In the context of A.C. (alternating current) fundamentals, the expression of impedance (Z) in terms of the fractional detuning factor (Δf/f) and the quality factor (Q) is typically used when discussing resonant circuits. A resonant circuit consists of an inductor (L) and a capacitor (C) connected in either series or parallel configuration. The impedance of such a circuit depends on the frequency of the applied alternating current.
The fractional detuning factor is defined as:
Δ
=
−
f
Δf
=
f
r
f
r
−f
Where:
f
r
is the resonant frequency of the circuit.
f is the frequency of the applied alternating current.
The quality factor (Q) of a resonant circuit is defined as:
=
Δ
Q=
Δf
f
r
Now, let's express the impedance (Z) of a resonant circuit in terms of the fractional detuning factor (Δf/f) and the quality factor (Q).
For a series resonant circuit, the impedance (Z) is given by:
=
+
(
−
)
Z=R+j(X
L
−X
C
)
Where:
R is the resistance in the circuit.
j is the imaginary unit.
=
2
X
L
=2πfL is the inductive reactance.
=
1
2
X
C
=
2πfC
1
is the capacitive reactance.
For a parallel resonant circuit, the impedance (Z) is given by:
1
=
1
+
1
(
−
)
Z
1
=
R
1
+
j(X
L
−X
C
)
1
In both cases, you can substitute the expressions for
X
L
and
X
C
and simplify the impedance equations. However, directly expressing impedance in terms of the fractional detuning factor (Δf/f) and quality factor (Q) might not yield a simple expression due to the complex nature of the impedance equations involving reactances.
If you're specifically looking for an expression of impedance in terms of
Δ
f
Δf
and
Q, it's best to start with the fundamental impedance equations and work towards substituting the expressions for
Δ
f
Δf
and
Q as provided above.