Coaxial Quaternions

Excerpts

• Coaxial Quaternions

  By a “quaternion” is meant the operator which changes one vector into another. It is composed of a magnitude and a turning factor.

• Coaxial Quaternions

  Let $A$ and $R$ be two coinitial vectors; the direction normal to the plane may be denoted by $\beta$. The operator which changes $A$ into $R$ consists of a scalar multiplier and a turning round the axis $\beta$.

• Coaxial Quaternions

  The resistance is the scalar part of the quaternion, and the inductance is the vector part.

• Coaxial Quaternions

  Etymologically “quaternion” means defined by four elements; which is true in space; in plane analysis it is defined by two.

• Coaxial Quaternions

  Note that the product is formed by taking the product of the magnitudes, and likewise the product of the turning factors.

• Coaxial Quaternions

  The angles are summed because they are indices of the common base $\beta$.

• Coaxial Quaternions

  This is the fundamental error in the Argand method.

Equations

• Coaxial Quaternions

  \beta^\theta = \cos\theta \cdot \beta^\theta + \sin\theta \cdot \beta^\frac{\pi}{2}

  The turning factor through angle theta is the sum of a turning through theta weighted by cos theta and a quadrantal turning (angle pi/2 about beta) weighted by sin theta.

• Coaxial Quaternions

  R = r\beta^\theta A

  The vector R is obtained from the vector A by a scalar multiplier r and a turning through angle theta about the axis beta, which is the quaternion that changes A into R.

• Coaxial Quaternions

  A = \dfrac{1}{r}\beta^{-\theta}R

  Conversely, A is obtained from R by the reciprocal multiplier 1/r and a turning through minus theta about beta.

• Coaxial Quaternions

  \dfrac{1}{A}R = r\beta^\theta

  The quotient of R by A is the quaternion r beta^theta that changes A into R.

• Coaxial Quaternions

  r = \sqrt{p^2 + q^2}

  The scalar multiplier r is the square root of the sum of the squares of p and q.

• Coaxial Quaternions

  \theta = \tan^{-1} \frac{p}{q}

  The book gives the angle theta as the arctangent of p/q. FLAG: from p = r cos(theta) and q = r sin(theta) the tangent of theta is q/p, so this appears to be a book error (or a typo for q/p); recorded as printed, not corrected.

• Coaxial Quaternions

  E = \left(r + 2\pi n l \cdot \beta^\frac{\pi}{2} \right) I

  For a sine alternating circuit, the impressed electromotive force equals the current operated on by the quaternion with resistance as scalar part and 2 pi n times self-induction as vector part.

• Coaxial Quaternions

  I^{-1} E = r + 2\pi n l \cdot \beta^\frac{\pi}{2}

  The operator that changes the current into the electromotive force is the quaternion with scalar part r (resistance) and vector part 2 pi n l (inductance).

• Coaxial Quaternions

  R = \left(p + q \cdot \beta^\frac{\pi}{2} \right) A

  The vector R is obtained from A by the quaternion with real part p and quadrantal part q.

• Coaxial Quaternions

  I = \left\{\frac{r}{r^2+(2\pi nl)^2} - \frac{2\pi nl}{r^2+(2\pi nl)^2}\cdot \beta^\frac{\pi}{2}\right\}E

  The current in the circuit is found from the impressed electromotive force by the reciprocal of the impedance quaternion.

• Coaxial Quaternions

  \sum R = \left\{\sum p + \left(\sum q\right) \cdot \beta^\frac{\pi}{2}\right\}A

  The resultant of several coaxial vectors R_i, each given as a quaternion times A, is the quaternion whose parts are the sums of the parts p_i and q_i, applied to A.

• Coaxial Quaternions

  A = \frac{\sum p - \left(\sum q \right) \cdot \beta^\frac{\pi}{2}} {\left( \sum p\ \right)^2 + \left( \sum q \right)^2}\sum R

  Conversely, the common vector A is recovered from the sum of the vectors R by the reciprocal of the summed quaternion.

• Coaxial Quaternions

  I_1 = \frac{r_1 - 2\pi nl_1 \cdot \beta^\frac{\pi}{2}}{r_1^2 + (2\pi n)^2 l_1^2}E

  For the first of several circuits in parallel, the current is the reciprocal impedance quaternion applied to the common electromotive force E.

• Coaxial Quaternions

  E = \frac{ \sum\left(\frac{r}{r^2 + (2\pi n)^2 l^2}\right) + 2\pi n\sum\left(\frac{l}{r^2 + (2\pi n)^2 l^2}\right) \cdot \beta^\frac{\pi}{2}} {\left(\sum\frac{r}{r^2 + (2\pi n)^2 l^2}\right)^2 + (2\pi n)^2\left(\sum\frac{l}{r^2 + (2\pi n)^2 l^2}\right)^2} \sum I

  For circuits in parallel, the common electromotive force is the reciprocal of the sum of the individual admittance quaternions, applied to the total current I. Attributed in the book to Lord Rayleigh (Phil. Mag., May 1886).

• Coaxial Quaternions

  R' = rr'\beta^{\theta+\theta'}A

  The product of two successive coaxial quaternions has magnitude equal to the product of the magnitudes and turning angle equal to the sum of the angles.

Problems