Load Line Calculator

The graphical method that explains every nonlinear-device bias problem: a device’s own I-V curve crossed with the straight line a supply and a series resistor impose, and the operating point where they meet — for a diode, an LED, a zener in reverse, or an NTC thermistor whose curve moves with its own self-heating.

Last updated 24 Sep 2026

Where the device curve and the load line cross

Device + supply + resistor → operating point

Which device?The first two share the same equation and differ only in their typical forward voltage and ideality factor. The zener is modelled as a hard knee with a slope resistance above it. The thermistor is the interesting one, because its curve moves as it warms itself up.

Supply voltage (V)The load line runs from (0 V, V_s ÷ R) to (V_s, 0). Those two intercepts are the whole line.

Series resistance (Ω)The slope of the load line is −1 ÷ R. A bigger resistor tilts the line flatter and makes the operating current much less sensitive to everything — which is the entire reason a series resistor is fitted.

Forward voltage at a stated current (V)From the data sheet, together with the current it is quoted at. Those two numbers fix the saturation current, so you never have to know it. 0.7 V at 10 mA is a small signal silicon diode; 2.0 V at 20 mA is a red LED; 3.2 V at 350 mA is a white one. Use the figures from your part’s datasheet; typical values vary widely between manufacturers.

The current that forward voltage is quoted at (mA)

Ideality factor nHow steeply the current rises with voltage: 1.0 for an ideal diffusion diode, 1.0 to 1.1 for a good silicon rectifier, 1.5 to 2.5 for an LED, where recombination in the active layer dominates. It sets the knee’s sharpness and therefore the device’s small-signal resistance.

Zener knee voltage V_z (V)The nominal zener voltage, quoted at a stated test current. Below this the model passes no current at all, which is why a zener fed from a supply below its knee simply sits at the supply voltage.

Zener slope resistance Z_z (Ω)How much the zener voltage rises per ampere above the knee. A 5 to 6 V zener is at its best here, a few ohms; a 3 V or a 30 V part is much worse. This is the number that decides how well a zener regulates. Use the figures from your part’s datasheet; typical values vary widely between manufacturers.

Thermistor resistance at 25 °C (kΩ)The nominal R₂₅. 10 kΩ is the commonest value by a wide margin.

Thermistor B constant (K)The B₂₅/₈₅ value from the data sheet. 3977 K is a very common figure for a 10 kΩ part. It sets both the sensitivity and, less obviously, how readily the part runs away when it heats itself.

Thermistor dissipation constant (mW/K)How many milliwatts raise the bead one kelvin in still air. 1 to 3 mW/K for a small leaded or 0805 part; the thermal resistance is its reciprocal, so 2 mW/K is 500 K/W. This is the number self-heating lives or dies by and it is strongly affected by mounting and airflow.

Ambient temperature (°C)Also used as the junction temperature for the diode, LED and zener branches, which do not model their own self-heating — see the notes.

Device dissipation rating (W)Used to draw the constant-dissipation hyperbola on the chart: the curve I = P_max ÷ V, which is the boundary the operating point must stay below. Where that hyperbola cuts the device curve is the most the device may be run at.

Supply change to test (V)How far the operating point moves for this much change in the supply. This is the sensitivity the picture exists to show, and for a diode fed from a small resistor it is alarming.

Temperature change to test (°C)For the diode, LED and zener this shifts the device curve sideways by the coefficient below. For the thermistor it is a change in ambient.

Device voltage temperature coefficient (mV/°C)How the device’s own voltage drifts at constant current. About −2 mV/°C for a silicon junction, −2 to −4 mV/°C for an LED, and for a zener it changes sign near 5 V: negative below, positive above, which is why a 5.6 V reference made of a zener and a forward diode can be made nearly temperature independent.

The circuit the two curves describe: a supply, a series resistor and whichever device you selected. The device turns amber as its dissipation passes four fifths of the rating you entered and red past it. The moving dots are the operating-point current — the one number the intersection of the two curves gives you — and everything written underneath is read off that same point. For the thermistor the device also has a temperature, which is shown because the curve it sits on moves with it.

23.03mAExample

a red LED specified at 2.0 V and 20 mA with an ideality factor of 1.8, fed from 5 V through 130 Ω at 25 °C

Two curves, one intersection

load line: I = (V_s − V) ÷ R, from (0, V_s/R) to (V_s, 0)
diode and LED: I = I_test·[exp((V − V_f)/nV_T) − exp(−V_f/nV_T)] ÷ [1 − exp(−V_f/nV_T)], r_d = nV_T ÷ (I + I_s)
zener: I = (V − V_z) ÷ Z_z above the knee, 0 below
thermistor: R(T) = R₂₅·exp(B(1/T − 1/298.15)), T = T_a + θ·V·I, r_d = R·(1 − g) ÷ (1 + g) with g = θ·P·B/T²
sensitivity: dV/dV_s = r_d ÷ (R + r_d), dI/dV_s = 1 ÷ (R + r_d)

r_d: the device’s small-signal (incremental) resistance at the operating point: the slope of its curve there, not V ÷ I. For a diode at 23 mA it is about two ohms, which is why a diode looks like a voltage source
g: the thermistor’s self-heating loop gain when driven from a current source. Above one, its incremental resistance goes negative and the device runs away without a series resistor to damp it
θ: the thermistor’s thermal resistance to its surroundings, which is the reciprocal of the dissipation constant a data sheet quotes in milliwatts per kelvin

Worked example

a red LED specified at 2.0 V and 20 mA with an ideality factor of 1.8, fed from 5 V through 130 Ω at 25 °C
nV_T is 1.8 × 25.6926 mV = 46.2466 mV, and the quoted point fixes the saturation current at 0.00000000331 pA — a number no data sheet gives and nobody needs to know
The load line runs from (0 V, 38.46 mA) to (5 V, 0). Twenty-six bisections find where it crosses the device curve: 2.007 V at 23.03 mA
The LED dissipates 46.2 mW there, which is 46.2% of the 100 mW rating, and its small-signal resistance is 2.008 Ω — two ohms, against 130 Ω of series resistor. That ratio is the whole story of the page
Because r_d is so much smaller than R, the resistor controls almost everything. A 0.5 V rise in the supply moves the LED voltage only 7.61 mV but moves the current 3.79 mA — 16.45%. Conversely a 25 °C rise, at -2.0 mV/°C, shifts the curve 50 mV to the left and changes the current by only 1.64%
And the number that matters for production: LEDs are binned, and 0.2 V of forward-voltage spread between parts changes the current by 1.515 mA — 6.58% — with this resistor. Fit a bigger resistor from a higher supply and that shrinks; drive the LED from a constant current and it disappears

The same LED, the same 23 mA, four different series resistors

Every row is close to the same LED current, reached from a different supply. The higher the supply and the larger the resistor, the flatter the load line and the less the current cares about the LED’s own forward voltage — and the more power is thrown away in the resistor. That trade is the reason constant-current drivers exist. Computed with this page’s own model.
Supply	Resistor	Current	Spread from ±0.1 V of V_f	Wasted in the resistor
3.0 V	43 Ω	23.1 mA	±2.222 mA	22.95 mW
5.0 V	130 Ω	23.03 mA	±0.758 mA	68.93 mW
12.0 V	430 Ω	23.24 mA	±0.231 mA	232.2 mW
24.0 V	950 Ω	23.15 mA	±0.105 mA	509.2 mW

The one picture that explains diode biasing

A resistor obeys Ohm’s law and a diode does not, so putting them in series is a problem with no algebraic answer. The graphical method solves it in one stroke: draw the device’s own current-versus-voltage curve, draw on the same axes the straight line that the supply and the resistor allow — from (0, V_s/R) down to (V_s, 0) — and the circuit must sit where they cross. That intersection is the operating point, and everything follows from it.

What the slope at the crossing tells you. The device’s small-signal resistance r_d is the slope of its curve at the operating point, and it is not V ÷ I. For a silicon diode it is n·V_T ÷ (I + I_s) — about 1.1 Ω at 23 mA for an ideal junction, a couple of ohms for an LED. Compare that with the series resistor and you know immediately how the circuit behaves: when r_d ≪ R the resistor sets the current and the device is close to a fixed voltage drop, which is why the schoolbook rule R = (V_s − V_f) ÷ I works. When r_d approaches R, the device starts controlling things and small changes in its curve move the current a lot.

Sensitivity is what the picture is for. Tilt the load line by changing the supply and the crossing slides along the device curve: the current changes by ΔV_s ÷ (R + r_d) and the voltage by only r_d/(R + r_d) of it. Shift the device curve sideways — which is what temperature does, at about −2 mV/°C for a silicon junction — and the current changes by ΔV_f ÷ (R + r_d) in the other direction. Both are just the resistive divider between R and r_d, and both are on this page as numbers. For an LED the third source of movement is the one that actually causes trouble in production: parts are binned over a range of forward voltage, and that spread goes straight into the current through the same divider.

The thermistor is the interesting case, because its curve moves. An NTC dissipates power, warms up, and its resistance falls — so the curve you cross the load line with depends on where you are on it. Sweeping the temperature rather than the voltage makes this explicit: at each temperature the power is (T − T_a) ÷ θ and the resistance is known, so V = √(PR) and I = √(P/R). Because P rises linearly with temperature while R falls exponentially, the voltage rises, peaks and then comes back down — the characteristic doubles back on itself, and the region beyond the peak has negative incremental resistance. The condition for the turning point is exactly θ·P·B/T² = 1, which is the self-heating loop gain this page reports. A thermistor driven from a current source past that point runs away; one in series with a large enough resistor does not, because the resistor damps the loop.

Where the design questions live. This page owns the graphical method. The design decisions belong to the pages that specialise in them: the LED series resistor calculator picks the resistor and its wattage and rounds it to a standard value, the zener regulator calculator sizes the series resistor across a whole input and load range and gives the worst-case zener dissipation, the NTC thermistor calculator does the resistance-to-temperature conversion with Beta or Steinhart–Hart coefficients and designs the divider around it, and the Thévenin and Norton calculator reduces any resistive network feeding the device to the single supply and single resistance this page needs. Read this page to understand why those four give the answers they do.

What the model leaves out. For the diode, LED and zener branches there is no self-heating: the junction is taken to be at the ambient temperature you enter. In reality dissipation raises the junction, which lowers the forward voltage, which raises the current — a positive feedback that is mild for a small LED and is not mild for a power LED or a zener near its rating. Enter a higher ambient to see the effect, or use the temperature-change field to bracket it. There is also no series resistance inside the diode model, so at currents far above the quoted test point the modelled curve is steeper than a real part’s.

Frequently asked questions

Why not just use (V_s − V_f) ÷ R?

Because V_f is not a constant — it is a point on a curve, and which point depends on the current, which is what you are trying to find. The simple rule works when the device’s small-signal resistance is much smaller than the series resistor, which it usually is, and this page tells you the two numbers so you can see whether that holds. It also tells you what the simple rule cannot: how much the answer moves when the supply, the temperature or the part changes.

What is the difference between the small-signal resistance and V ÷ I?

V ÷ I is the static resistance — the slope of the line from the origin to the operating point. The small-signal resistance is the slope of the curve AT the operating point. For the LED in the worked example they differ by a factor of forty-three: 87.1 Ω static, 2.008 Ω incremental. The static value tells you the DC dissipation; the incremental value tells you how the circuit responds to change, and it is the one that matters for every sensitivity question.

The zener says no current flows. Is that right?

Yes, if the supply is below the knee voltage. The model passes no current below V_z, so the whole supply appears across the zener and nothing is regulated. It is the correct answer and a common real fault: either the series resistor is too big, the load is taking all the current, or the supply has sagged below the knee. Real zeners do leak a little below the knee, typically microamps, which this model ignores.

How much does the thermistor’s self-heating actually matter?

It depends entirely on the current you run it at. In a 10 kΩ divider from 3.3 V the power is microwatts and the rise is a small fraction of a degree, which is why most thermistor circuits ignore it. Push milliamps through the same bead and the rise is tens of degrees and the reading is meaningless. The self-heating rise is reported here; if it is a significant fraction of your measurement accuracy, lower the current or pulse the excitation.

Can I use this for a transistor or a MOSFET?

The method is the same and it is exactly how an output characteristic is read against a load resistor, but this page does not carry those device models. The four here are the two-terminal devices whose curves have closed forms. A transistor’s I-V curve is a family, one per base current or gate voltage, and the load line is drawn across all of them.

Why does the chart squeeze so many points near the knee?

Deliberately. Half the sweep covers the region just below the operating point and half covers everything above it, because an exponential device does all its interesting behaviour in a few tens of millivolts and an evenly spaced sweep would show a flat line and then a wall. The chart places each point at its own voltage rather than at its position in the sweep, so the shape is right even though the sampling is not uniform.

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References

W. Shockley, “The theory of p-n junctions in semiconductors and p-n junction transistors”, Bell System Technical Journal, vol. 28, no. 3, July 1949, pp. 435–489. The origin of the exponential diode equation used here, and of the ideality factor that is added to it in practice.
A. S. Sedra and K. C. Smith, Microelectronic Circuits, 8th edition, Oxford University Press 2020, chapter 4 (Diodes). The graphical analysis of a diode with a resistive load, the load line, the small-signal resistance r_d = nV_T/I and the constant-voltage-drop and piecewise-linear approximations are all set out there.
P. Horowitz and W. Hill, The Art of Electronics, 3rd edition, Cambridge University Press 2015, §1.6 and §1.7. The practical treatment: why a diode is driven through a resistor, what the Ebers–Moll relation implies for temperature drift (about −2 mV/°C at constant current), and the zener’s slope resistance.
IEC 60539 (directly heated negative temperature coefficient thermistors) defines the dissipation constant and the B value used in the thermistor branch. Cited by number; it is copyrighted and no values from it are reproduced here. The B-value relation and the self-heating loop gain condition θ·P·B/T² = 1 derived above were both checked numerically rather than taken from a table.