Triode Valve / Tube Formulas & Theory

The amplification factor, anode or plate resistance; and the transconductance are some of the key factors associated with the triode valve / tube theory and formulas.

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When designing, repairing, or servicing triode valve / triode vacuum tube circuits it is very useful to have an understanding of the theory and what the different performance specifications mean.

The voltage and current relations in the triode for both anode and grid are of importance along with figures like triode amplification factor, the anode or plate resistance and the transconductance.

All of these give an understanding of the performance of a particular triode vale or triode vacuum tube.

Triode voltage & current relations

The number of electrons that reach the anode of a triode valve or vacuum tube under space charge limited conditions is primarily governed by the electrostatic field in the cathode grid region.

Once the electrons have passed through the grid they travel on towards the anode very rapidly and space charge effects can normally be ignored, especially to a first approximation which is normally good enough for most calculations.

The critical area of the triode valve is within the cathode grid space. It is here that the theory needs to be examined to determine its operation.

In the cathode – grid area the electrostatic field is determined by both the grid and anode or plate.

Electrostatic shielding theory shows that the electrostatic field in the vicinity of the cathode of a triode is proportional to (Ec + Eb/µ), where Ec and Eb are the grid and anode voltages respectively. The voltages are measured with respect to the cathode. µ is the amplification factor of the valve.

Triode amplification factor µ

The value µ is the constant known as the amplification factor of the valve or vacuum tube – it applies to triodes and is not really applicable to tetrodes or pentodes. It is independent of the voltages on the grid and anode and is determined by the geometries of the elements within the valve. Typically of the grid is placed close to the cathode this will give it a high amplification factor. For most triodes the amplification factor falls within the region of 10 to 100.

The amplification factor µ of a triode valve / vacuum tube is a measure of the relative effectiveness of the grid and anode voltages in producing the electrostatic fields at the surface of the cathode.

In more practical terms the amplification factor, µ of a triode can be considered to be the theoretical maximum gain that can be obtained. The amplification factor is based on the variation of anode voltage to grid voltage, but it is measured with the anode current held constant.

µ = Δ V a / Δ V g

    µ = amplification factor
    ΔVa = change in anode voltage
    ΔVg = change in grid voltage

Triode characteristic curves

The performance and characteristics of triode valves or vacuum tubes are often represented by a number of graphs detailing their performance.

The characteristic curves or graphs are normally plotted for the relationship of the grid voltage and anode or plate current, and for the relationship of the anode or plate voltage and the corresponding current.

Typical triode grid voltage and anode current characteristic
Typical triode grid voltage and anode current characteristic (Eb = anode voltage)

The various curves of grid voltage against anode current all have approximately the same shape, differing mainly in the displacement from each other. This results from the fact that the anode current is determined by the equation (Ec + Eb/µ).

Typical relationship between anode current and anode current for a triode valve / tube
Typical relationship between anode current and anode current for a triode valve / tube (Ec= grid voltage)

In a similar way that the curves for the grid voltage and anode current are similar, so too are those for anode voltage and current, although it can be seen that the curves for positive grid voltage are rather different.

Anode resistance

The anode resistance or plate resistance is more exactly described as the dynamic anode or plate resistance. It represents the resistance that the anode circuit offers to a small change in voltage.

Therefore when a small increment in anode voltage ΔEb produces a small change in anode current ΔIb the anode resistance can be calculated as follows:

rp = Δ E b / Δ I b = δ E b / δ I b

    rp = dynamic anode resistance

Triode mutual conductance or transconductance

The transconductance or mutual conductance gm of a triode is defined as the rate of change of anode current with respect to the grid voltage.

It is possible to express this as a simple equation:

Δ I b = Δ Ec gm
gm = δ I b / δ E c = µ / rp

    µ = mutual conductance / transconductance
    rp; = anode resistance

The transconductance or mutual conductance is a form of conductance, i.e. the inverse of Ohms. As a result the units in which they were quoted where mhos (Ohm spelt backwards). Nowadays the unit of conductance is the Siemens (S), but for valves / tubes the unit mho is still used.

For valves the figures were normally quoted in µmhos, so be aware as this would make gain figures enormous if the µ was missed.

In very much later calculations associated with valves, gain figures started to be given in terms of mA/ V, where the voltage (V) was applied to the grid, and the current (mA) was the change of plate current for a 1V change of grid current.

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