Equations of a circle

Equation (6.78) can also be expressed as an equation of a circle of the form... 

To determine the shape of the pressure profile it is necessary to express h as a function of x. From the equation of a circle it may be seen that... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Exercise Solve the problem stated at the beginning of this section to obtain the equation of a circle. Note that / = y and g — Vl + y 2. 

The equation of a circle with its center at the origin and with a radius of 5 is x2 + y2 = 25. Replacing the x in this equation with 2y and solving for y,... 

This is the equation of a circle, centre [(es + e00)/2,0], radius (es- eoc)/2, so that a plot of e" against e should give a semicircle, as shown in Fig. 3.3. Experimental results for many polar liquids give excellent agreement with this theoretical curve, their relaxation times being of the order of 10" 11 s. 

By expanding (2) it is seen that the general equation of a circle is ... 

This semicircle shape can be algebraically evidenced in showing that the real and imaginary parts satisfy the equation of a circle centered at a point on the real axis at the distance corresponding to the circle radius R/2 ... 

The equation of a circle in a phase diagram of T versus < ) can be written as... 

This is the equation of a circle with its center on the real axis at... 

We indeed obtain the equation of a circle with center (R/2-, 0) and radius R/2. 

In the complex representation (Nyquist diagram), it corresponds to the equation of a circle centered on R/2 with a radius of R/2. The impedance Z tends toward the material intrinsic resistance R when the frequency tends toward 0 and toward 0 when the frequency tends towards the infinite. The maximum of amplitude of the imaginary part Z" is obtained at the resonance ... 

Then, the equation of a circle for charge-control rnechanism becomes [22]... 

FIGURE 8.20 Schematic representation of the method used to determine the voitages on the retarding ring eiectrodes of a curved-field reflectron. The voltages placed on the reflectron lenses are determined from the equation of a circle = + x where V is a voltage, x is distance, and... 

This is the equation of a circle of radius %I1 centered aroimd the point +/t/2/ e" = 0 therefore a Debye-type process should produce a semicircle (no negative e" values are possible). 

This is the equation of a circle with its centre on the 2 -axis at Z = R + 0.5 Ret and radius 0.5 Ref From the intersection point of the semicircle with the R -axis, one obtains the value Rq + R t and the high-frequency ohmic component is equal to Rq. Let be the frequency at which 2" exhibits a maximum vs Z. For the maximum -2" = Z -Rq it follows that ... 

Exercise 2.5 A very inefficient way to compnte the value of % is as follows Consider a circle of radius Vi circumscribed by a unit sqnare (each side is 1 unit). Recall that the equation of a circle is... 

This is the equation of a circle in the Re(Z), lm(Z) plane, centered at R/2, of radius R/2 Note RC (or t) are positive, so that only half of the circle has a physical meaning. Measurements of Z as a function of the frequency scan this semi-circle when plotting -lm(Z) as a function of Re(Z) This is the Cole-Cole plot (Fig. 13.21). 

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