Transformer turns ratio calculator

What a turns ratio does to voltage, current and impedance, from the one fact Faraday's Law gives: every winding on a core carries the same volts per turn. And the other half of a transformer design, how many turns the core needs so the pulse it sees stays inside the allowed flux swing.

Fig 1 — 10:30 turns, 1.20 V per turn: 12.0 V in becomes 36.0 V out; 500 mA drawn on the secondary is 1.50 A on the primary.

Primary voltage V_p (V)

Primary turns N_p

Secondary turns N_s

Secondary load (0 for none) (Ω)

Pulse voltage on the primary (V)

Pulse duration (µs)

Allowed flux swing ΔB (mT)

Core area A_e (mm²)

Turns ratio N_s / N_p: 3 · 1 : 3, step-up
Volts per turn, on every winding: 1.20 V
Secondary voltage: 36.0 V
Secondary current · primary current: 500 mA · 1.50 A
Power through the transformer: 18.0 W
Load as the primary sees it, R / n²: 8.00 Ω · 72.0 Ω behind 1 : 3
Volt-seconds per pulse, 48.0 V for 5.00 µs: 240 µV·s
Primary turns for 200 mT on 50 mm²: 24 turns (24.00 exact) · swing 200 mT with 24
Secondary turns at this ratio: 72 turns · 1.20 V per turn at 12.0 V would be 2.00 V per turn during the pulse

The 10 primary turns entered above are fewer than the 24 the flux swing needs at 48.0 V for 5.00 µs: the core would swing 480 mT per cycle, past the 200 mT allowed.

How this is calculated

Standard: TI SLUP123 — Magnetics Design for Switching Power Supplies, Section 1 (Dixon)

$\frac{V_s}{V_p} = \frac{N_s}{N_p} = n$: SLUP123: Faraday's Law "equates the flux rate of change through a winding to the volts/turn applied to the winding" and "operates bilaterally", so a second winding linked to the same flux carries the same volts per turn.
$\frac{I_p}{I_s} = n, \qquad R_{p} = \frac{R_{load}}{n^2}$: An ideal transformer passes power through unchanged, V_s·I_s = V_p·I_p, so the current scales inversely with the turns and a load on the secondary is seen through the square of the ratio.
$N_{p} = \frac{V\, t}{\Delta B\, A_e}$: SLUP123 Eq 2 in SI, ∫E dt = N·ΔΦ, with ΔΦ = ΔB·A_e and the volt-seconds of a rectangular pulse V·t. The minimum turns that keep the swing inside the allowed ΔB; the note names saturation and core loss as the two limits that follow from the swing.

Assumptions

An ideal transformer for the ratio relations: all flux links both windings, no magnetising current, no winding resistance or capacitance. SLUP123 lists each of these among the parasitics of a real part.
The turns calculation is for a rectangular pulse of known voltage and duration. A sinusoid or a partial reset changes the volt-seconds, and the allowed ΔB depends on the topology and the material at temperature.
A_e and the allowed flux swing come from the core and material datasheets; nothing here estimates them.
AC only. Faraday's Law concerns changing flux; a DC component walks the core towards saturation and is outside the ratio relations.

What a transformer's turns ratio sets

A transformer is two windings that share one magnetic flux. SLUP123 states the law that makes it work: Faraday's Law "equates the flux rate of change through a winding to the volts/turn applied to the winding", and it "operates bilaterally: if 2.5 volts/turn is applied to winding A, the flux through A will change by 2.5 Webers/second. If a second winding, B, is linked to all of the flux produced by winding A, then 2.5 Volts/turn will be induced in B." Every winding on the core therefore carries the same volts per turn, and the voltage ratio is the turns ratio: V_s/V_p = N_s/N_p. That is the first of the three relations, and the only one that is Faraday's directly.

The other two follow from an ideal transformer passing power through without storing or losing any. If the secondary delivers V_s·I_s, the primary must supply the same, so the current ratio is the turns ratio inverted: I_p/I_s = N_s/N_p. A step-up in voltage is a step-down in current by the same factor, which is why the primary of a step-up transformer is the winding with the thick wire. And a load on the secondary, seen from the primary, is scaled by the square of the ratio: R_p = R_load / n², because the voltage went one way by n and the current went the other way by n. That is the impedance transformation an audio output transformer or an RF matching transformer is bought for, and it is why a 1:3 turns ratio is a 1:9 impedance ratio.

None of this says how many turns. The ratio fixes only their proportion; the absolute count is set by the core, and SLUP123 gives the relation for that too. Its Eq 2 in SI units is ∫E dt = N·ΔΦ: the volt-seconds applied to a winding equal the turns times the flux change they produce. With the flux change written as the flux density swing times the core area, ΔΦ = ΔB·A_e, the minimum primary turns for a pulse of V volts lasting t seconds is N = V·t / (ΔB·A_e). Fewer turns than that push the core past the allowed swing; the note is direct that in switching supplies "the major core material limitations are saturation and core losses, both of which depend upon flux swing." The calculator does both halves: the ratio from the turns entered, and the turns the core needs from the pulse and the swing.

Turns ratio chart

The common ratios with what each does, computed by the calculator above with 12 V on the primary and a 100 Ω load on the secondary. Read the three right-hand columns together: the voltage goes one way by n, the current the other way by n, and the impedance the primary sees by n².

N_p : N_s	12 V in becomes	Current, secondary to primary	Impedance factor	100 Ω load seen as
1 : 1	12.0 V	× 1	× 1	100 Ω
1 : 2	24.0 V	× 0.5	× 0.25	25.0 Ω
1 : 3	36.0 V	× 0.333	× 0.111	11.1 Ω
1 : 5	60.0 V	× 0.2	× 0.04	4.00 Ω
1 : 10	120 V	× 0.1	× 0.01	1.00 Ω
2 : 1	6.00 V	× 2	× 4	400 Ω
3 : 1	4.00 V	× 3	× 9	900 Ω
5 : 1	2.40 V	× 5	× 25	2.50 kΩ
10 : 1	1.20 V	× 10	× 100	10.0 kΩ

The impedance column is the one that trips people up. A 1:10 step-up makes a 100 Ω load look like 1 Ω to whatever drives the primary, which is why a small step-up transformer can stall a source that drove the same load happily through a 1:1; and a 10:1 step-down turns 100 Ω into 10 kΩ, which is how an audio transformer lets a low-impedance driver match a high-impedance line.

Worked example: 10 : 30 turns from 12 V into 72 Ω, and the turns a core needs

The defaults: 12 V across a 10-turn primary, a 30-turn secondary into 72 Ω; and for the core, 48 V applied for 5 µs per cycle with 200 mT of allowed swing on 50 mm² of core area.

ratio         = 30 / 10                          = 3       (1 : 3, step-up)
volts/turn    = 12 / 10                          = 1.2 V   on both windings
V_s           = 1.2 × 30                         = 36 V
I_s           = 36 / 72                          = 0.5 A
I_p           = 0.5 × 3                          = 1.5 A   (12 × 1.5 = 36 × 0.5 = 18 W)
reflected     = 72 / 3²                          = 8 Ω     (12 V / 8 Ω = 1.5 A, the same answer)

volt-seconds  = 48 V × 5 µs                      = 240 µV·s
N_p minimum   = 240 µV·s / (0.2 T × 50 mm²)      = 24 turns
N_s           = 24 × 3                           = 72 turns

The two halves are independent, and that is the point of showing them side by side. The ratio is fixed by what the circuit needs; the turn count is fixed by what the core can take. Ten primary turns carry 12 V perfectly well as a ratio, but if the same core sees the 48 V pulse in the second half, ten turns would swing it 480 mT and saturate it, and the calculator says so.

Where the ideal transformer stops being valid

Magnetising current. A real core needs some current to establish the flux, drawn by the primary whether or not the secondary is loaded. SLUP123 notes that in a ferrite transformer it "is small enough to be of less concern than the flux swing", but it is why an unloaded primary still draws current and why the current ratio is only exact once the load current dominates it.

Leakage inductance. Not all of the primary's flux links the secondary. The part that does not behaves as an inductor in series with each winding, and it is the reason a switching transformer rings and needs a snubber or a clamp, and the reason the voltage ratio droops under load at the edges. The note's equivalent circuit puts it among the parasitics that "cause a variety of circuit problems". The flyback calculator works the clamp that leakage demands.

Saturation. The turns computed from the flux swing assume the swing is symmetric about zero, as it is in a push-pull or bridge. A forward converter resets from one side and a flyback stores energy in a gap; both use the same Faraday relation for the turns but with their own limits on ΔB, and the material's saturation flux density falls with temperature.

Not for DC. Faraday's Law is about changing flux. A steady voltage on a winding produces a steadily rising flux until the core saturates, which for a mains transformer on DC takes milliseconds; the ratio relations apply to the AC part of a waveform only.

Winding resistance and capacitance. The copper drops voltage under load, which is what a transformer's regulation figure reports, and the winding capacitance sets the resonance that limits the high-frequency response. Neither is in the ideal relations here.

Common turns ratio mistakes

Reading the turns ratio as the impedance ratio. Impedance goes with the square: 1:3 in turns is 1:9 in ohms, and the source sees the load divided by nine, not by three.
Sizing the primary wire for the secondary current. In a step-up the primary carries n times the secondary current; in a step-down it is the secondary that needs the thick wire.
Choosing turns from the ratio alone. Any pair with the right proportion gives the right voltage, but only pairs with enough turns keep the core below its flux swing. The volt-seconds decide the count; the ratio then decides the other winding.
Using the marked mains voltage as the pulse voltage. The turns calculation wants the voltage actually across the winding during the interval it is applied, which in a converter is the input during the on-time, not an RMS figure.
Expecting the ratio to hold to the last percent. Regulation, leakage and magnetising current all move the secondary voltage under load; a few per cent of droop is normal and is why a feedback loop exists.