode sheet (+1c).
However, the diameter of the outer diaphragm during the winding process is always only 1 more than the thickness of the inner diaphragm and the thickness of the negative electrode sheet (+1s+1a).
At this time, every time the negative electrode sheet is prerolled one turn, the core diameter increases by 4 layers of separator + 2 layers of negative electrode sheet + 2 layers of positive electrode sheet thickness (+4s+2s+2a).
Table 3: Changes in the diameter of the positive electrode sheet during the infinite winding process
Positive winding  Positive sheet winding initial diameter  Negative electrode sheet winding initial diameter 
Positive electrode winding 1st turn  p+4(m+1)s+4(n+1)s+2(n+1)a  p+4(m+1)s+4(n+1)s+2(n+1)a+c+s 
Positive winding 2nd turn  o+4(m+1)s+4(n+1)s+2(n+1)a+4s+2a+2  p+4(m+1)s+4(n+1)s+2(n+1)a+5s+2a+3c 
Positive winding 3rd turn  p+4(m+1)s+4(n+1)s+2(n+1)a+8s+4a+4c  p+4(m+1)s+4(n+1)a+2(n+1)a+9a+4a+5c 
… 


Positive electrode winding x+1 turn  p+4(m+1)s+4(n+1)s+2(n+1)a+4xs+2xa+2  p+4(m+1)s+4(n+1)s+2(n+1)a+(4x+1)s+2xn+(2x+1)c 
Merger of similar items  1(m+n+x+2)s+2(n+x+1)a+2xc  p+(4m+4n+4x+9)s+2(n+x+1)a+(2x+1)c 
Through the above analysis of the infinite winding process of the diaphragm and the pole piece, we have obtained the change law of the winding core diameter and the length of the pole piece.
This layerbylayer analytical calculation method is conducive to the precise arrangement of tab positions (including single tabs, multiple tabs and full tabs). But the winding process has not ended so far, and the positive electrode sheet, the negative electrode sheet and the separator are in a flush state at this time.
The basic principle of battery design is to require the separator to completely cover the negative electrode sheet, and the negative electrode sheet should also completely cover the positive electrode sheet.
Figure 3: Schematic diagram of cylindrical battery core structure and finishing process
Therefore, it is necessary for us to further explore the winding problem of the core negative electrode sheet and the separator. Apparently, since the positive electrode sheet has already been wound, and before that, the initial diameter of the positive electrode sheet is always equal to the end diameter of the previous coil. So at this time:
 The initial diameter of the inner diaphragm supersedes the ending diameter of the previous lap.
 The initial diameter of the negative plate is increased by the thickness of a separator (+1s) on this basis.
 The initial diameter of the outer separator is increased by the thickness of the negative electrode sheet (+1s+1a).
Table 4: Changes in the diameter and length of pole pieces and diaphragms during the winding process of cylindrical batteries
Initial Diameter of Rewinding Inner Diaphragm  Rewinding inner diaphragm length 
p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c  π [p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c]*β/360 
Initial diameter of winding negative electrode sheet  Rewinding negative electrode piece length 
p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c+s  π [p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c+s]*β/360 
Initial Diameter of Rewinding Outer Diaphragm  Rewinding outer diaphragm length 
p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c+s+a  [p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c+s+a]*β/360 
Diameter after winding 

p+4(m+n+x+3)s+2(n+x+2)a+2(x+1)c+2s+a 
So far, we have obtained the mathematical expressions of the length of the positive electrode, negative electrode, and separator under any number of winding turns.
Assumptions:
 Diaphragm prerolled m+1 circle
 Negative sheet preroll n+1 circle
 Positive electrode winding x+1 circle
 The central angle of the winding circle of the negative electrode sheet is θ°
 The central angle of the isolation film winding circle is β°
Then there is the following relationship:
Positive electrode sheet length:
Negative electrode sheet length:
Inner diaphragm length:
Outer diaphragm length:
The number of pole pieces and diaphragm layers not only determines the length of the pole pieces and diaphragm, which in turn affects the capacity design, but also determines the final diameter of the winding core. And that greatly reduces the risk of assembly of the winding core. Although we obtained the core diameter after winding, we did not consider the thickness of tabs and the issue of tape at the end.
Assuming that:
 the thickness of the positive tab is tabc
 the thickness of the negative tab is taba
 the finishing glue is 1 circle and the overlapping area avoids the position of the tab, and the thickness is g
Then the diameter of the final winding core is:
① When the positive and negative lugs pass through the center of the circle:
② When the positive and negative lugs do not pass through the center of the circle:
The above formula is the general solution relationship for the design of cylindrical battery pole pieces.
It identifies pole piece length issues, diaphragm length issues, and core diameter issues. Moreover, the relationship between them is described quantitatively, which greatly improves the design accuracy and has great practical application value.
Finally, what we need to solve is the arrangement of the tabs. Usually, there are 1 tab, 2 tabs, or even 3 tabs on one pole piece, which is a small number of tabs. The tablead is welded on the surface of the pole piece. Although it will affect the accuracy of the length design of the pole piece to a certain extent (it has no effect on the diameter), the tablead is usually narrow and has little effect.
Therefore, the general solution formula for the size design of cylindrical batteries proposed in this paper ignores this problem.
Figure 4: Location layout of positive and negative tabs
The figure above is a schematic diagram of the location of the tabs. According to the general solution relationship of the pole piece size proposed before, we can clearly understand the length and diameter changes of each layer of pole piece during the winding process.
Therefore, when arranging the tabs, in the case of a single tab, the positive tab and the negative tab can be accurately arranged at the target position of the pole piece.
In the case of multiple tabs or all tabs, alignment of multiple tabs is usually required. We only need to deviate from a fixed angle for each layer of tabs on this basis, so as to obtain the arrangement position of each layer of tabs.
Due to the gradual increase in the diameter of the core during the winding process, the pitch of the lug arrangement generally changes in an arithmetic sequence relationship with a tolerance of π(4s+2a+2c).
In order to further study the influence of pole piece and separator thickness fluctuations on the core diameter and pole piece length, a 4680 large cylindrical full tab cell is taken as an example.
Assuming that:
 the diameter of the coil needle is 1mm
 the thickness of the tape at the end is 16um
 the thickness of the separator is 10um
 the thickness of the positive electrode sheet is 171um when it is coldpressed
 the thickness when it is wound is 174um
 the thickness of the negative electrode is 249um when it is coldpressed
 both the separator and the negative electrode sheet were prerolled 2 turns
It is calculated that the positive electrode sheet is wound 47 times, with a length of 3371.6 mm, and the negative electrode sheet is wound 49.5 times, with a length of 3449.7 mm and a diameter of 44.69 mm after winding.
Figure 5: Effect of pole piece and diaphragm thickness fluctuations on core diameter and pole piece length
It can be seen intuitively from the above figure that the fluctuation of the thickness of the pole piece and the diaphragm has a certain influence on the diameter of the winding core and the length of the pole piece.
When the thickness of the pole piece is 1um thicker, the diameter of the winding core and the length of the pole piece will increase by about 0.2%.
When the thickness of the diaphragm is 1um thicker, the diameter of the core and the length of the pole piece increase by about 0.5%.
Therefore, in order to control the consistency of the winding core diameter, the fluctuation of the pole piece and the diaphragm should be minimized. And it is also necessary to collect the relationship between the cold pressing and the winding of the pole piece rebound with time, so as to assist the battery design process.
Summarize
1. Capacity design and diameter design are the bottomlevel design logic of cylindrical lithium batteries. The key to capacity design lies in the design of the pole piece length. While the key to diameter design lies in the analysis of the number of layers.
2. The location of the tabs is also very important. For multitab or alltab structures, the alignment of the tabs can be used as the evaluation standard for the design capability of the battery cell and the process control capability. The method of layerbylayer analysis can better meet the requirements of tab position layout and alignment.