Relationships between Frequency, Capacitance, Inductance and ...

LPC Physics Relationships between f, C, L and X.
I
o
q( t)
= ∫ I
o
sin( 2 π ft) dt = cos( πft)
2πf
2
thus Vc can be written as:
I
o
VC
= cos( 2πft) = VC
cos( 2πft)
.
max
2πfC
We define the reactance of the capacitor as:
Thus,
this implies that we can calculate Xc by Eq. 3.
X
C
VCmax
X
C
= Eq. 3
I
max
⎛ I
o
⎞ 1
= ⎜ ⎟ I
o
=
Eq. 4
⎝ 2πfC
⎠ 2πfC
If an inductor is placed in a series circuit with a sinusoidal voltage supply, then the
current is given by:
ε
o
I
o
= where X
L
= 2πfL
X
L
In the same way, an inductor can be considered an AC resistor, with effective resistance
.
X L
Proof:
The voltage drop across an inductor is given by
di
V L
= L .
dt
If
i( t)
= I
o
sin( 2πft)
,
then
VL
= 2πfLI
o
cos 2πft
and we can define the inductive resistance X L by:
thus,
X
L
max
( )
VL
max
X
L
= Eq. 5
I
2πfLI
C
= = 2πfL
. Eq. 6
I
This implies that we can calculate Xc using Eq. 5 by measuring the voltage drop across
the inductor and the current through the circuit. We can also calculate X L from Eq. 6 with
knowledge of f and L.
o
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