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Figure 27. Circuit with a capacitor
Figure 28. Circuit with inductor
For the equation 
idt =Cdu
, integrating both sides yields the following formula:
∫
u
0
u
Cdu =
∫
t
0
t
idt
, or 
u(t)=
1
C
∫
t
0
t
idt + u
0
(
t
0
)
If 
t
0
=0
, above equation can be simplified to
u(t)=
1
C
∫
t
0
t
idt + u
0
(0)
and for 
t
0
= −∞
, above equation reduces to
u(t)=
1
C
∫
−∞
t
idt
The initial voltage in above equation, 
u
0
(
t
0
)
, is usually defined with the same polarity as 
u
,
which means 
u
0
(
t
0
)
is a positive quantity. If the polarity of 
u
0
(
t
0
)
is in the opposite direction,
then 
u
0
(
t
0
)
is negative.
Advances in Bioengineering
210

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4.3. Inductor
An inductor in figure 28 is a passive element that is to store energy in magnetic field and is
made by winding a coil of wire around a core that is a insulator or a ferromagnetic material.
A magnetic field is established when current flows through the coil. The symbol 
is utilized to represent the inductor in a circuit. The unit of measurement for inductance is the
Henry or Henries (H). The relationship between voltage and current for inductor is given by
u = L
di
dt
The convention for writing the voltage drop across an inductor is similar to that of a resistor.
Physically, current cannot change instantaneously through a inductor since an infinite voltage
required. Mathematically, a step change in current through an inductor is possible by applying
a voltage. For convenience, when a circuit has just DC currents (or voltages), the inductors can
be replaced by short circuits, since voltage drops across the inductors are zero.
After producing current on the both sides of equation, the following expression can be acquired
after integration:
∫
0
t
uidt =
∫
0
i
Lidi =
1
2 L i
2
Above expression demonstrates that magnetic energy increases with the increase of current
through inductor component. In this course, electrical energy could be converted into magnetic
energy, namely inductor acquires energy from the source. Formula 
1
2 L i
2
is the magnetic
energy of inductive element. When current decreases, magnetic energy decreases and then is
converted into electric energy, namely inductor releases energy to the source. Hence, inductor
is not a dissipative element, but a energy storage element, too.
For the equation 
udt = Ldi
, integrating both sides yields the following formula:
∫
t
0
t
u(t)dt =
∫
i
(
t
0
)
i(t)
Ldi
, or, 
i(t)=
1
L
∫
t
0
t
u(t)dt + i
(
t
0
)
If 
t
0
=0
, above equation can be simplified to
i(t)=
1
L
∫
t
0
t
u(t)dt + i(0)
and for 
t
0
= −∞
, above equation reduces to
i(t)=
1
L
∫
−∞
t
u(t)dt
Biomedical Sensor, Device and Measurement Systems
http://dx.doi.org/10.5772/59941
211

---

The initial current in above equation, 
i
(
t
0
)
, is usually defined with the same polarity as 
i
, which
means 
i
(
t
0
)
is a positive quantity. If the polarity of the initial current 
i
(
t
0
)
is in the opposite
direction, then 
i
(
t
0
)
is negative.