How a vacuum tube amplifies: in the equivalent circuit it is a JFET, and op amp circuits can be built from tubes

Thermionic electrons, plate and grid: controlling current with voltage

A triode is a vacuum glass tube holding three electrodes and a heater that warms them. Heat the cathode to roughly 700-800°C and thermionic electrons boil off its surface into a cloud. Put the plate (anode) at +100-300V relative to the cathode and the electrons fly to it. That is the plate current. By convention current runs opposite to electron flow, so on a schematic it goes from plate to cathode.

Right in front of the cathode sits a mesh electrode, the grid. Make the grid negative relative to the cathode and electrons are pushed back, so fewer pass through; more negative still and the current stops (cutoff). The grid voltage is, in effect, a tap on the plate current. Because the grid is kept negative, electrons do not collect on it and almost no current flows. The input impedance is set by an external grid-leak resistor (around 1MΩ), and like a JFET or MOSFET the device controls current with voltage.

Read off the 12AX7 curves at 250V plate (approximate): about 2mA at −1V grid, 1.2mA at −2V, and 0.5mA range at −3V. The datasheet g_m = 1.6mA/V is the value near 1.2mA.

Turning current into voltage with the plate resistor: the phase inverts

The plate resistor R_a turns plate current changes into voltage changes. The plate voltage is E_p = B+ − R_a·I_p, so when the grid moves positive, I_p rises, the drop across R_a rises, and the plate voltage falls. Input and output are in opposite phase, an inverting amplifier like the common-emitter or common-source stage.

With B+ = 250V, R_a = 100kΩ and an operating current of 0.87mA (next section), the drop across R_a is 87V and the plate-to-cathode voltage is about 161V. A ±1V grid swing moves the current between 0.47 and 1.42mA and the plate voltage between 203V and 108V, about ±47V. Voltage gain is about 47 (33.5dB). The output carries about 160V DC, so a coupling capacitor passes only the AC to the next stage.

The three constants μ, gm, rp, and μ = gm × rp

A triode is described by three numbers: transconductance g_m (change in plate current per 1V of grid voltage, mA/V), plate resistance r_p (the inverse of the current change per 1V of plate voltage, Ω), and amplification factor μ (the plate voltage needed to match the effect of 1V at the grid). They are related by μ = g_m × r_p. μ is the upper limit of voltage gain with no load, and the actual gain is set by the R_a and r_p divider: A = μ·R_a ÷ (R_a + r_p) (the same as g_m × (R_d ∥ r_ds) for a JFET).

A tube with large μ has high r_p and passes little current; one with small μ has low r_p and passes more. Choosing between gain and current drive is the choice of tube type.

The values are typical for about 250V plate and 1-10mA. g_m and r_p move a lot with the operating point, so read them from the characteristic curves at your operating point.

Ep-Ip curves and the load line: finding the operating point and gain

Plot plate voltage E_p on the horizontal axis and plate current I_p on the vertical axis, one curve per grid voltage, and you have the Ep-Ip characteristic. Overlay the load line set by the circuit. For a 12AX7 with B+ = 250V, R_a = 100kΩ and cathode resistor R_k = 1.5kΩ (self-bias), E_p = 250 − (100k + 1.5k)·I_p: a straight line through 250V at I_p = 0 and 2.46mA at E_p = 0.

With self-bias, the cathode rises with current, so the grid becomes relatively negative. The operating point satisfies V_g = −R_k·I_p: I_p ≈ 0.87mA, E_p ≈ 161V, grid at −1.3V. It is the same idea as JFET self-bias with a source resistor.

Gain depends on whether the cathode has a bypass capacitor. With the datasheet typical values (μ = 100, r_p = 62.5kΩ), bypassed gain is A = μ·R_a ÷ (R_a + r_p) ≈ 61.5. Unbypassed, R_k looks like (μ+1) times its value, 151kΩ, from the plate, so A = μ·R_a ÷ (R_a + r_p + (μ+1)·R_k) ≈ 31.8 (table). This is negative (current) feedback: the gain halves, but the effect of tube variation and distortion shrinks.

The typical 61.5 and the operating-point 47 differ because the datasheet's 1.6mA/V applies near 1.2mA, and g_m falls when the current is reduced to 0.87mA. Even R_a = 1MΩ gives only 94, and never exceeds μ = 100.

The Ep-Ip curves in the figure come from an approximate model fitted to 12AX7 curves (a Koren-type equation with adjusted parameters), and differ from a real tube by a few percent to about 20%. For design, reread them from the datasheet curves.

In the equivalent circuit, a tube is a JFET with a low drain resistance

In the small-signal equivalent circuit a triode can be drawn two ways: a voltage source μ·v_gk in series with r_p, using the grid-cathode voltage v_gk, or its Norton equivalent, a current source g_m·v_gk in parallel with r_p. The input side is open.

The small-signal circuit of an N-channel JFET also has an open input and an output of a current source g_m·v_gs in parallel with r_ds. The form is the same as a triode, with g_m and r_ds corresponding to r_p. Both are biased with the gate negative, draw almost no gate current, and conduct the most near V_GS = 0.

A BJT (hybrid-π) has r_π at the input, so input current flows. The output has the same form, but g_m = I_C ÷ V_T is 38.7mS at 1mA, more than an order of magnitude above a tube's 1.6mS, and in return r_π = β ÷ g_m ≈ 2.6kΩ appears at the input. An enhancement MOSFET has the same form as a JFET, but is turned on with the gate positive, the opposite direction.

Terminal mapping, and where the pentode sits

A tube circuit reads directly as a JFET circuit using the table below.

A triode has a low r_p (62.5kΩ for a 12AX7), so its output is closer to a voltage source with a high internal resistance than to a current source. Think of it as an FET with a low drain resistance, and you see why gain is set by the R_a and r_p divider and caps at μ however large R_a gets.

A pentode adds a screen grid between the grid and the plate so that plate current barely changes with plate voltage. Its r_p reaches hundreds of kΩ to MΩ, giving an output close to the current source of a saturated MOSFET or BJT.

Classic circuits in tubes (1): grounded stages and followers

The common-emitter (common-source) stage corresponds to the common-cathode stage, which is the circuit of Figure 2. Gain is A = −μ·R_a ÷ (R_a + r_p), output resistance is R_a in parallel with r_p, and the input is high impedance.

The emitter (source) follower corresponds to the cathode follower: the plate goes to B+ and the output is taken from the cathode. Gain is A = μ·R_k ÷ (r_p + (μ+1)·R_k), approaching 1 for large μ. For a 12AU7 (μ = 17, r_p = 7.7kΩ) with R_k = 10kΩ, A ≈ 0.906, and the output resistance is r_p ÷ (μ+1) in parallel with R_k, about 410Ω (close to 1/g_m = 455Ω).

A BJT emitter follower (I_C = 1mA) has an output resistance of 26Ω, and the cathode follower is about 16 times higher, but still enough to drive 500pF of cable (410Ω and 500pF make a 776kHz LPF). For a 12AX7 with R_k = 100kΩ, A = 0.984 and the output resistance is about 615Ω.

Classic circuits in tubes (2): differential pair and cascode

The differential pair becomes the long-tailed pair: two triodes share a cathode joined through a tail resistor R_t. Feed a signal to one grid and ground the other for AC, and the two plates give outputs in opposite phase, which is why it is the standard phase-inverter stage in guitar amplifiers.

With two 12AX7s and R_a = 100kΩ, the gain at one output is A_d ≈ μ·R_a ÷ (2(R_a + r_p)) = 30.8, and the common-mode gain with R_t = 47kΩ is A_c ≈ 1.04, so the common-mode rejection ratio (CMRR) is 20 log(30.8 ÷ 1.04) ≈ 30dB (23dB for R_t = 22kΩ). That is far below an op amp's 80-100dB: a long-tailed pair is for inverting the phase rather than for cancelling common-mode signals.

The cascode connects the cathode of an upper triode (input resistance 1/g_m, a few hundred Ω) as the plate load of a lower triode. The lower stage's voltage gain is held to about 2.5, so the Miller capacitance is 7.6pF, one tenth of the usual 84pF. The upper output resistance is about 6.4MΩ and the gain approaches g_m·R_a (158 for R_a = 100kΩ, 2.6 times the single stage's 61.5). The upper grid needs a separate bias voltage (tens of volts or more).

Op amps in tubes: the 1950s vacuum-tube op amp

The name "operational amplifier" comes from a 1947 paper by Ragazzini and others, for a high-gain direct-coupled amplifier used for analog computing. The implementation then was vacuum tubes. In the 1950s general-purpose tube op amps such as the Philbrick (GAP/R) K2-W were sold, using two 12AX7s (four triodes), with open-loop DC gain around ten thousand and supplies of about ±300V.

The structure is the same as the internal blocks of an IC op amp: a differential input stage (long-tailed pair), a gain stage (common cathode), and an output stage (cathode follower). Feed back from the output to the inverting input through R_f and put the input through R_i, and you have an inverting amplifier. The closed-loop gain is −R_f/R_i, exactly the same equation as an IC op amp.

Take A_OL = 15000 (the order of magnitude in K2-W material). The gain error is 1/(1 + A_OL·β). For a closed-loop gain of −10, the loop gain is 15000 ÷ 11 ≈ 1364 and the error is 0.073%; for −100 it is 0.67%, and for −1 it is 0.013%. No tube characteristic remains in this calculation, and the inverting and non-inverting VCVS reasoning applies as is.

Compared with an IC op amp, the supply is high, the offset voltage is large and drifts with temperature, the output current is small, and heaters are needed. The idea of an op amp, that feedback makes the circuit's behavior depend only on the feedback resistors, was already in place in the tube era.

Practical cautions: B+ shock, heaters, grid leak, microphonics

These are items a semiconductor circuit does not have, with numbers.

A grid-leak resistor is needed to set the grid's DC potential. The grid floats beyond the coupling capacitor, so without leakage stray electrons accumulate, the potential drifts negative, and the bias becomes undefined. A larger resistor shifts the bias by grid current × resistance, so about 1MΩ is the guide (0.1µA gives 0.1V).

The floating grid node is a high-impedance node, like the MOSFET gate in the millennium bypass, and shows the same symptoms: hum when touched, bias drifting with flux contamination. Keep the wiring short, shield it, and clean the board.

Heater AC hum enters through leakage between heater and cathode. Ground one side of the heater, lift the center tap to about +20V, or run the heater on DC.

The Miller effect and guitar pickups: why a high input impedance suits them

A triode has a capacitance C_gp between grid and plate: about 1.7pF for a 12AX7, with C_gk ≈ 1.6pF between grid and cathode. In an inverting stage, C_gp sees the input on one end and the inverted, A-times-larger output on the other, so from the input it looks (A+1) times larger (the Miller effect). C_in = C_gk + C_gp(1 + A). For A = 47, C_in ≈ 1.6 + 1.7 × 48.4 ≈ 84pF, and for A = 61.5 it is 108pF.

With source resistance R_s and C_in (94pF including 10pF of wiring) forming an LPF, the −3dB frequency is 169kHz for R_s = 10kΩ, 25kHz for 68kΩ, 6.8kHz for 250kΩ and 1.7kHz for 1MΩ (from 1/(2πRC), see the RC filter calculator). To keep the audio band, the source resistance should be under 100kΩ, so designers drive the stage from a cathode follower or use a cascode.

A guitar pickup is an LCR resonant circuit: a single coil with L = 2.5H and C = 500pF (self-capacitance plus cable) resonates at 4.5kHz. Adding the 94pF Miller capacitance lowers that to 4.13kHz. The peak height (Q) depends on the load resistance, and a light load (like a BJT's r_π = 2.6kΩ) shrinks the resonant peak. That a tube (or a JFET) can look at it with 1MΩ is why a high input impedance suits guitars.

Why tubes are still used: only measurable facts

Only measurable facts follow.

1. How it clips. The waveform of a single common-cathode stage is asymmetric. In the approximate 12AX7 model, a ±0.5V grid input (about 24V peak output) gives a second harmonic of about 3.5%, ±1V (about 47V output) gives 7.0%, and the third harmonic is 0.15%. The second harmonic appears first at small amplitudes, and the third stays much smaller. Beyond ±1.3V the grid goes positive and the bias shifts.

2. Output stage and output transformer. The power stage is often push-pull, so the even-order distortion of the two tubes cancels in the output transformer, clipping becomes symmetric, and odd orders dominate. The output transformer adds its own peaks and droops to the frequency response and compresses large signals through magnetic saturation. The output impedance, a few ohms, is also higher than a solid-state amplifier's.

3. How distortion appears is set by the circuit. Transistors and op amps can produce similar distortion with shallow feedback or diode clipping. What is distinctive in a tube is that the device characteristic alone gives a gentle clipping (soft clip). Where low distortion, low output impedance or low voltage is needed, semiconductors win.