Triode operation, characteristic curves, and operating point

A triode can be understood as a three-electrode device in which a small variation of grid voltage controls a much larger variation of plate current.

To build a credible SPICE model of the 6BX7-GT it is worth starting from here: understanding what the curves represent, how the operating point is defined, and which local quantities describe the tube behavior.

1. What a triode is

A triode contains three active electrodes: cathode, control grid, and anode, also called the plate. The cathode is brought to emission temperature by a filament or by a heater; in the 6BX7-GT the heating is indirect, so the heater is electrically distinct from the cathode.

When the cathode is hot, thermionic emission makes electrons available in the surrounding space. If the anode is positive with respect to the cathode, the electrons are attracted toward it. The grid, placed between cathode and anode, modifies the electric field and therefore controls how many electrons reach the plate.

At a normal amplifier operating point, the grid is negative with respect to the cathode. Making it more negative reduces plate current; making it less negative increases plate current. This control action is the basic principle of triode amplification.

2. Which voltages matter: always with respect to the cathode

In real circuits the cathode is not necessarily at ground. For this reason it is useful to write the interelectrode voltages explicitly:

\[
V_{ak}=V_a-V_k
\]
\[
V_{gk}=V_g-V_k
\]

\(V_{ak}\) is the anode-to-cathode voltage; \(V_{gk}\) is the grid-to-cathode voltage. The latter is the true control variable of the triode.

A very simple example: if the grid is at 0 V and the cathode sits at +16 V because of the cathode resistor, then

\[
V_{gk}=0-16=-16\ \mathrm{V}.
\]

The grid is therefore negatively biased with respect to the cathode even though, when measured to ground, it is at 0 V.

3. A real example: 6BX7-GT data

The 6BX7-GT is an octal twin triode: two triode sections are contained in the same envelope. The General Electric datasheet gives, for each section in a typical operating condition, an amplification factor of about 10, an internal plate resistance of about 1300 \(\Omega\), and a transconductance of about 7600 \(\mu\)mho, i.e. 7.6 mS.

These values allow an immediate check of the fundamental triode relation:

\[
\mu \approx g_m r_p
\]

Using the datasheet values:

\[
7.6\ \mathrm{mS}\times1.3\ \mathrm{k\Omega}=9.88\approx10.
\]

The same datasheet also reports, for a typical condition at 250 V with a 390 \(\Omega\) cathode resistor, a plate current of about 42 mA. The voltage drop across the cathode resistor is approximately

\[
V_k=I_aR_k=0.042\times390\approx16.4\ \mathrm{V}.
\]

If the grid is at 0 V, the effective bias is therefore about \(V_{gk}=-16.4\) V. This example shows why, in tube circuits, thinking in terms of grid-to-cathode voltage is more useful than speaking simply of “grid voltage”.

4. How to read plate curves

The most important graph of a triode is the family of curves \(I_a=f(V_{ak})\) for different values of \(V_{gk}\). The horizontal axis shows anode-to-cathode voltage, the vertical axis shows plate current, and each curve corresponds to a different grid bias.

To read the graph, one can proceed in three steps: choose the curve corresponding to the desired \(V_{gk}\), move to the required \(V_{ak}\), and read the corresponding \(I_a\). If, for example, anode voltage is held constant and the grid is made less negative, one moves toward curves with higher current.

The curvature is not a drawing artifact: it represents the actual nonlinearity of the device. The local slope changes with the operating point, so \(g_m\), \(r_p\), and the effective gain are not constant over the whole graph.

5. The transfer curve

Another way to observe the same device is to keep plate voltage fixed and plot current versus grid voltage:

\[
I_a=f(V_{gk})\quad\text{at}\quad V_{ak}=\text{constant}.
\]

For very negative grid voltages the current becomes small: this is the region near cutoff. As \(V_{gk}\) becomes less negative, current rises rapidly. The transition is not an ideal threshold: it is gradual and also depends on plate voltage.

6. The three local parameters: μ, gm, and rp

6.1 Amplification factor μ

The amplification factor \(\mu\) expresses, locally, how much a change in plate voltage must compensate a change in grid voltage in order to keep current constant:

\[
\mu=-\left.\frac{\Delta V_{ak}}{\Delta V_{gk}}\right|_{I_a}.
\]

It is dimensionless. A high \(\mu\) indicates strong grid control compared with the anode.

6.2 Transconductance gm

Transconductance measures how much plate current changes when grid voltage changes while plate voltage is kept constant:

\[
g_m=\left.\frac{\partial I_a}{\partial V_{gk}}\right|_{V_{ak}}.
\]

It is measured in siemens. Saying that \(g_m=7.6\) mS means that, locally, a 1 V variation at the grid produces about 7.6 mA of plate-current change, if \(V_{ak}\) remains constant.

6.3 Plate resistance rp

Plate resistance describes instead how much plate voltage must change to produce a current change at constant grid voltage:

\[
r_p=\left.\frac{\partial V_{ak}}{\partial I_a}\right|_{V_{gk}}.
\]

For small signals around the same operating point, the three quantities are related by \(\mu\approx g_mr_p\).

7. Load line and operating point

The tube does not operate by itself: the external circuit imposes an additional relation between voltage and current. In the simplest case, with the cathode at the reference node and a plate resistor \(R_a\) fed by \(B^+\), one has

\[
I_a=\frac{B^+-V_{ak}}{R_a}.
\]

On the plate curves this equation is a straight line. The two endpoints are immediate: with \(I_a=0\), \(V_{ak}=B^+\); with \(V_{ak}=0\), \(I_a=B^+/R_a\).

With the fitted model used for this plot, the quiescent point of the example is about \(V_{ak}=236\) V and \(I_a=19.3\) mA. Changing grid bias moves the point to another curve; changing \(R_a\) changes the load-line slope; changing \(B^+\) changes its voltage-axis intercept.

8. Small signal and amplification

Around the Q point, a small variation of the grid signal produces a variation of current. The plate resistor then converts this current variation into a voltage variation. Because an increase in current increases the voltage drop across \(R_a\), the plate voltage decreases: the common-cathode stage is therefore inverting.

In the simplest linear model, neglecting the following load and assuming the cathode is at AC ground, the voltage gain is approximately

\[
A_v\approx-\mu\frac{R_a}{r_p+R_a}.
\]

Using \(\mu=10\), \(r_p=1.3\) k\(\Omega\), and \(R_a=3.3\) k\(\Omega\) as an example gives an ideal gain of about \(-7.2\). This is only a local approximation: farther away from the Q point, the curve nonlinearity becomes increasingly important.

9. What changes in a twin triode

The 6BX7-GT contains two triode sections in the same envelope. The physical principle does not change: each section has its own cathode, grid, and anode. Sharing the envelope and heater makes the component compact, but it does not mean the two sections are identical down to the last decimal place. Datasheets indeed report slightly different interelectrode capacitances for the two sections.

A twin triode can be used for two separate stages, differential circuits, oscillators, drivers, or other functions. In the case of the 6BX7-GT, its ability to operate at relatively high current also made it suitable for deflection and driver applications.

10. Datasheet curves, real tube, and model

Datasheet curves are typical or average curves, not an exact photograph of every specimen. A real tube varies with manufacturer, production lot, age, temperature, and operating conditions. In addition, the printed graph has limited resolution and its digitization introduces a further source of error.

For this reason, the mathematical model used here should not be interpreted as a “perfect tube”. Its role is more concrete: to provide a continuous and coherent function that reproduces, within the data domain, the chosen family of reference curves.

11. What should remain from this article

To read a triode correctly, one must reason in terms of \(V_{ak}\), \(V_{gk}\), and \(I_a\); understand that the curves form a family parameterized by grid bias; use \(\mu\), \(g_m\), and \(r_p\) as local quantities; and remember that the operating point arises from the intersection between tube behavior and the external circuit.

The next step is to transform this family of curves into a continuous mathematical function. That is exactly what the fitted Koren-style model does in the second article.

Technical sources

• General Electric, 6BX7-GT Twin Triode — Description and Rating, ET-T804, 2-52.
• RCA, 6BX7-GT Medium-Mu Twin Triode, tentative data, 6-56.
• Digitized fitting data and plots: GX-TXT / Giux-Lab pipeline, dataset curve_family.primary.