Calculation of Inductance and Capacitance

Transformer-coupled amplifier performance is dependent at low frequencies upon transformer OCL, and at high frequencies upon leakage inductance and winding capacitance. Calculation of these quantities is essential in design and useful in tests for proper operation. Inductance formulas are repeated here for convenience, along with capacitance calculations.

[38]

where

N = turns in winding
A_c = core area in square inches
l_g = total length of air gap in inches
l_c = core length in inches
μ = permeability of core (if there is unbalanced direct current in the winding, this is the incremental permeability).

[33]

where

N = turns in that winding
MT = mean length of turn for whole coil
a = total winding height
b = winding width
c = insulation space
n = number of insulation spaces
= number of primary-secondary interleavings (see Fig. 57).

Winding capacitance is not expressible in terms of a single formula. The effective value of winding capacitance is almost never measurable, because it depends upon the voltages at the various points of the winding. The capacitance current at any point is equal to the voltage across the capacitance divided by the capacitive reactance. Since many capacitances occur at different voltages, in even the simplest transformer, no one general formula can suffice. The major components of capacitance are from

1.  Turn to turn.

2.  Layer to layer.

3.  Winding to winding.

4.  Windings to core.

5.  Stray (including terminals, leads, and case).

6.  External capacitors.

7.  Vacuum-tube electrode capacitance.

These components have different relative values in different types of windings. Turn-to-turn capacitance is seldom preponderant because the capacitances are in series when referred to the whole winding. Layer-to-layer capacitance may be the major portion in high-voltage single-section windings, where thick winding insulation keeps the winding-to-winding and winding-to-core components small. Items 5, 6, and 7 need to be watched carefully lest they spoil otherwise low-capacitance transformers and circuits.

If a capacitance C with E₁ volts across it is to be referred to some other voltage E₂, the effective value at reference voltage E₂ is

[70]

By use of equation 70 all capacitances in the transformer may be referred to the primary or secondary winding; the sum of these capacitances is then the transformer capacitance which is used in the various formulas and curves of preceding sections.

In an element of winding across which voltage is substantially uniform throughout, capacitance to a surface beneath is

[71]

where

A = area of winding element in square inches
ε = dielectric constant of insulation under winding = 3 to 4 for organic materials
t = thickness in inches of insulation under winding. This includes wire insulation and space factor.

Fig. 128. Transformer winding with uniform voltage distribution.

If the winding element has uniformly varying voltage across it, as in Fig. 128, the effective capacitance is the sum of all the incremental effective capacitances. This summation is

[72]

where

C = capacitance of winding element as found by equation 71
E₁ = minimum voltage across C
E₂ = maximum voltage across C
E = reference voltage for C_e.

If E₁ is zero and E₂ = E, equation 72 becomes

[73]

or the capacitance, say, to ground of a single-layer winding with its low-voltage end grounded is one-third of the measured capacitance of the winding to ground. Measurement should be made with the winding ungrounded and both ends short-circuited together, to form one electrode, and ground to form the other.

In a multilayer winding, E₁ is zero at one end of each layer and E₂ = 2E/N_L at the other, where E is the winding voltage and N_L is the number of layers. The effective layer-to-layer capacitance of the whole winding is

[74]

where C_L is the measurable capacitance of one layer to another. The first and last layers have capacitance to other layers on one side only, and this is accounted for by the term in parentheses in equation 74.

Because the turns per layer and volts per layer are greater in windings with many turns of small wire, such windings have higher effective capacitance than windings with few turns. In a transformer with large turns ratio, whether step-up or step-down, this effective capacitance is often the barrier to further increase of turns ratio. With a given load impedance across the low impedance winding, there is a maximum effective capacitance C_m which can be tolerated for a given frequency response. If layer and winding capacitances have been reduced to the lowest practicable figure C₁ the maximum turns ratio is Appreciable amounts of capacitance across which large voltages exist must be eliminated by careful design.

Since effective capacitance is greater at higher voltages, in step-down transformers the capacitance may be regarded as existing mainly across the primary winding, in step-up transformers across the secondary winding. The effect of this on frequency response has been discussed in High-Frequency Response.

The input capacitance of a triode amplifier is given by ⁽¹⁾

[75]

where

C_G-F = grid-to-cathode capacitance
C_G-P = grid-to-anode capacitance
α = voltage gain of the stage.

C_G-F and C_G-P are given for many tubes in the tube handbooks. They can be measured in any tube by means of a capacitance bridge.

(1)	See Principles of Radio Communications, by J. H. Morecroft, John Wiley & Sons, 2nd ed., New York, 1927, p. 511.