What's the math behind the Zobel network on LM386 audio amplifier?

Question

(Image source: Texas Instruments - LM386 datasheet)

This is an 8 Ω speaker. I want the math on how they got 0.05 µF and 10 Ω.

Answer 1

I think the actual math is buried in an old National Semiconductor app note or handbook, but everything I have is just the conclusions, as cited in other answers.

For the math, there's this:

https://en.wikipedia.org/wiki/Zobel_network#Zobel_networks_and_loudspeaker_drivers

Answer 2

The math cannot be given as it depends on many things about what the amplifier chip needs and what load it is driving, and those are unknown.

The values given in the datasheet are just typical values you should use in general, unless they don't work in your specific case.

The basic idea just is that the speaker looks like the rated (8 ohm) load on low frequencies and no load on high frequencies.

The Zobel network (or Boucherot cell) is the exact opposite, the network given looks like 10 ohm load on high frequencies and no load on low frequencies.

This just gives the amplifier a load that is present on the whole frequency range, so that the amplifier is stable, instead of introducing overshoots and ringing.

With the given R and C values, the RC cutoff frequency is 318 kHz.

As it happens, the bandwidth of the amplifier is also 300 kHz.

Amplifier is also said to work with 4 to 32 ohm loads.

At 300 kHz, the 50nF capacitor has an impedance of roughly 11 ohms. Adding the 10 ohm resistor means the network is a 21 ohm load at 300 kHz.

This fits well with the requirement of load being below 32 ohms on the whole frequency range.

Answer 3

This RC series network is a possible solution to the chip's tendency to internally oscillate: Is it possible for an opamp to oscillate at a frequency greater than its GBP?

The LM386 does use the compound PNP/NPN transistor pair on the bottom-side of Class AB output stage. This kind of output stage tends toward self-oscillation at high frequency. Oscillation amplitude is often small, since amplifier negative feedback operates to limit the swing. If you look carefully, this high-frequency oscillation may look triangular.

At such high frequencies, the 0.05uf capacitor is a short-circuit, while at audio frequencies it is nearly open-circuit. So not much math involved. I'm guessing that at 5 MHz, you don't want power amp output to appear inductive.

from Audio Handbook Dennis Bohn, National Semiconductor Corp., 1976:

...The addition of the optional \$0.05 uF\$ capacitor and \$10\Omega\$ resistor is for suppression of the "bottom-side fuzzies" (i.e. bottom side oscillation occurring during the negative swing into a load drawing high current - see Section 4.5.5)

Section 4.5.5: The normal power supply decoupling precautions should be taken when installing the LM380. If \$V_s\$ is more than 2" to 3" from the power supply filter capacitor it should be decoupled with a \$0.1 uF\$ disc capacitor at the \$V_s\$ terminal of the IC.
The \$R_c\$ and \$C_c\$ shown as dotted line components on figure 4.5.9 and throughout this section suppresses a 5 to 10MHz small amplitude oscillation which can occur during the negative swing into a load which draws high current. The oscillation is of course at too high a frequency to pass through a speaker, but it should be guarded against when operating in an RF sensitive environment.

The first paragraph (above) refers to the LM386 chip while section 4.5.5 involves the LM380 audio power amplifier - a slightly more powerful version of LM386. The components shown for LM380 amplifier are \$ R_c=2.7\Omega, C_c=0.1uF\$