Resistor in parallel with an inductor, and a resistor in parallel with a capacitor.
When a resistor, R, is in parallel with an inductor, L, or a capacitor, C, the resultant impedance can be found in the Z-plane, which is the right half-plane of the complex number field, x + jy, where j is the "imaginary" square root of -1, as follows:Represent the resistor, R, as a vector directed along the x-axis as shown by the Red arrow in the figure below. Represent the inductor, L, by a vector along the positive jy axis, shown in blue. Its length will be the product of the angular frequency (omega) of the a.c. excitation voltage for the parallel combination times the inductance, L. This product is its inductive reactance.
Join the heads of the inductive reactance vector to the head of the resistance vector with the straight line, shown in green.
Find the vector, Z (subscript, L), which begins at the origin and terminates on the green line such that it intersects the green line at a right angle. The yellow circle, with diameter, R, contructed around the red R vector will also intersect the green line at the same point, and is the locus of such intersections as L, or omega, is changed.
Resistor in parallel with an inductor, and a resistor in parallel with a capacitor.
The proof of this is shown by writing an equation for the green line, and for the line on which the vector, Z, lies. The slope of the line for the Z vector is the negative inverse of the slope for the green line, because they are perpendicular to each other. The point of intersection of the the two lines is found by the simultaneous solution of the two equations for the two lines, and the coordinates of this point correspond to the real and imaginary parts of the impedance of the parallel combination.
For a resistor in parallel with a capacitor, the capacitive reactance is the reciprocal of the product of capacitance, C, and the angular excitation frequency, omega, as indicated in the figure.
The angular phase difference between the applied a.c. voltage and the a.c. current will be the angle between the Z vector and the real x-axis. Most elementary electrical engineering texts show the resultant vector sum of series connected resistances and reactances as the (parallelogram) vector sum. The parallel combinations are shown as vector sums in the admittance-plane, or Y-Plane, which is the complex inverse of the Z-plane. The advantage of the visualization shown here is that both series and parallel combinations can be treated in the same diagram. The value of this is in the synthesis of a.c. circuits which will be shown in subsequent web pages.