**Magnitude Error:**

In the above equation A_{0} is the dc gain, ω_{ρ} is the 3-dB frequency of the open loop gain in radians/sec and A_{0} ω_{ρ} represents the gain-bandwidth-product (GBP) or the unity gain bandwidth of the op-amp in radians/sec. For a 741-type IC op-amp, typical values are

A_{0} = 2 × 10^{5} and f p = ω_{p}/2π = 5 Hz , therefore, providing a GBP of 1 MHz. Eq. represents what is known as the one-pole model of the op-amp while Eq. (7.74(b)) represents an integrator model of the op-amp.

The circuits that are designed on the basis of an ideal op-amp (possessing infinite gain) operate as intended only for a restricted frequency range (usually some tens of kHz) because at higher frequencies the gain of the op-amp falls down (Figure (b)). Thus, at higher frequencies, the performance of such circuits deviates from the desired one. Before discussing circuit techniques for compensating for this deterioration, let us initially evaluate the effect of the op-amp pole on the behaviour of a well known circuit. Assume the non-inverting amplifier of Figure for which usual analysis, supposing infinite gain, gives

v_{0} / v_{0} = 1 + (R_{2} /R_{1}) = = 1 + k

where k = R_{2} /R_{1} ------- (75)

If the circuit is reanalyzed assuming the op-amp gain A to be represented by

Eq. ( (b)), we obtain

v_{0} / v_{1} = (k + 1) {1/ (1+( s (k + 1)/ ω_{t} ) ------ (76)

where ωt = A_{0} ω_{p} is the GBP of the op-amp. A comparison of Eqs. (75) and (76) shows that the consideration of the op-amp pole has resulted in an additional frequency-dependent term ε in the gain of the circuit viz.

ε (s) = {1/ s(k+1)/ ω_{t}} = 1/ 1 + s T -------- (77)

ε (s) is called the error function because it gives an estimate of the magnitude error (γ)

and phase error (φ) made by the pole of the op-amp. We see that to get back Eq. (74) the error function of Eq. (77) should have ideally unity magnitude and zero phase. It means that the magnitude error made by the op-amp pole may be represented by [| ε ( j ω) | - 1] while the phase error is merely equal to the phase of the error function, i.e. ∠ε ( j ω) .

The magnitude error is given by :

γ = | ε ( j ω) | - 1 ≅ - 1 ω^{2 }τ^{2} ; for ωτ < < 1 ------- (78)

while the phase error is given by

φ=∠ ε ( j ω) ≅ - ωτ; for ωτ << 1 ----------- (79)

From Eqs. (78) and (79) this is seen that the magnitude error and the phase error go on enhancing at higher frequencies and it is responsible for the deterioration in the performance of the circuit at higher frequencies and that, out of γ and φ, the latter is the dominant error.