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Series and Parallel Equivalent Circuits
The impedances of Actual resistors, inductors and capacitors are combinations of resistance,
inductance, and capacitance. The simplest models for actual inductors and capacitors are the series
and parallel equivalent circuits shown in Figure 1-1.
For example, the complex impedance of an inductor is
Z = R + jX = R + jωL
=
R p ⋅ jωL p
R p + jωL p
8a - Inductor Series
Equivalent
=
R p + jR p (R p / ωL p )
1 + (R p / ωL p )
2
8b - Inductor Parallel
Equivalent
(series equivalent circuit)
(8a)
(parallel equivalent)
(8b)
9a -Capacitor Series
Equivalent
9b - Capacitor Parallel
Equivalent
Figure 1-1 Equivalent circuits for inductors and capacitors.
The complex impedance of a capacitor is
Z = R + jX = R − j / ωC
=
R p ⋅ 1 / jω C p
R p + 1 / jωC p
=
(series equivalent circuit)
Rp
1 + jω C p R p
=
R p (1 − jωC p R p )
1 + (ωC p R p )
2
3
(parallel equivalent)
(9a)
(9b)