Introduction
A transimpedance amplifier is the configuration of choice when high-bandwidth and low noise operation is required. A transimpedance amplifier (TIA) converts an input current to a voltage. A common application is for use with a photodiode to convert a photocurrent into a readily measurable signal. Transimpedance amplifiers are useful because they can deliver an order of magnitude or more speed up relative to a simple photodiode plus load resistor configuration.
The characteristics of noise in transimpedance amplifiers is significant because photodiodes are frequently used to detect faint sources. Some noise contributions are fundamental in nature, e.g., thermal or Johnson noise associated with resistors and Poisson noise due to quantization of charge. Other noise sources depend on the specific op-amp device used in the design.
Design tools such as SPICE allow calculation of noise and commercial implementations, such as LTspice, include models for the manufacturer's op-amps. Characteristics of a specific device such as the input capacitance, current noise, and voltage noise factor into the achieved performance but it can be unclear which of these factors dominate the combined noise in a given design, which complicates op-amp selection.
Here we show how to compute the various noise terms present in a transimpedance amplifier. Existing resources on this topic are available in various texts and op-amp datasheets. However, the information is frequently fragmentary, and results are often quoted without derivation so that the assumptions and domain of applicability are unclear. The purpose of this post is to provide an explicit and accurate review of this topic.
The Ideal Transimpedance Amplifier
The basic configuration of a transimpedance amplifier is shown in Figure 1. In contrast to a load resistor configuration, the TIA holds the input at the inverting input at a fixed voltage (ground in this example) which minimizes photodiode dark current and improves linearity.
Unlike op-amp circuits that are used as voltage amplifiers, the transimpedance amplifier has no input resistor. For a perfect op-amp configured as in Figure 1, the output is \(V_o = I_D R_2\) and the current gain (volts per amp) is measured in ohms.
A Practical TIA Circuit
The circuit shown in Figure 1 is not sufficiently detailed to analyze performance. Figure 2 shows a more realistic layout with a feedback network that includes both resistive and capacitive elements.
Real photodiodes have finite shunt resistance, \( R_1\), and capacitance, \(C_1\). Real op-amps also have common mode input capacitance, typically listed as \(C_{\rm {IN}} \) in datasheets, which will also contribute in parallel to \(C_1 .\) We also need to take account of limited bandwidth and finite DC gain of the op-amp.
The existence of significant photodiode capacitance explains why transimpedance amplifiers are necessary. If we used a biased photodiode with load resistor \(R_2\) then the -3 dB bandwidth is \(1/(2 \pi R_2 C_1)\).
The TIA Transfer Function or Signal Gain
It is a little more complicated to find the relationship between the photodiode input current, \(I_D\), and the output voltage, \(V_o\), for Figure 2.
First let's consider an op-amp with finite gain and bandwidth. If \(A(\omega\)) denotes the dependence of gain on frequency, \(f\), where \(\omega = 2 \pi f\), the output voltage, \(V_o\), is
\[ V_o = ( V_P - V_M)\; A(\omega), \tag{1} \]where \(V_P\) and \(V_M\) are the voltages at the non-inverting and inverting inputs, respectively. Op-amps are typically characterized by the gain bandwidth product, GBW, and the open-loop DC gain, \(A_0\). GBW is functionally equivalent to the frequency where the open loop-gain drops to unity. (See Appendix for our model of \(A(\omega)\)).
Next, to keep the expressions compact, write the parallel combination of \(R_1\) // \(C_1\) as \(Z_1\) and \(R_2\) // \(C_2\) as \(Z_2\). Referring to Figure 2, where \(I_1\) is the current flowing in \(Z_1\) and \( I_D\) is the photodiode current, the Kirchoff rules for voltage and current are
\[ \begin{eqnarray} I_2 & = & I_1 + I_D \\ V_M + I_2 Z_2 &=& V_o \\ I_1 Z_1 &=& V_M \\ V_P & =& 0. \end{eqnarray}\tag{2} \]Eliminating \(I_1\), \(I_2\), and \(V_M\) yields the TIA transfer function as a function of frequency,
\[ \boxed{ H(\omega) =\frac{V_o}{ I_D} = A(\omega)\; \frac{Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2} .} \tag{3}\]In the limit of large open-loop gain when \(A(\omega ) \gg 1\) we have \( V_o = I_D Z_2 \). To evaluate Eq. 3 we need an expression for the impedance of a resistor and capacitor in parallel, for which we use \[ Z_i = \left( R_i ^{-1} + \mathbb{i} \omega C_i \right)^{-1} , \tag{4}\] with \(i\) = 1 or 2.
Gain Peaking and Phase Shift
Figure 3 provides examples of \( H(\omega) = V_o/ I_D\) as a function of frequency for a feedback resistor of \(R_2 \) = 10 k\(\Omega\) and two different values of input capacitance. Initially, assume that the photodiode shunt resistance, \(R_1\), is infinite and the feedback capacitance, \(C_2\), is zero. The op-amp has a GBW (500 MHz) and a DC gain (250,000) typical of a high performance device.
With input capacitance \(C_1 \) = 0 pF (red) the signal gain mirrors the performance of the op-amp and is flat up to the GBW frequency and then rolls off at 6 dB per octave. This result suggests that designing a high-bandwidth TIA is simply a matter for finding an op-amp with sufficient large GBW. When the input capacitance is increased to \(C_1 \) = 10 pF (blue), a value characteristic of the combined capacitance of a typical photodiode and op-amp. This shows a dramatically different outcome with pronounced signal gain peaking accompanied by a -180\(^\circ\) phase shift at about 28 MHz.
TIA Overshoot and Ringing
Figure 4 shows the step response associated with the blue transfer function plotted in Figure 3. The step response exhibits overshoot and ringing. The ringing appears at 28 MHz and persist for several hundred ns.
Figure 4 implies that the ideal TIA circuit in Fig. 1 is unusable. The combination of the feedback resistor and input capacitance has a characteristic frequency of \( f_{R_2 C_1} = 1/( 2 \pi R_2 C_1 )\). In this example, \( f_{R_2 C_1} \) = 1.59 MHz, which is much lower than the op-amp's gain-bandwidth product GBW of 500 MHz. The \( R_2 \)-\( C_1\) network in combination with the op-amp gain, \( A( \omega) \), introduces a quadratic term in frequency in the denominator of the transfer function (see below), which causes a prominent peak in the gain and large phase lag at a frequency of approximately \( \sqrt{ f_{R_2 C_1} {\rm GBW} }\).
We'll see below that adding a small amount of capacitance in parallel with the feedback resistor (i.e., non-zero \(C_2\)) can suppress these oscillations. Damping can also be achieved by adding some resistance in series with the photodiode. Typically the former option is chosen as the feedback resistor already has some parasitic capacitance; moreover, Johnson noise from a small series resistance can dominate the noise budget at high frequencies.
A Damped Harmonic Oscillator Model for the Non-Ideal TIA
A convenient way to analyze overshoot and ringing is to rewrite the transfer function in a standard from that lets us analyze the system as a damped harmonic oscillator. This allows us to identify the two key parameters that define the system: the natural frequency and damping ratio. This in turn allows us to find the location of gain peaking, the degree of overshoot, and the -3 dB point. By adjusting the damping ratio we will be able to eliminate the ringing seen in Figure 4.
The transfer function of Eq. (3) can be written as \[ \frac{V_o}{I_D} = R_2 \left[ 1 + 2 \mathbb{i} \zeta \frac{ \omega}{\omega_{n}} - \frac{\omega^2}{\omega_{n}^2} \right]^{-1} , \tag{5} \]
where \(\omega_n \) is the natural frequency (where the gain peak occurs), and \(\zeta\) is the dimensionless damping ratio. When \(\zeta = 0\), oscillations are undamped and when \(\zeta = 1\), the system is critically damped with no overshoot. Figure 5 shows some examples of this behavior.
Solving Eq. (5) for the half-power or -3 dB point, i.e., \( |H|^2 = 1/2\), yields
\[ \omega^2_{-3{\rm dB}} = \left[ 1 - 2 \zeta^2 + \sqrt{2 + 4\zeta^2 (\zeta^2-1)}\right] \; \omega^2_n . \tag{6} \]The -3 dB point according to Eq. (6) is shown on right panel of Fig. (5). Figure 6 illustrates how increasing \(\zeta\) decreases the bandwidth. When \(\zeta = 1/\sqrt 2\) then we have the simplification that \( \omega_{-3{\rm dB}} =\omega_n \). If \(\zeta = 1 \), which is the case when the system is critically damped with no overshoot, then \( \omega_{-3{\rm dB}} = (2^{1/2}-1)^{1/2} \omega_n \simeq 0.64 \omega_n\). 1
The Non-Ideal TIA
Substituting for \(Z_1\) and \(Z_2\) from Eq. (4) in the transfer function (Eq. 3) and by grouping terms of powers of \(\omega\) we find that the coefficient of the quadratic term is
\[ \omega_n^2 = \frac{ 2\pi \; { \rm GBW}}{(C_1+C_2)R_2} \left[1+\frac{1}{A_0}\left(1+\frac{R_2}{ R_1}\right) \right] .\tag{7}\]Typically, the op-amp DC gain is very high, \(A_0 \gg 1\), and the photodiode shunt resistance exceeds the feedback resistor value by orders of magnitude, \(R_1 \gg R_2\). Hence, to a good approximation
\[ \boxed {\omega_n \simeq \sqrt{ \frac{2\pi \; \rm GBW}{ (C_1+C_2)R_2 } } .} \tag{8}\]In terms of frequency
\[ f_n \simeq \sqrt{ {\rm GBW}\; f_{\rm FB} , \tag{9} }\]with \(f_{\rm FB} = 1/\left[ 2\pi R_2 (C_1+C_2)\right]\), which can be interpreted as saying that the natural frequency is the harmonic mean of the op-amp GBW and the corner frequency of the feedback network. Figure 3 shows that \(f_n\) accurately locates the gain peak and the \(180^\circ\)phase shift.
The coefficient of the linear term in \(\omega\) gives the damping ratio 2
\[ \boxed{\zeta = \frac{ 1+R_2 /R_1 + 2\pi \;C_2 R_2 \; {\rm GBW}\; } {2 \sqrt{2\pi (C_1 +C_2)R_2 \; {\rm GBW} }}, } \tag{10} \]where we have assumed that \(A_0 \gg 1\).
As \(\zeta\) approaches 1 ringing disappears, so Eq. (10) implies that for fixed feedback resistor and input capacitance, we can reduce ringing by increasing the op-amp GBW or adding a finite capacitance is parallel with the feedback resistor.
Choosing \(C_2\) and the Case of \(\zeta=1/\sqrt{2}\)
We can now choose \(\zeta\) by specifying an acceptable degree of overshoot and find the appropriate value of \(C_2 \) for fixed op-amp properties (GBW and \(A_0\)), feedback resistor, and input capacitance.
If \( V_{\rm eq}\) is the asymptotic equilibrium value of the step response, and \( V_{\rm max} \) is the peak overshoot, then \[ V_{\rm max}/V_{\rm eq} = \exp\left( - \frac{\pi \;\zeta}{\sqrt{1-\zeta^2} }\right) . \tag{11}\]
According to Eq, (11) the conventional choice of \(\zeta = 1/\sqrt{2}\) \(\simeq\) 0.7071, corresponds to an overshoot of \(e^{-\pi} \) or 4.3%. Using Eq. (10) and (11) we plot in Figure 7 the degree of overshoot corresponding to \(C_2\) for our current example. This plot also shows how increasing \(C_2\) and the corresponding increase in \(\zeta\) diminishes the bandwidth (see Fig. 6 and the dots marking the -3 dB point on right hand panel of Fig. 5).
Adding feedback capacitance increases \(\zeta\) and decreases the TIA bandwidth.
If we choose \(\zeta =1/\sqrt{2} \) then
\[ C_2 = \frac{(4 \pi \;{\rm GBW} \; C_1 R_1^2 R_2 - R_1^2 -2R_1 R_2 )^{1/2}-R_2 } {2 \pi\; {\rm GBW} R_1 R_2} \tag{12} \]so long as
\[ \frac{4 \pi\; {\rm GBW}\; C_1 R_1^2 R_2 }{(R_1+R_2)^2 } > 1. \]To a good approximation when \(R_1 \gg R_2\)
\[ \boxed{C_2 \simeq \sqrt{\frac{C_1}{\pi \; {\rm GBW}\; R_2 }}; \;\;\;\; \zeta = 1/\sqrt{2} .} \tag{13} \]Also, as \(\omega_{-3db} =\omega_n\) when \(\zeta =1/\sqrt 2\) the resultant -3 dB signal bandwidth for this choice of \(C_2\) is
\[ f_{-3dB } = \sqrt{\frac{\rm GBW}{2 \pi (C_1+C_2)R_2}} . \tag{14}\]As \(C_1\) and \(R_2\) may be fixed by choice of detector and maximum photocurrent, and \(C_2\) set by practical limits on parasitic capacitance, Eq. (14) can be rearranged to determine the op-amp GBW required to achieve a specific signal bandwidth.
The TIA Speed Advantage
A biased photodiode with capacitance \(C_1\) and load resistor \(R_2\) achieves a -3 dB bandwidth of \(f_{\rm LR } = 1/(2 \pi C_1 R_2 )\). Comparison with Eq. (14) shows that the TIA has a bandwidth advantage of approximately \( \sqrt{ {\rm GBW}/f_{\rm LR }}\). In the current example (\(R_2\) = 10 k\(\Omega\) and \(C_1\) = 10 pF), the speed advantage factor for a 500 MHz GBW op-amp is \(\times \)11.
Noise in Transimpedance Amplifiers
By noise we mean noise generated in the op-amp and its associated components and not interference or unwanted external signals. Several noise sources contribute to the signal-to-noise achieved by a transimpedance amplifier. Specific to transimpedance amplifiers are internal current and voltage noise contributions that are present in non-ideal op-amps. There are also fundamental and ubiquitous noise contributions due to quantization of charge and the associated Poisson noise and the thermal or Johnson noise that arises in resistors (usually dominated by the feedback resistor).
These noise sources are normally uncorrelated. If they exhibit no temporal correlation, i.e. no covariance, each noise source can be evaluated independently and the total noise evaluated as root sum squares.
Input Referenced Op-Amp Voltage Noise, \(e_n\)
The contribution of the op-amp's voltage noise spectrum is denoted as \(e_n(\omega)\).3 Typically, \(e_n\) is listed on manufacturers' data sheets and is the input referenced voltage noise density with units of \( {\rm V}/\sqrt{\rm Hz}.\) Common values for high-performance op-amps are \(e_n\) = 1–10 nV\(/\sqrt{\rm Hz}\).
Figure 8 depicts an ideal, noise-free op-amp with an external voltage noise source labeled \(e_n\) connected to the inverting input. This configuration represents the conventional approach to modeling op-amp voltage noise.
Analysis of \(e_n\) noise
Applying Kirchoff's circuit laws to Figure 8 and writing \(Z_i \) as the parallel combination \(R_i\) // \(C_i\) we have \[ \begin{eqnarray} V_1 & = & I Z_1 \\ V_M & = & V_1 + e_n (\omega) \\ V_o & = & I Z_2 + V_1 \\ V_o & = & A(\omega) (V_P-V_M) \end{eqnarray}. \tag{15} \]
Eliminating \(I\) and \(V_1\) we find the corresponding output voltage noise spectrum as
\[ e_{nV} (\omega) = - \frac{A(\omega)(Z_1 + Z_2) }{Z_1 + A(\omega)Z_1 + Z_2 } e_n(\omega) . \tag{16} \]When the op-amp gain is large, the feedback resistor is the dominant contribution to \(Z_2\), and the diode shunt resistance is effectively infinite, we have
\[ e_{nV} ( \omega) \simeq (1+ \mathbb{i} \omega C_1 R_2) e_n(\omega).\tag{17} \]Thus, at low frequencies the op-amp in Fig. 8 acts like a follower and the output voltage noise equals \(e_n\). As frequency increases the voltage noise source interacts with the input capacitance, \(C_1\), because feedback drives a current, \(e_n/Z_1\) so that the voltage at the inverting remains close to the ground potential of the non-inverting input. The associated noise current is \( \mathbb{i} \omega C_1 e_n\) corresponding to an output voltage of \( \mathbb{i} \omega C_1 R_2 e_n \). The noise induced by combination of \(e_n\) and input capacitance is designated \(e_n C\) noise, and increases with increasing frequency and input capacitance.
As an example, consider a range values of \(C_1\) and a flat input voltage noise spectrum with \(e_n \) = 5 nV/\(\sqrt{\rm Hz}\). The curves for the output noise voltage in Figure 9 show a prominent peak which increases in magnitude in proportion with the value of \(C_1\) and shifts to lower frequencies at \(f_n\) given by Eq. (9).
The dotted lines in Figure 9 show the impact of adopting the value of \( C_2\) given by Eq. (13) so that the damping ratio is \(1/\sqrt{2}\). Adding capacitance in parallel with the feedback resistor reduces but does not eliminate the \( e_N C \) noise.
Input Referenced Op-Amp Current Noise \(i_n\)
Figure 10 shows the TIA configuration of an ideal op-amp combined with a current noise source \(i_n(\omega) \) that represents the op-amp's input referenced current noise. The input referenced current noise can range from fA/\(\sqrt{\rm Hz}\) to pA/\(\sqrt{\rm Hz}\), depending on op-amp architecture (e.g., BJT, JFET, or CMOS).
The current noise appears between the inverting input and ground and therefore in parallel to the photodiode current in Fig. 2. Consequently, the signal gain relation in Eq. (3) also describes the relation between \(i_n(\omega)\) and the output voltage noise,
\[ e_{nI}= \; \frac{ A(\omega) Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2} i_n(\omega) . \tag{18}\]As the voltage output associated with \(i_n\) is indistinguishable from input signal current and Fig. 5 (right) shows the shape of the corresponding output voltage noise. If \(i_n\) is flat, then the associated output voltage noise rolls over at \(f_n = \sqrt{ {\rm GBW} f_{\rm FB} }\) according to Eq. (9)..
Johnson Noise
Johnson noise is a fundamental noise source associated with thermal agitation of charge carriers in a resistor at any temperature above absolute zero. At radio frequencies (\( \nu \ll k_B T/h \simeq 6 \) THz at room temperature) the spectrum of Johnson noise is flat. The TIA further limits the output voltage noise at high frequencies.
Johnson noise in a resistor can be modeled as a noiseless resistor in series with a voltage noise power spectral density with \( e^2_{nJ} = 4k_B T R \), with units \( \rm{V} ^2 \ \rm{Hz}^{-1} \) and Boltzmann constant \(k_B\). For the Johnson noise associated with the TIA feedback resistor, \(R_2\), it is convenient to consider the complementary circuit of a noiseless resistor in parallel with a current source with noise power spectrum,
\[ i^2_{nJ} = 4 k_B T /R \ \ \ [ \rm{A} ^2 \ \rm{Hz}^{-1} ] . \tag{19}\]Figure 11 illustrates this circumstance where the Johnson noise current appears in parallel with \(Z_2\), where \(R_2\) and \(C_2\) in parallel comprise \(Z_2\).
Using the currents and voltages labeled on Figure 11, Kirchoff's circuit laws require \[ \begin{eqnarray} I_1 & =& I_2 + I_J\\ V_o & = & I_2 Z_2 + V_1 \\ V_1 & = & I_1 Z_1 . \tag{20} \end{eqnarray} \] Combining these relations yields the output voltage corresponding to \(I_J\) as \[ e_{nJ}(\omega) = -A(\omega) \frac{ Z_1 Z_2}{Z_1 + A(\omega) Z_1 +Z_2 } I_J . \tag{21} \]
The transfer function for photodiode current Eq. (3) and Johnson noise current are identical.
In terms of the output voltage power spectral density, the Johnson noise contribution is \[ e_{nJ}^2(\omega) = \left[ A(\omega) \frac{ Z_1 Z_2}{Z_1 +A(\omega) Z_1 +Z_2 }\right]^2 \ i^2_{nJ} . \tag{22} \]
For an ideal op-amp with \(A\gg 1\), the output voltage Johnson noise is \( e_{nJ} = - i_{nJ} Z_2 \). As \( Z_2 \) is made up of the feedback resistor \(R_2\) and the parallel feedback capacitance \(C_2\), the output voltage noise is \(e_{nJ} = -i_{nJ} /(1/R_2 + \mathbb{i} \omega C_2 )\) and the Johnson noise rolls over at a 3dB point with frequency \(1/(2\pi R_2 C_2) \).
An Example of Op-Amp Noise: LTC6268
We adopt the LTC6268 op-amp as an example for noise analysis. This device has GBW = 500 MHz, a high open loop gain (\(A_0 = 250,000\)), a FET-input with low input bias current (3 fA) and low input capacitance (0.45 pF). It also has low input referenced voltage and current noise noise making it suitable for fast transimpedance amplifier designs. Figure 12 shows the measured input referenced voltage noise for an LTC6268.
Figure 13 shows \(i_n(\omega)\) for the LTC6268 op-amp.
The contributions of \(e_n\), \(i_n\), and \(i_{nJ}\) to the output voltage noise are plotted in Figure 14 for a TIA using an LTC6268 with a 10 k\(\Omega\) feedback resistor and a total input capacitance of 10 pF. A compensating capacitor of \(0.8\ {\rm pF}\), is placed in parallel with the feedback resistor to achieve a damping ratio of \(\zeta = 1/\sqrt{2}\). The Johnson noise from the feedback resistor is also plotted.
Equations (16), (18), (19), and (22) are used to find the individual noise contributions. The combined noise is found assuming that these sources are uncorrelated so that the root-sum-square yields the total. Figure 14 shows that at low frequencies ( \(\lt \) 4 MHz) Johnson noise from the feedback resistor (13 nV/ \( \sqrt{\rm Hz}\) ) dominates. At high frequencies the \(e_n C\) noise peak is dominant because of the relatively large value of input capacitance (10 pF). Input current noise makes a relatively small contribution.
Figure 14 also shows the noise prediction using the manufacturers model for the LTC6268 op-amp built in to LTspice (dot-dashed lines). The LTspice model generates the total noise and the Johnson noise contribution but not the individual contributions of \(e_n\) and \(i_n\).
In the LTspice TIA circuit model we assume that the value of the op-amp common mode input capacitance quoted in the datasheet of \(C_{in} \) = 0.45 pF and a photodiode capacitance of 9.55 pF, for a total \(C_1 \) = 10 pF to match the analytic calculation. The agreement of Eqs. (16), (18), (19), and (22) and LTspice is excellent over the full range of frequencies. Above 20 MHz a discrepancy is noticeable where LTspice predicts a total noise that is about 5% lower than our results. Comparisons of LTspice calculations for lower values of \(C_1\) where \(e_n\) is less dominant suggests that the difference arises because the model for the LTC6268 in LTspice adopts a lower value for \(i_n\) than that plotted in the datasheet and used here.
Signal-to-Noise Ratio and NEP
If the signal is photocurrent \( I_D(\omega) \), then the signal-to-noise ratio is output voltage given by Eq. (3) divided by the total noise combined as root-sum-square, i.e.,
\[ SNR = \frac{ I_D H(\omega) }{ \sqrt{ e_{nP}^2 + e_{nV}^2 + e_{nI}^2 +e_{nJ}^2 }} . \tag{23}\]The first term in the denominator of Eq. (23) accounts for the Poisson or shot noise associated with the photodiode current, which includes not only the photocurrent but also any dark current, \(I_{\rm DC}\), contribution. Photon counting statistics sets the fundamental quantum limit on SNR when all other noise sources have been eliminated. The current shot noise power spectral density is \( 2 q_e (I_D + I_{\rm DC}) \) with units A\(^2/\rm{Hz}\) and \(q_e\) is the elementary charge. The corresponding output voltage noise, \(e_{nP}\), is found using Eq. (3) or Eq. (18). The other three noise terms are the op-amp voltage noise (Eq. 16), op-amp current noise Eq. (18), and Johnson noise Eq. (22), respectively.
The noise-equivalent power (NEP) is a measure of the sensitivity of a photodiode and TIA combination defined as the signal power that gives unity signal-to-noise ratio in a one hertz bandwidth. To compute the NEP we need the system responsivity, \(\mathcal{R} = I_D/P \), which gives the photocurrent \(I_D\) in terms of the incident radiant power \(P\). As the photon energy is \( h \nu\) then the photocurrent is \[ I_D = \frac{\eta P q_e}{h\nu } \] and \[ \mathcal{R} = \frac{\eta\; q_e}{h\nu },\]
where we have included a photon detection efficiency factor, \(\eta\), known as the quantum efficiency. The NEP is then computed as the ratio of the total noise spectrum divided by the responsivity.
Appendix: Operational Amplifier Gain Parameterization
A convenient parameterization describes the gain as a function of frequency, \(\omega = 2 \pi f\), as a single pole, low-pass filter with open-loop DC gain \(A_0\) \[ A(\omega) = \frac{A_0}{1+ \mathbb{i}( \omega /\omega_p)} \] and the location of the pole is given by \( \omega_p\). The op-amp's bandwidth is usually specified in manufacturers' datasheets as the gain-bandwidth product, GBW, or equivalently where the magnitude of the gain falls to unity. Hence, \[ |A(\rm{GBW})|^2 = 1 \] or \[ \frac{A_0^2}{1+ ( 2\pi \; {\rm GBW} /\omega_p)^2} = 1 . \] Hence, \[ \omega_p = 2 \pi \frac{ {\rm GBW}}{\sqrt{A_0^2-1}} .\] As the open loop, DC gain is very large \[ \boxed{ \omega_p \simeq 2\pi \frac{ {\rm GBW}}{A_0} , } \tag{A1} \] is an excellent approximation.
Notes
1. Horowitz and Hill in The Art of Electronics: The x-Chapters introduce a quantity \(\zeta\) in § 4x.3.2, which they call the damping ratio. The caption to their Fig. 4x.20 states \(\zeta =1 \) corresponds to critically damped, in agreement with common usage. A critically damped system will just fail to overshoot; however, the step response for \(\zeta=1\) (Fig. 4x.21, Case b), shows pronounced overshoot. Inspection of the numerically computed transfer functions in Fig 4x.20 and step responses in Fig. 4x.21 suggests that Case b is not \(\zeta = 1\) but \(\zeta = 1/\sqrt{2} \simeq 0.707\) and Case c is the critically damped case with \(\zeta = 1\). Case-a (identified as 1.25 dB gain peaking in Fig. 4x.20) is for a damping ratio of \(\zeta = 1/2\). ↩
2. An expression for \(\zeta\) appears on p.14 of the LTC6268 datasheet . This expression as presented is dimensionally inconsistent and must contains typographical errors. ↩
3. The squared quantity, \(e^2_n (f) \), is the voltage power spectrum with units of \({\rm V^2\, Hz^{-1}}\). For example, the Johnson noise associated with a resistor, \(R\), at absolute temperature, \(T\), is \(e^2_n = 4 k_B T R \). The quantity \(e_n \) is convenient because if the spectrum is flat (a constant) like Johnson noise, the RMS voltage measured in bandwidth \(\Delta \nu\) is \( e_n \sqrt{\Delta\nu}\). ↩
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