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When using a small value capacitor, a nonlinear situation exists. The signal quickly charges the capacitor. This creates DC charging for each value of the signal and makes it follow the capacitor charging function. Does anybody follow? Ideally, you want all of the signal without the nonlinear function of the charging capacitor.

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Kevin,
Connect a resistor in series with a capacitor, so that the resistor can charge and discharge the capacitor. Connect a signal to the resistor. You now have a linear low-pass filter, with its output across the capacitor.
It doesn't matter that the capacitor charging is exponential instead of linear, unless you need a linear triangle wave or sawtooth.
If you feed the low-pass filter with a pure sine wave, then the output will be just as pure (very linear) as the input, except at high frequencies when its output level is reduced and is purer (less distortion and even more linear).

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Ante,
A triangle wave can be extremely linear if you use an integrator that is made from an opamp that has a voltage-gain of 100,000 or more, and use a low frequency.
Of course the linearity drops at higher frequencies.
Gross linearity error can be seen as a departure from a straight line on a scope.

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Kevin,
A transistor can produce a very linear ramp if cofigured as a constant current source. But to make a triangle wave you need 2 transistors (1 ramping up, the other ramping down) and a complicated circuit to switch them on and off. An opamp has all those things built-in.
You could also use a transistor as an integrator, but linearity would not be as good as an opamp because its gain is no where near as high as an opamp.

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