The Ideal Integrator
The ideal integrator, shown in Fig. 1, with symbol shown in Fig. 2, makes use of an ideal operational amplifier with , , and . The current through the resistor, , can be expressed as:
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(1)
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Thus, we can write the integrator output voltage, , as:
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(2)
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In the Laplace domain:
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(3)
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Or equivalently:
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(4)
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The magnitude and phase response of an ideal integrator is shown in Figs. 3 and 4.
Rewriting the transfer function as:
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(5)
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We can then define the quality factor of an ideal integrator:
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(6)
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Since . Fig. 5 shows a multiple-input integrator, with output voltage:
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(7)
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Integrator Noise
Fig. 6 shows an integrator where the output is fed back to one of its inputs, giving us:
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(8)
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Ignoring the noise from the amplifier, the output noise of the integrator in Fig. 6 can be expressed as:
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(9)
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The total integrated noise is then:
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(10)
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Integrator Non-Idealities
In practice, integrators are limited by the characteristics of non-ideal amplifiers: (1) finite gain at DC, and (2) non-dominant amplifier poles.
Finite Gain
Non-Dominant Poles
Capacitor Non-Idealities