Introduction:

Consideration has been confined to purely resistive feedback circuits and op-amps have been largely taken as ideal, apart from the treatment of a finite CMRR. The other significant practical limitation of op-amps is their finite bandwidth and the related frequency response. Internally, the bandwidth of op-amp is restricted by the consequences of frequency dependent elements of the transistors. The parasitic capacitances present restrict the frequency of operation of individual gain phases.

Frequency Response of a Single Inverting Gain phase:

The simplified, equivalent, small-signal model of a MOS transistor-based inverting amplifier phase is as shown in figure below. The input resistance is supposed to be very high and is thus omitted from the model. The parasitic capacitive elements have been grouped altogether and appear as single equivalent capacitors at the input, C_i, and output, C_o, and as a feedback capacitance, C_f, among output and input of the phase. The amplification is as shown as current source generating an output current controlled by the small-signal trans-conductance, g_m.

Figure: Equivalent Circuit of a MOS Transistor Amplifier phase

To find out the frequency dependent gain, the Kirchhoff’s Current Law can be applied to output node taking V_gs1 = V_i. Then:
i₁ = i₂ + i₃ + i₄

(V_i - V_o) j ω C_f = g_m1 V_i + (V_o/r_o) + V_o j ω C_o

j ω C_f V_i - g_m1 V_i = (V_o/r_o) + j ω C_o V_o + j ω C_f V_o

V_i (j ω C_f - g_m1) = V_o [(1/r_o) + (j ω C_o) + j ω C_f]

V_o/V_i = [j ω C_f - g_m1]/[(1/r_o) + (j ω C_o) + (j ω C_f)]

V_o/V_i = [- g_m1 {1 – jω(C_f/ g_m1)}]/[ (1/r_o) + j ω(C_o + C_f)]

V_o/V_i = (- g_m1 r_o) [{1 - j ω (C_f/ g_m1)}/{1 + j ω (C_o + C_f)r_o}]

The low frequency gain is as:

V_o/V_i| _{ω →0} = - g_m1 r_o

This is dimensionless and points out the inverting amplification phase. This response is of the general form:

V_o/V_i| _{ω →0} = - g_m1 r_o

V_o(ω)/V_i(ω) = K[1 – j(ω/ω_Z)]/ [1 + j(ω/ω_P)] with K = - g_m1 r_o

The numerator has a value of:

ω_Z = gm₁/C_f

The denominator has a value of: 

ω_P = 1/( C_o + C_f)r_o

In pole and zero terms, the negative coefficient in numerator points out that the zero is positioned in the right half plane whereas the positive coefficient in the denominator points out that the pole is positioned in the left half plane.

Let consider the numerator frequency dependent term:

1 – j (ω/ω_Z) = |M_Z| ∠Φ_Z = √1 + (ω/ω_Z)² ∠tan^-1 (- ω/ω_Z)

The phase and magnitude responses for the numerator term of frequency response are shown plotted in the Bode diagram form as functions of frequency in figure below:

Figure: Magnitude and Phase of numerator term of Frequency Response

Let consider the denominator frequency dependent term:

Bode plots of phase and magnitude of the denominator term of the frequency response is as shown in figure below:

Figure: Magnitude and phase of denominator of Frequency Response

The numerator expression provides a break frequency at ω_z and it can be observe that the gain mounts at a rate of 20dB/dec above this frequency, being +3dB if ω = ω_z. The phase on the other hand is negative going among 0 and -90^o being -45^o if ω = ω_z.  It must be noted that in the Bode approximations, the gain only starts to modify at the break frequency whereas the effect of the phase starts at a decade beneath the break frequency and levels off at a decade above the break frequency. Note as well that the negative phase contributed by the numerator term comes from negative coefficient of the imaginary term. This in turn comes from the feedback, output to input through C_f.

The denominator expression provides a break frequency at ω_p and it can be observe that the gain falls off at a rate of -20dB/dec above this frequency, being -3dB if ω = ω_p. The phase response on the other hand is similar to that of the numerator term going negative among 0 and -90^o being -45^o if ω = ω_p. The negative phase in this instance is due to the fact that, it is contributed by the denominator inverse term that gives the negative coefficient of the imaginary term whenever rationalised. 

For typical values of stray capacitances present in a MOS transistor gain stage employed in an op-amp, that is the related values of dc gain and the frequencies of pole and zero are obtained as:

g_m1r_o = 100 ≡ 40 dB; f_P = 5.3 MHz; f_Z = 3.2 GHz

The phase and magnitude plots for the frequency response of the amplifier phase having such characteristics as shown in figure below, where it can be observe that the pole takes place at a much lower frequency than zero.

Figure: Gain and Phase Response for MOS Transistor Single Gain phase

As the frequency is raised, the first noticeable consequence is the phase shift due to the first pole that starts at a decade beneath the frequency at which this pole is positioned. Once the frequency of the pole is reached the magnitude starts to fall off at a rate of -20dB/decade and the phase reaches -225^o, that is, -180^o-45^o = -225^o with the inclusion of inversion in the amplifier. Whenever the frequency reaches ten times that of the pole, then the phase levels off at – 270^o, however the magnitude continues to fall. At a frequency of ten times, that at which the zero is positioned, the phase starts to fall further due to the effect of zero. At frequency of zero, the phase reaches – 315^o, whereas the magnitude levels off as the rising consequence of the zero cancels the falling result of the pole. At frequency of ten times that of zero the phase reaches its lowest point of – 360^o and levels off here. The total phase is asymptotic to -360^o.

Feedback and Stability:

Consider a feed-back amplifier for which the closed loop frequency-dependent gain is provided as:

A_V = V_o/V_i = A(ω)/1 + A(ω)β

If A(ω) β >> 1, V_o/V_i → 1/ β

Then the closed loop gain is independent of A(ω).

From the stability view point it is the loop gain A(ω)β that is significant. 

If A(ω)β = - 1 ∠0^o or ∠180^o  then:

A_V = V_o/V_i = A(ω)/1-1 = A(ω)/0 → ∞

This gives mount to instability in that the amplifier becomes self-sustaining needing no input to generate an output. In practice, it begins to oscillate.

When β is real and fractional then the amplitude response of loop A(ω)β is simply that of A(ω) scaled down by the factor of β. Phase response of the loop is similar to that of A(ω).

In order to make sure stability of the amplifier the feedback loop should be prevented from reaching the condition above. To do this, it should be ensured that the magnitude of the loop gain is less than unity whenever the total loop phase ∠A(ω)β (comprising the inversion via the amplifier) is 0^o or 360^o. If the amplifier is inverting, this applies to a feedback loop phase shift of 180^o.

Two-Stage Non-Inverting Amplifier:

Consider two phases identical to the prior single phase amplifier placed in cascade and hence the combined gain is non-inverting. This consists of frequency response of the form:

V_o(ω)/V_i(ω) = A_o[1 – j(ω/ω_Z1)][ 1 – j(ω/ω_Z2]/ [1 + j(ω/ω_P1)] [1 + j(ω/ω_P2)]

Where A₀ >>1  and  ωp1 <  ω_p2 <  ω_z1 <  ω_z2

Figure below shows the open-loop response and the feedback-loop response for this amplifier where A₀ = 10⁵, β = 0.1 and A_V = 10. The pole and zero pair of each individual phase are separated by almost 3 decades; however the zero of the first phase is close to the pole of second phase. This has the effect of arresting the 40dB/dec fall in the gain while raising the slope of the phase characteristic. The consequence of this is that a loop phase of -180^o is reached prior to the magnitude of the loop gain and has been decreased to unity, thus allowing instability to take place. This consequence is due to primarily existence of the negative-coefficient zeros in the right half plane.

As a result, high gain amplifiers usually necessitate to be compensated, either to make sure that the gain is decreased to unity prior to a phase of -180^o being reached, or to lift the phase characteristic and hence the phase of -180^o takes place at a higher frequency where the loop gain has fallen under unity. A very simple approach to this is frequently adopted in low and medium grade operational amplifiers by comprising a dominant pole at a very low frequency to guarantee sufficient gain and phase margins. The gain and phase responses of such a compensated amplifier are shown in figure below. The gain and phase margins are measures of how far away the amplifier response is from the boundary of instability and are properly defined below.

Figure: Gain and Phase responses of two-Stage Negative feedback amplifier

Figure: Gain and Phase Responses of a Compensated two Stage amplifier

Gain Margin: It is defined as the difference between the magnitude of loop gain A(ω)β and unity at frequency at which the loop phase is -180^o (that is, neglecting the inversion of amplifier).

Gain margin = 1/|A(ω)β|_{∠Φ = -180^o} ≡ 0- A(ω)β dB|_{∠Φ = -180^o}

The design suggestion is that this must be at least 6dBs (a factor of 2) to make sure adequate stability.

Phase Margin: It is defined as the difference between the loop phase shift ∠Φ_Aβ and -180^o at frequency at which the loop gain A(ω)β is unity.

Phase margin = - 180^o - ∠Φ_Aβ|_{|Aβ| = 1} Where ∠Φ_Aβ is negative.

This is generally quoted as a positive quantity and the design proposal is that this must be at least 30^o and is safer whenever greater than 45^o to guarantee sufficient stability.

Often, manufacturer’s list op-amps as being unity-gain stable. This signifies that full feedback with β = 1 can be applied about the amplifier and stability maintained. Note that beneath such conditions |A(ω)β | = |A(ω)|, the open loop gain of op-amp. Sometimes, though, this condition can’t be satisfied and manufacturers will specify a minimum closed loop gain for op-amp. This is really a manner of specifying the maximum value of β that can be tolerated whereas maintaining stability. This is essential if the open-loop gain |A(ω)|, of the amplifier it is above unity whenever the phase ∠A(ω) is 180^o. The feedback network itself can as well be made frequency-dependent and given a phase shift ∠β(ω) to correct the total phase response.

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