1.4 Polynomial Equations, Conic Sections & Complex Numbers

Polynomial Equations and Root Finding

Welding engineering problems produce polynomials whenever a cross-sectional area, volume or cost is expressed in terms of a single linear dimension. A fillet weld area grows with the square of the leg size; a groove volume grows with the square of thickness for a fixed included angle; a bead deposited as a parabolic cap contributes a term in the cube of the plate thickness when integrated over length.

The Quadratic Case

For $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant $b^2 - 4ac$ decides the character of the roots: positive gives two distinct real roots, zero gives one repeated root, and negative gives a complex conjugate pair. In a physical sizing problem a negative discriminant means no real leg size satisfies the requirement, which is a design answer, not an arithmetic error.

Worked case. An equal-leg fillet weld must supply a cross-sectional deposit area of $A = 45\text{ mm}^2$ including a 10% reinforcement allowance. For an equal-leg fillet, $A = 0.5 w^2 (1.10)$, so $0.55 w^2 - 45 = 0$ and $w = \sqrt{45/0.55} = 9.05\text{ mm}$. The negative root is discarded on physical grounds.

Higher-Order Polynomials

Newton-Raphson iterates

$x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}$

and converges quadratically near a simple root. It is the standard method when an expression such as a Rosenthal peak-temperature relationship cannot be inverted in closed form.

Curve Fitting and Interpolation

Clause 8.1.1 also lists graphs and equations, interpolation and extrapolation. Hardness-versus-cooling-time data, flux basicity-versus-toughness data and deposition-rate-versus-current data are all routinely fitted by low-order polynomials. Two rules keep the fit honest:

1. Interpolate freely, extrapolate cautiously. A cubic fitted to five data points can swing wildly just outside the fitted range.
2. Do not fit a higher-order polynomial than the data supports. A fourth-order fit through five points passes through every point exactly and tells you nothing about the underlying physics.

Conic Sections in Weld Geometry

The Parabola

A parabola has the form $y = ax^2 + bx + c$, or in vertex form $y - k = a(x - h)^2$. Parabolas appear throughout welding:

• Penetration profile. A carbon-dioxide-shielded GMAW arc produces a broad, deep, roughly parabolic penetration profile, in contrast to the narrow finger of an argon-rich spray arc.
• Weld bead cap. A reinforcement cap is commonly modelled as a parabolic segment of width $w$ and height $h$, with cross-sectional area $A = \tfrac{2}{3} w h$. This is the standard approximation used in consumable take-off calculations.
• Projectile and deposition trajectories. Droplet trajectory across an arc gap under gravity and drag is parabolic to first order.

The Hyperbola

A rectangular hyperbola has the form $xy = k$, meaning one variable is inversely proportional to the other. The most important welding instance is the relationship between travel speed and heat input at fixed arc power:

$H = \frac{60 V I}{1000 S}$

With $V$ and $I$ held constant, $H \cdot S = \text{constant}$, so heat input and travel speed trace a rectangular hyperbola. Doubling travel speed halves heat input. A candidate who recognises the hyperbolic form can answer a whole family of exam questions without recomputing.

The general conic form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ is classified by the discriminant $B^2 - 4AC$: negative gives an ellipse, zero a parabola, positive a hyperbola.

The Ellipse

An elliptical weld pool of semi-axes $a$ and $b$ has area $\pi a b$. Pool elongation is the physically meaningful quantity: as travel speed rises the pool stretches from near-circular into a long teardrop, and the resulting columnar grain orientation drives centreline solidification cracking.

Complex Numbers and AC Welding Circuits

Complex numbers exist on the CWEng Body of Knowledge because AC welding power is an AC circuit problem. Electrical engineering uses $j$ rather than $i$ for the imaginary unit to avoid collision with current.

A complex number $z = R + jX$ has:

$|z| = \sqrt{R^2 + X^2} \qquad \theta = \arctan\!\left(\frac{X}{R}\right)$

and the polar form $z = |z| \angle \theta$.

Impedance, Not Just Resistance

An AC welding circuit contains resistance $R$, inductive reactance $X_L = 2\pi f L$ and capacitive reactance $X_C = 1/(2\pi f C)$. The total impedance is

$Z = R + j(X_L - X_C)$

$|Z| = \sqrt{R^2 + (X_L - X_C)^2}$

Ohm's law in AC form is $V = I |Z|$, and the current lags the voltage by the phase angle $\theta$ when the circuit is net inductive — which a welding transformer always is.

Power Factor

$\text{PF} = \cos\theta = \frac{R}{|Z|} = \frac{\text{true power (W)}}{\text{apparent power (VA)}}$

Worked case. A welding transformer draws from a supply presenting $R = 0.80\ \Omega$ and $X_L = 0.60\ \Omega$.

$|Z| = \sqrt{0.80^2 + 0.60^2} = \sqrt{0.64 + 0.36} = 1.00\ \Omega$ $\theta = \arctan(0.60/0.80) = 36.87^\circ \qquad \text{PF} = \cos 36.87^\circ = 0.80$

A power factor of 0.80 means the utility must supply 125 VA for every 100 W of real power actually converted at the arc. That is why power factor correction capacitors are fitted to large transformer-rectifier installations: they reduce $X_L - X_C$, shrink $|Z|$ toward $R$, and drive the power factor toward unity.

Exam Trap 1: Adding resistance and reactance arithmetically. Impedance magnitude is the square root of the sum of squares, never $R + X$. In the example above, $0.80 + 0.60 = 1.40\ \Omega$ would overstate the impedance by 40%.

Exam Trap 2: Confusing power factor with efficiency. Power factor is the phase relationship between voltage and current; efficiency is the ratio of useful output to input energy. A machine can have an excellent power factor and poor thermal efficiency, or the reverse.

Exam Trap 3: Keeping a negative root. Quadratics for leg size, thickness or diameter always produce one negative root. It is mathematically valid and physically meaningless, and Descartes' rule of signs tells you to expect it.