In complex numbers, resistance R is a real term and reactance is a j term. Thus, an 8-Ω R is 8; an 8-Ω XL is j8; an 8-Ω XC is -j8. The general form of a complex impedance with series resistance and reactance, then, is ZT = R ± jX in rectangular form.
The same notation can be used for series voltages where V = VR ± jVX.
For branch currents IT = IR ± jIX, but the reactive branch currents have signs opposite from impedances. Capacitive branch current is and inductive branch current is -jIL.
The complex branch currents are added in rectangular form for any number of branches to find IT.
To convert from rectangular to polar form: <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072988215/363997/24_05sum.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a>. The angle is 0Z. The magnitude of ZT is <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072988215/363997/24_05bsum.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> Also, 0Z is the angle with tan = X/R.
To convert from polar to rectangular form, <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072988215/363997/24_06sum.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> where R is ZT cos 0Z and the j term is ZT sin 0Z. A positive angle has a positive j term; a negative angle has a negative j term. Also, the angle is more than 45°for a j term larger than the real term; the angle is less than for a j term smaller than 45° the real term.
The rectangular form must be used for addition or subtraction of complex numbers.
The polar form is usually more convenient in multiplying and dividing complex numbers. For multiplication, multiply the magnitudes and add the angles; for division, divide the magnitudes and subtract the angles.
To find the total impedance of a series circuit, add all resistances ZT for a the real term and find the algebraic sum of the reactances for the j term. The result is ZT = R ± jX . Then convert ZT to polar form for dividing into the applied voltage to calculate the current.
To find the total impedance ZT of two complex branch impedances Z1 and Z2 in parallel, ZT can be calculated as Z1Z2/(Z1 + Z2).
