Busbar Layout for Parallel Strings

Cross-String Current Sharing introduced two busbar layouts in passing, daisy-chain and centre-fed, to show that physical busbar geometry sets a systematic component of the resistance spread between parallel strings, on top of the random cell-to-cell and joint-to-joint variation covered there. This page works through the full set of layouts an engineer is actually choosing between, with the resistance math behind each one, so the choice can be made on numbers rather than a rule of thumb.

Model and Notation

Take N strings tapped at even intervals off a busbar, and let r be the resistance of one inter-tap segment of that busbar (the same on the return rail, so each string picks up resistance from both the + and − side of the loop). Index the taps by hop count k, the number of segments between a given tap and the feed point, running from k = 0 at the feed end up to the farthest tap. R_k below is the busbar’s own contribution to a string’s total loop resistance R_i from Cross-String Current Sharing‘s I_i = (V_OCV,i − V_bus)/R_i; it adds to, rather than replaces, the cell- and joint-level variation covered there. The six layouts below are compared on the resulting spread in R_k across strings, for a fixed amount of copper wherever that comparison is fair.

Six busbar layout options compared: daisy-chain, center-fed, star, ring, H-tree, and tapered cross-section, each showing cell connections and relative resistance to the feed point — The six layouts compared, N = 6 strings shown. Badge colour marks each string’s resistance relative to the feed (green low, red high); see the numbers below for what each layout actually achieves.

The Six Layouts

1. Daisy-Chain (Single-End Feed)

Both rails run the full span and are fed from the same end (Panel 1). Each tap simply picks up one more segment of resistance than the last:

R_k^{\text{daisy}}=k\,r,\quad k=0,\dots,N-1

The farthest string sees the full rail resistance, R_max = (N−1)r, growing linearly with pack size. This is the layout you get by default if strings are just bussed together in physical order without thinking about where the feed cable lands. It produces the worst-case gradient of anything in this comparison, and it’s the one Worked Example 2 on Cross-String Current Sharing is built around.

2. Center-Fed (Dual-End)

Moving the feed to the middle of the same two rails changes the picture substantially, and it costs no extra copper, just a different attachment point for the external cable:

R_k^{\text{center}}=\left|k-\frac{N-1}{2}\right|r

R_max drops to about (N−1)/2·r, half of daisy-chain’s for the same rail. Because it costs nothing beyond deciding where to land the supply cable, center-feeding is close to a free improvement over daisy-chaining any time the physical layout allows the feed to reach the middle of the string run.

3. Star / Radial

Star topology gives every string dedicated + and − leads running back to a common bus block (Panel 3), rather than sharing a length of rail with its neighbours. If those leads are cut to equal length, R_k is the same for every string by construction, and the systematic gradient this page is about disappears entirely. What star spends instead is copper and parts count: N separate conductor pairs and N sets of connectors, and in practice small differences in individual lead length, crimp quality or connector contact resistance reintroduce some of the random (rather than systematic) spread that Worked Example 1 on Cross-String Current Sharing already covers.

4. Ring / Loop

Closing both rails into a loop (Panel 4) gives every tap two parallel paths back to the feed, one each way around the ring:

R_k^{\text{ring}}=\frac{k(N-k)}{N}\,r

For even N the worst case sits at k = N/2, giving R_max ≈ (N/4)r, a quarter of daisy-chain’s peak for the same per-segment resistance, and better than center-fed on both the worst case and the average across strings (0.97r vs. 1.5r mean at N = 6). The cost is a longer rail, with enough extra length to route both ends back together, plus a mechanical loop that has to be closed somewhere, which daisy-chain and center-fed don’t need.

5. H-Tree / Fractal Branching

An H-tree branches the + and − conductors together, halving the number of parallel paths at each level, until every string sits at the tip of a branch (Panel 5). Every leaf is reached by the same number of branch levels, so like star, R_k is nominally equal across strings by construction. Unlike star, most of the conductor length is shared trunk rather than dedicated per-string runs, so a given uniformity costs less copper than star for the same N. That comes at the price of a routing shape (recursive binary splits) that’s straightforward to lay out on a PCB but awkward to bend into sheet copper or cable form for a large-format cell pack, which is why it shows up far more often in on-die and PCB power distribution than in traction battery packs.

6. Tapered Cross-Section (a technique, not a topology)

Tapering doesn’t change which taps a rail visits. It changes the rail’s cross-section along the way, and can be applied to a daisy-chain, center-fed or ring rail alike. Size each segment’s cross-sectional area to the current it actually carries (the standard busbar-sizing rule of constant current density) and, on a daisy-chained rail, the resistance of the segment carrying current for the (N−j) strings still downstream of it becomes r/(N−j) instead of a constant r:

R_k^{\text{taper}}=r\sum_{j=1}^{k}\frac{1}{N-j}

R_max works out to r·H_N−1, where H_n is the nth harmonic number. For N = 6 that’s about 2.28r, roughly half of the untapered daisy-chain’s 5r for the same feed-end layout. Better still, it scales with ln(N) rather than N: at N = 24 the tapered rail’s R_max (≈ 3.73r) is already below the ring’s (≈ 6r). The catch is manufacturability: the cross-section ratio between the thickest and thinnest ends of the bar is (N−1):1, and beyond some N that either makes the feed end unreasonably bulky or the far end too thin to meet its own current rating, so tapering has a practical ceiling that ring and center-fed don’t.

How They Compare

R_max in units of r, for three pack sizes:

Layout	N = 6	N = 12	N = 24	Scaling
Daisy-chain	5r	11r	23r	Linear in N
Center-fed	2.5r	5.5r	11.5r	Linear in N (½ slope)
Ring	1.5r	3r	6r	Linear in N (¼ slope)
Tapered	2.28r	3.02r	3.73r	Logarithmic in N
Star	≈0	≈0	≈0	Constant
H-tree	≈0	≈0	≈0	Constant

Star and H-tree are shown as ≈0, reflecting the ideal case before connector and lead-length tolerance are accounted for.

Choosing a Layout

Peak versus average matters here, and the two don’t always favour the same layout. Ring beats tapered daisy-chain on worst-case R_max at N = 6 (1.5r vs. 2.28r), but tapering beats ring on the average across strings (0.83r vs. 0.97r). That’s because tapering pushes most strings’ resistance very low near the thick feed end and concentrates the penalty on the last string or two, while ring spreads a smaller penalty more evenly. If the failure mode that matters is a single string overheating or undercharging at the extreme, optimise for R_max; if it’s aggregate pack-level mismatch, the mean is the more relevant number.

For most packs, center-feeding a daisy-chained rail is worth doing by default: it halves the worst case for zero extra copper. Ring is worth the extra length and the loop closure when the layout allows it and the worst-case string matters. Tapering is worth it when the rail is already a single machined or stamped bar, since grading its cross-section doesn’t add a manufacturing step, but it stops paying off on its own once N gets large enough that the taper ratio becomes impractical. Star and H-tree earn their systematic-gradient-free result mainly when the residual cell- and joint-level variation from Worked Example 1 on Cross-String Current Sharing is the dominant term left to protect against. Beyond that point, the added parts count is hard to justify over a well center-fed or ring bus.