import numpy as np
import matplotlib.pyplot as plt
from matplotlib import ticker
import audio_dspy as adsp
import random as r
import scipy.optimize as opt
plt.style.use('dark_background')
Bad Circuit Modelling: Op Amps¶
So far in this bad circuit modelling series, we've been assuming that any op amps in our circuits are perfect, ideal components. Today, we're going to flip that paradigm, and assume that all of the components in our circuit are ideal except for the op amps. In this writing, we'll examine the popular Texas Instruments LM741 op amp, but the same principles could be applied to any similar op amp.
Op Amp Limitations¶
There are a lot of ways to look at op amps in a non-idealized way. For instance, op amps are themselves made up of components including resistors, capacitors, transistors, and diodes. It would be possible to develop a model of non-ideal op amps based exclusively on the non-ideal behaviors of these internal components, however this would prove to be quite complex, and given the vast experience of engineers and circuit designers working with op amps as isolated components in and of themselves, in this article, we'll focus on op amps as holistic, indivisible components.
In circuit design we usually assume that op amps have zero input current, infinite input impedance, and zero output impedance. While the LM741 does allow a tiny bit of input current (~80 nA), and does not have perfect impedance characteristics, these imperfections are usually negligible for audio circuits. Further, op amps have a "slew rate" that limits their behavior at extremely high frequencies, much lower than the frequencies we typically work with in audio circuits.
In this article, we'll focus on two main aspects of non-ideal op amps: the effects of temperature and aging on the op amp's bandwidth and noise characteristics.
Circuit Analysis¶
As usual, we'll start with our circuit: the Sallen-Key lowpass filter.
from IPython.display import Image
Image(filename='../CMTolerance/Pics/Sallen-Key.png')
Once again, the filter cutoff frequency and Q value are as follows:
$$ f_c = \frac{1}{2 \pi R C} $$ $$ Q = \frac{1}{2 - \frac{R_2}{R_1}} $$
And again, we'll design our filter for $f_c = 1 \text{ kHz}$, and $Q = 2$; $C = 4.7 \text{ nF}$, $R = 33.8 \text{ kOhms}$, $R_2 = 1.5 \text{ kOhms}$, and$R_1 = 1 \text{ kOhms}$.
def design_SKLPF(R, C, R1, R2, fs, age=1, temp=300):
fc = 1.0 / (2 * np.pi * R * C)
Q = 1.0 / (2 - (R2 / R1))
return adsp.design_LPF2(fc, Q, fs)
fs = 44100
b, a = design_SKLPF(4.7e-9, 33800, 1000, 1500, fs)
adsp.plot_magnitude_response(b, a, fs=fs)
plt.ylim(-60)
plt.title('Sallen-Key LPF')
Op Amp Temperature Dependence¶
In 2002, NASA released an interesting paper documenting the effects of extreme temperatures on Op-Amp performance. While the paper examines the now defunct OP181 op amp, we can scale their results to apply to the LM741. Most notably, the paper finds that the op-amp displays a temperature-bandwidth dependence described by the following chart.
Image(filename='./Pics/NASA.jpg')
We can approximate this bandwidth limitation as a first-order lowpass filter, with cutoff frequency dependent on the operating temperature. Let's take a look at the distribution of the cutoff frequencies as a function of temperature.
x = np.array([90, 25, -125, -170, -185]) + 273
y = np.array([6, 6, 5, 3, 1.5])
plt.plot(x,y,'o')
plt.title('Cutoff Frequency vs. Temperature')
plt.xlabel('Temerature [K]')
plt.ylabel('Cutoff Frequency [kHz]')
Now we can fit this data to a curve, in this case we'll use a binding function.
def binding(x,kd,bmax):
return (bmax*x)/(x+kd)
param=opt.curve_fit(binding, x,y)
plt.plot(x,y,'o',np.arange(500),binding(np.arange(500),*param[0]))
plt.title('Cutoff Frequency vs. Temperature')
plt.xlabel('Temerature [K]')
plt.ylabel('Cutoff Frequency [kHz]')
print(param[0])
After scaling the data to be appropriate for the LM741, we get something like the following:
$$ f_c = \frac{30.96 T}{T + 290.48} $$
where T is the temperature in Kelvin, and $f_c$ is the cutoff frequency in kHz.
x2 = np.array([90, 25, -125, -170, -185]) + 273
y2 = np.array([6, 6, 5, 3, 1.5]) * 2.7
param2=opt.curve_fit(binding, x2,y2)
plt.plot(x,y,'o',np.arange(500),binding(np.arange(500),*param[0]))
plt.plot(x2,y2,'o',np.arange(500),binding(np.arange(500),*param2[0]))
plt.title('Cutoff Frequency vs. Temperature (LM741)')
plt.xlabel('Temerature [K]')
plt.ylabel('Cutoff Frequency [kHz]')
plt.legend(['OP181 Data', 'OP181 Curve', 'LM741 Scaled Data', 'LM741 Curve'])
print(param2[0])
Then we can add the following frequency response to the ideal frequency response of the lowpass filter.
fs = 96000
x = np.array([25, -125, -185, -225]) + 273
legend = []
for T in x:
fc = binding(T, *param[0])*1000
b, a = adsp.design_LPF1(fc, fs)
adsp.plot_magnitude_response(b, a, fs=fs)
legend.append('Temperature: {}K'.format(T))
plt.ylim(-15)
plt.xlim((10, 50000))
plt.legend(legend)
plt.title('Temperature-Frequency Response Dependence')
Op Amp Aging¶
R.G. Keen has a wonderful article about op amp aging. Essentially as op amps grow old, and are subjected various sorts of "zaps", they begin to develop a noise characteristic, and lose a bit more of their bandwidth. The noisep power should rise with age, with a characteristic similar to the chart shown below.
x = np.array([0, 10, 30, 50, 100])
y = np.array([-120, -60, -45, -30, -20]) + 120
param=opt.curve_fit(binding, x, y)
plt.plot(x,y-120,'o',np.arange(100), binding(np.arange(100),*param[0])-120)
plt.title('Noise Level vs. Age')
plt.xlabel('Age [yrs]')
plt.ylabel('Level [dB]')
print(param[0])
From my own personal experience, I've found that as op amps get old and start to fail, they tend to display a distortion characteristic similar to the well-known "dropout" nonlinearity (shown below).
Image(filename='../../Complex_Nonlinearities/Paper/Pics/dropout.png')
Implementation¶
As usual, I have implemented a model of the Sallen-Key lowpass filter with variable temperature and aging parameters, to show how the imperfections explained above affect the overall sound of the filter. Source code for this implementation is available on GitHub, and a video demo can be found on YouTube.