Handbook for Sound Engineers

Resistors, Capacitors, and Inductors 267

I is the current in amperes.

The inductance of single-layer, spiral, and multi-
layer coils can be calculated by using either Wheeler’s
or Nagaoka’s equations. The accuracy of the calculation
will vary between 1% and 5%. The inductance of a
single-layer coil, Fig. 10-25A, may be found using
Wheeler’s equation

(10-37)

For the multilayer coil, Fig. 10-25B, the calculations are

(10-38)

For the spiral coil, Fig. 10-25C, the calculations are:

(10-39)

where,

B is the radius of the winding,
N is the number of turns in the coil,
A is the length of the winding,
C is the thickness of the winding,

L is in PH.

Q. Q is the ratio of the inductive reactance to the
internal resistance of the coil. The principal factors that
affect Q are frequency, inductance, dc resistance, induc-
tive reactance, and the type of winding. Other factors
are the core losses, the distributed capacity, and the
permeability of the core material. The Q for a coil
where R and L are in series is

(10-40)

where,
f is the frequency in hertz,

L is the inductance in henrys,
R is the resistance in ohms.

The Q of the coil can be measured as follows. Using

the circuit of Fig. 10-26, Q of a coil may be easily

measured for frequencies up to 1 MHz. Since the

voltage across an inductance at resonance equals Q×V,

where V is the voltage developed by the oscillator, it is

necessary only to measure the output voltage from the

oscillator and the voltage across the inductance.

The voltage from the oscillator is introduced across a

low value of resistance R, about 1% of the anticipated

radiofrequency resistance of the LC combination, to

assure that the measurement will not be in error by more

than 1%. For average measurements, resistor R will be

on the order of 0.10:. If the oscillator cannot be oper-

ated into an impedance of 0.10: , a matching trans-

former may be employed. It is desirable to make C as

large as convenient to minimize the ratio of the imped-

ance looking from the voltmeter to the impedance of the

test circuit. The voltage across R is made small, on the

order of 0.10 V. The LC circuit is then adjusted to reso-

nate and the resultant voltage measured. The value of Q

may then be equated

. (10-41)

The Q of a coil may be approximated by the equation

(10-42)

where,

f is the frequency in hertz,

L is the inductance in henrys,

R is the dc resistance in ohms as measured by an

ohmmeter,

XL is the inductive reactance of the coil.

Time Constant. When a dc voltage is applied to an RL

circuit, a certain amount of time is required to change

the voltage. In a circuit containing inductance and resis-

Figure 10-25. Single- and multilayer inductors.

L B

(^2) N 2
9 B+ 10 A
-----------------------=
L 0.8B
2
N
2
6 B++ 9 A 10 C
=-------------------------------------
L B
(^2) N 2
8 B+ 11 C
=-----------------------
A. Single layer B. Multilayer C. Spiral
A A
B B C B C
Q^2 SfL
R
------------=
Figure 10-26. Circuit for measuring the Q of a coil.
Osc V
L
C
R Voltmeter
Q Resonant voltage across C
Voltage across R
=----------------------------------------------------------------
Q^2 SfL
R
------------=
XL
R
----- -=