518
N. Yadav and R. Dhar
where, ϕ is the phase difference between current and voltage. It is zero for purely
resistive circuit. Both current (I
* ) and voltage (V
* ) are functions of frequency and
time, but for brevity, I
* (f, t) and V
* (f, t) shall be written as I and V in the following
text. Fourier transform over n periods is given by:
I = I
∗
( f, t) = I
+ j I
=
2
nT
nT
∫
0
I 0 e
jωt dt
(3)
where,
I 0 =
(I )
2
+ (I )
2
; tan(ϕ) =
I
I
(4)
In the time domain, the connection between the properties of the system and its
corresponding response to external stimuli is very complex. So, it requires a solution
of a system of differential equations. The output of the inductive and capacitive
elements is given in the form of V L = L(dI/ dt) and I C = C(dV/ dt). Here, L and C
refer inductance and capacitance respectively.
These elements or their combinations can lead to many difficult complex problems. Therefore, Fourier transform is required because it can considerably simplify
the mathematical treatment of the system. In addition to the phase detection, the
Fourier transform suppresses all frequency components in I(f,t) except a narrow band
centered around the generator frequency. This improves accuracy, reduces noise and
direct current drifts by several orders of magnitude. Finally, the impedance (Z) or
admittance (Y ) are calculated by the following relations [13]:
Impedance, Z = Z
∗
( f ) = Z
+ j Z
= V
∗
/I
∗
(5)
Admittance, Y = Y
∗
( f ) = Z
∗−1
= Y
+ jY
(6)
It is important to mention that in the case of series combination of circuit elements,
impedance analysis given by Eq. (5) whereas in the case of parallel combination of
circuit elements, admittance analysis given by Eq. (6) is easy to handle. V, I and Z
are considered to be complex in the complex domain and can be written as I = YV
or V = ZI.
Impedance is a complex vector quantity which may be represented with either
polar or rectangular coordinates in a plane (refer Fig. 2). It is real and totally frequency
independent only for purely resistive response i.e. when phase angle is equal to zero.
When it varies with frequency, the Kronig-Kramers equations which link the real
and imaginary parts guarantee that ϕ = 0 over the entire range of frequencies but
should change with frequency also [6].
N. Yadav and R. Dhar
where, ϕ is the phase difference between current and voltage. It is zero for purely
resistive circuit. Both current (I
* ) and voltage (V
* ) are functions of frequency and
time, but for brevity, I
* (f, t) and V
* (f, t) shall be written as I and V in the following
text. Fourier transform over n periods is given by:
I = I
∗
( f, t) = I
+ j I
=
2
nT
nT
∫
0
I 0 e
jωt dt
(3)
where,
I 0 =
(I )
2
+ (I )
2
; tan(ϕ) =
I
I
(4)
In the time domain, the connection between the properties of the system and its
corresponding response to external stimuli is very complex. So, it requires a solution
of a system of differential equations. The output of the inductive and capacitive
elements is given in the form of V L = L(dI/ dt) and I C = C(dV/ dt). Here, L and C
refer inductance and capacitance respectively.
These elements or their combinations can lead to many difficult complex problems. Therefore, Fourier transform is required because it can considerably simplify
the mathematical treatment of the system. In addition to the phase detection, the
Fourier transform suppresses all frequency components in I(f,t) except a narrow band
centered around the generator frequency. This improves accuracy, reduces noise and
direct current drifts by several orders of magnitude. Finally, the impedance (Z) or
admittance (Y ) are calculated by the following relations [13]:
Impedance, Z = Z
∗
( f ) = Z
+ j Z
= V
∗
/I
∗
(5)
Admittance, Y = Y
∗
( f ) = Z
∗−1
= Y
+ jY
(6)
It is important to mention that in the case of series combination of circuit elements,
impedance analysis given by Eq. (5) whereas in the case of parallel combination of
circuit elements, admittance analysis given by Eq. (6) is easy to handle. V, I and Z
are considered to be complex in the complex domain and can be written as I = YV
or V = ZI.
Impedance is a complex vector quantity which may be represented with either
polar or rectangular coordinates in a plane (refer Fig. 2). It is real and totally frequency
independent only for purely resistive response i.e. when phase angle is equal to zero.
When it varies with frequency, the Kronig-Kramers equations which link the real
and imaginary parts guarantee that ϕ = 0 over the entire range of frequencies but
should change with frequency also [6].