From guitar to speaker: voltage, current and impedance through a buffer, booster, overdrive, amp and effects loop

The chain and the circuit solved here

We built the common chain guitar → buffer → booster → overdrive → amp (preamp → send/return → power amp) → speaker as one ngspice circuit. The guitar is a single coil (L = 2.5 H, R = 6.5 kΩ), the cable from guitar to pedals is 3 m (300 pF) and from pedals to amp 6 m (600 pF).

The buffer is an op-amp voltage follower (1 MΩ in, 100 Ω out), the booster has a gain of 4 (+12 dB) and the overdrive is the TS808 from the earlier article (Drive 50 kΩ, Tone centre). The amp has a 1 MΩ input, a two-stage 12AX7 preamp, a send (reduced 1:10, 1 kΩ output), a return (100 kΩ input), a power amp (2 Ω output impedance) and the equivalent circuit of a 12-inch guitar speaker.

A 0 V source in series at each device's output measures the current flowing there. The input is one strummed E5 chord.

How much each stage raises the amplitude

Figure 2 plots the peak voltage and current at each point on a log scale, with the ratio between neighbouring points. The guitar's 0.43 V is ×1 through the buffer and ×4.0 through the booster, then dips to ×0.94 in the TS808 because it clips. The same 1.64 V reaches the amp's first grid and becomes 106 V at the plate, ×65.

That is too much for the next stage, so the Gain pot brings it down to 1/50 before the second stage raises it ×38 again. The second stage gains less than the first because 2.1 V on its grid is clipping it too. The send brings it down 1/58 to line level, and the power amp raises it ×11 to 15.4 V. The voltage goes up and down repeatedly; only the power amp raises the current.

Up to the speaker, signals are passed as voltage

At every connection from the guitar to the send, the receiving input impedance is 100 to 10,000 times the sending output impedance. The voltage passes on almost undiminished and the current is in microamps. From the buffer's 100 Ω output into the booster's 1 MΩ input, for example, the voltage loss is only 0.0009 dB.

Passing on the shape of the voltage while drawing almost no current is the idea behind this kind of connection (bridging). Maximum power transfer would mean matching the impedances, but instrument signals are not handled that way. Only the final connection, from power amp to speaker, passes power by driving current.

Waveforms at each point

The voltage changes shape in only two places: the TS808 and the preamp. The buffer and booster change only the size (the booster by 4×); the TS808 rounds off the peaks and rolls off the highs. Each tube stage flips the waveform, because the plate moves opposite to the grid, and grid current squashes the peaks slightly. The send and speaker voltages are scaled versions of the second plate's waveform.

The current does not always look like the voltage. At the guitar's output and the TS808's output, much of the current charges and discharges the cable capacitance. Current into a capacitance is proportional to how fast the voltage changes (dV/dt), so the high harmonics are emphasised and the waveform looks spiky. At the TS808's output, large current flows at the corners the diodes have flattened. The speaker current differs a little from the terminal voltage because the impedance changes with frequency (Figure 7).

On the real scale: DC bias and amplitude

Figure 4 draws the same 25 ms as Figure 3 on each point's real voltage scale. The red dashed line is the DC voltage (bias).

The pedal outputs, amp input, send and speaker all swing around 0 V. The op-amps inside the pedals run on a 9 V single supply around a 4.5 V bias, but capacitors at the inputs and outputs block the DC, so the jacks sit at 0 V (this article's simulation uses a ±4.5 V split supply from the start).

The tube stages are different. The cathode is lifted to 1.48 V by the plate current through 1.5 kΩ, which leaves the grid biased 1.48 V below it, at −1.48 V. The plate sits at 201 V, about 1 mA × 100 kΩ below B+ 300 V, and swings 106 V peak on top of that. A signal that was 0.43 V is now swinging ±106 V on top of 201 V DC.

The guitar: a voltage source with high internal impedance

A pickup is a voltage source: the string's EMF in series with the coil's L and R. Seen from the cable, its output impedance rises from 6.5 kΩ at 100 Hz to 16.7 kΩ at 1 kHz and 53 kΩ at 3 kHz, reaching 235 kΩ at 10 kHz where it resonates with the coil's self-capacitance (pots included).

Amp and pedal inputs are 500 kΩ to 1 MΩ because they must be well above this output impedance, or the voltage is divided down more and more toward the highs. Even strumming hard, the guitar puts out only 0.77 µA at the peak (see the pickup equivalent circuit).

Cables and buffers

Cable capacitance (around 100 pF/m) forms a resonant circuit with the pickup's L. Going straight into the amp over 9 m of cable, 900 pF in total pulls the resonant peak down to 2.8 kHz, and 10 kHz ends up 20 dB below 100 Hz.

With a buffer 3 m from the guitar, the guitar sees only the first 3 m (300 pF): the peak stays at 5 kHz and 10 kHz drops only 9.5 dB. The remaining 6 m is driven by the buffer's 100 Ω output; 100 Ω with 600 pF has a corner at 2.7 MHz, far outside the audio band.

A buffer's voltage gain is 1, so its output is the same 0.43 V as the guitar's. What changes is the output impedance (tens of kΩ → 100 Ω), which is why the cable capacitance no longer matters.

Booster and overdrive

The booster is a non-inverting stage with a gain of 4, taking 0.43 V to 1.75 V without changing the waveform. Its output is 100 Ω, and 3.4 µA flows into the TS808's 510 kΩ input.

The TS808 rounds off the 1.75 V it receives with the diodes in its feedback, trims the highs in its tone stage and puts out 1.65 V. Because the booster raised the level first, the TS808 clips well even at Drive 50 kΩ. The current from the TS808 to the amp is 5.0 µA, part of which charges and discharges the 6 m of cable.

A real TS808 has a volume pot (100 kΩ) at the output, which raises the output impedance to as much as 25 kΩ near the middle. We solved it at full volume (about 100 Ω). A long cable after a pedal with its volume turned down can lose some treble.

The amp input and the tube preamp

The amp input is 1 MΩ (the grid-leak resistor) and goes through a 68 kΩ grid stopper to the 12AX7's grid. The 12AX7 draws plate current from B+ 300 V through 100 kΩ and self-biases with about 1.5 V across a 1.5 kΩ cathode resistor. At rest the plate sits at 201 V with about 1 mA of plate current.

The first plate swings 106 V peak on top of its 201 V DC. The 1.65 V input exceeds the 1.5 V bias, so grid current flows on the peaks and squashes them a little (the booster and TS808 are "pushing" the amp). The Gain pot brings it down by 1/50 before the second stage amplifies again, whose plate swings 79 V peak.

Inside the pedals it is a few volts and microamps; inside the tubes it is tens to hundreds of volts and around 1 mA. Tube circuits amplify with large voltage swings, so the level has to be brought well down before it is passed on.

Watching it clip: TS808 and 12AX7

Figure 6 overlays one cycle of the output of the TS808 and of one 12AX7 stage of the amp, for a 196 Hz sine at several levels.

Raising the TS808's input 40-fold, from 0.05 V to 2 V, raises its output only 9-fold, from 0.25 V to 2.3 V. The gain falls 5.0, 2.9, 1.8, 1.3, 1.2 times as the peaks round off and flatten. The feedback diodes conduct gradually, so the clipping is smooth.

The 12AX7's plate is a small sine of 197–209 V (about 59 times) at 0.1 V in, but swings 75–290 V at 4 V in. At the top it approaches B+ 300 V (plate current almost stops: cutoff); at the bottom it flattens around 75 V (grid current flows and the plate current cannot rise further). From the roughly 203 V bias it goes +87 V up and −128 V down, so the two halves clip differently and even harmonics appear. With large inputs the cathode also rises from 1.45 V to 1.60 V: the operating point itself moves.

Send and return

The send/return (effects loop) jacks let you insert external gear between the end of the preamp and the start of the power amp. The signal after the preamp's tone circuit is reduced 1:10 and sent out through a buffer with a 1 kΩ output. The send is 1.36 V peak (about 1 V rms), a level line-level gear such as delay and reverb can take.

The current from the send into the return (100 kΩ input) is 27 µA. Putting delay or reverb here adds the ambience after the distortion, so it stays clearer than when placed before the preamp (between guitar and amp).

Power amp and speaker: the first place current flows

The power amp raises the roughly 1 V at the return to around 15 V and drives current through the speaker. At the chord's peak the terminal voltage is 15.4 V and the current 2.2 A. Here, for the first time, the sending side (the power amp's 2 Ω output) and the receiving side (the speaker) have impedances of the same order.

Even a nominal 8 Ω speaker varies widely with frequency. In this equivalent circuit it is 46 Ω at the cone's mechanical resonance (100 Hz), a minimum of 6.5 Ω at 448 Hz, and 23 Ω at 5 kHz from the voice-coil inductance. Per 1 V at the return, the terminal voltage barely moves (11.5 to 14.6 V) while the current changes six-fold, from 0.31 A at 100 Hz to 1.76 A at 448 Hz (Figure 7).

The lower the power amp's output impedance, the more constant the voltage is regardless of the speaker's peaks and dips. Many tube amps have a fairly high output impedance, so the voltage rises at the resonance and the bass comes through more.

A hundred-million-fold in power

Comparing the product of peak voltage and current, the guitar's output is 0.43 V × 0.77 µA ≈ 0.33 µW and the speaker 15.4 V × 2.2 A ≈ 34 W, about 100 million times (80 dB). The pedals and preamp mainly raise the voltage; only the power amp raises the current.

Looking at voltage alone, it goes 0.43 V (guitar) → 1.75 V (booster) → 106 V (preamp) → 1.36 V (send) → 15.4 V (speaker). Each device hands on a level the next one can take.

What the simulation leaves out

• The 12AX7 uses Koren's model (widely used parameters), with grid current approximated by a diode. B+ is an ideal supply; voltage sag at high volume is not modelled.
• The tone stack is replaced with a fixed divider (with a capacitor that keeps a little treble).
• The power amp is a behavioural model with only voltage gain, output limiting and a 2 Ω output impedance. The output transformer and output-tube nonlinearity are not modelled.
• The speaker is an electrical equivalent circuit with 6.4 Ω DC resistance, 0.7 mH voice-coil inductance and a 100 Hz mechanical resonance (46 Ω peak). These are not the values of any particular product.
• The buffer, booster and TS808 use op-amp models (GBW, slew rate, output swing). Noise is not modelled.