# Real Resistors, Capacitors and Inductors.

Cuthbert Nyack
The equivalent circuits of real Resistors, Inductors and Capacitors are shown below. The Resistor has a capacitor Cr across its leads. The produces 1 break frequency fr = 1/(2 p R Cr) above which the impedance of the real resistor decreases by 20 dB/dec.

The real Inductor has a capacitor Cl across its leads and a resistor Rl of the coils of the wire. The Impedance has 2 break frequencies. at the low frequency end the impedance does not go to zero as jwL suggests but is limited by Rl below the break frequency fl1 = Rl/(2 p L).
At high frequencies, the impedance does not increase indefinitely but has a second break frequency fl2 = 1/Ö(L Cl) where the Inductor has a parallel(maximum impedance) self resonating frequency with Cl. At this break frequency the impedance slope changes from +20dB/dec to -20dB/dec.

The capacitor has a lossy resistor resistor RL in parallel with it which is important for charge storage and low frequency applications but is not included in the analysis here. Because of the resistor Rc, the impedance of the real capacitor does not decrease indefinitely but is limited by Rc for frequencies higher than the break frequency fc1 = 1/(2 p Rc C). At still higher frequencies, the impedance of the inductor Lc becomes larger than Rc and the capacitor has a second break frequency at fc2 = Rc/(2 p Lc). If fc2 is less than fc1 then the capacitor has a series(minimum impedance) self resonating frequency at the average of fc1 and fc2. (Arithmetic average of logs, geometric average of frequencies). Applet below show the behaviour of the Impedance of the Components with frequency. The numerical values of the impedances are given in terms of R, C and L. In practice the break frequencies will be measured and the components calculated as shown on the applet.