It is shown that even two element circuits may involve important dynamic derivation. The transimpedance amplifier analysis in this paper provides a general design template for such circuits, and also provides a guiding example to explain how to analyze the dynamic characteristics of the amplifier. In the first part of this series, the gain of the transimpedance amplifier circuit is derived from the finite gain monopole amplifier to the infinite gain monopole amplifier, as shown in Figure 1. In the second part of this paper, we will study its consequences.

Figure 1: a seemingly simple circuit has only two components: an operational amplifier and a feedback resistor.

From the first part, we know that the gain, i.e. trans impedance, is derived as follows:

The poles are:

Amplifier gain gives us an opportunity to apply control theory to circuits. This example will illustrate the importance and practicability of control theory in understanding the dynamic characteristics of circuits. We hope that we can have a deep understanding of the control technology and its application.

Pole pair (quadratic) polynomials are usually expressed as:

Resonant time constant of amplifier τ n = 1/ ω n = 1/（2 x π X FN) and damping ζ They are as follows:

When ζ《 1, the pole becomes a complex pole pair, and the pole angle is:

For real poles, ζ 》 1 and φ = 0。

For a constant group (or envelope) delay (maximum flat envelope delay / mfed or Bessel) response, the phase decreases linearly with frequency and occurs at φ = 30o. The time delay of all frequencies is the same, keeping the waveform unchanged. then:

For transimpedance amplifier mfed response:

For critical damping (the fastest step response without overshoot), ζ = 1 and τ T = 4 x τ I or ft = FI / 4. Both poles are FI / 2.

As RR increases and fi decreases, the amplifier shows more overshoot in VIX. To some extent, this is advantageous for z-meter because of the polar angle φ = forty-five °， damping ζ = cos（ φ） = Cos (45o) ≈ 0.707, and the frequency (or amplitude) response is constant or flat, close to the bandwidth frequency. This is the maximum flat amplitude (MFA) frequency response. For steady-state (frequency domain) applications, MFA response is optimal. For transient (time domain) applications with ideal step response, mfed response is optimal（ In the design of vertical amplifier for oscilloscopes, the two standards for optimizing the response are conflicting.)

Operational amplifier speed and amplifier stability

The slow op amp has low ft and high efficiency τ T 》》 τ i. The result is that the two real poles are far away. At the limit value:

This is the origin and the pole at fi. FT must be small enough to keep ft < fi. However, with the decrease of FT, the loop gain decreases, which may not be enough to maintain the gain error of op amp. In this case, accuracy requires a certain speed.

With the increase of operational amplifier ft, the damping and stability of ZM decrease. For a given ς And fi:

If ft = 1MHz and G0 = 105, FG = 10Hz and critical damping loop（ ζ = 1) Fi = 40Hz. Suppose CI = 10PF, then RR = 398m Ω， In this way, fi “40Hz can be maintained for any small value.

Figure 2 shows how the closed-loop poles move as ft (faster operational amplifier) increases. At the origin and fi (– 1/ τ i) The pole of separation is at FI / 2 (at this point) π = 1) Then it becomes a pair of complex poles. With the increase of FT, the polar angle increases and ζ reduce. The amplifier becomes unstable and the response is more oscillatory.

Figure 2: the closed loop poles move with the increase of ft.

As long as the parameter changes (FT in Figure 2 or τ T) If it appears in the S2 and s terms of the polynomial at the same time, the position or trajectory of the pole movement will be shown in the graph. The damping of the amplifier is the smallest at infinite ft, and the damping of the amplifier is the smallest at infinite ft τ When t → 0s, the pole position is at the limit value

In JX ω There are two values on the axis, and the response is stable (not oscillatory): the origin and ± JX ∞. Both are infinite, (zero (0) is infinitesimal). When τ When t → 0s, the two terms of s in the pole polynomial are close to zero, leaving a constant term and not affected by frequency. In the limit case, the pole is located at JX ω On the shaft, ζ= But at the finite value of S, their amplitude is zero. The pole frequency is very high and damping is no longer important. They are too far away from fi to affect loop dynamics. This is the condition of ideal operational amplifier. Therefore, we can conclude that for very slow or very fast operational amplifiers, the poles are sufficiently separated to make the response stable. Only in the range of FT, when the op amp and CI poles are too close, the damping is excessively reduced at the sufficiently low pole frequency FN, and a considerable oscillation occurs in the amplifier.

And back to the transimpedance amplifier, if the op amp is almost ideal, that is, fast enough τ T ≈ 0s, then the pole polynomial is about 1. If the operational amplifier is fast enough and the poles are separated, there will be a stable loop. In order to provide additional damping so that the operational amplifier ft (and loop gain) is not too low, the capacitor CF needs to be shunted through RR. Then we use the circuit algebra containing CF to calculate:

The pole pair parameters are as follows:

The function of CF is to reduce the temperature in the quadratic coefficient τ F added to τ i. What’s more important is the sum added to the linear term τ T. This will increase the damping. because τ i = τ T. So:

For the critical damping, let π = 1; that τ T = （3 + 2 x √2） x τ i ≈ 3.414 x τ I and τ n ≈ 1.848 x τ i。 If there is no CF (CF = 0pf), as previously calculated, τ T = 4 x τ i。 If there is CF, the operational amplifier can be faster with the same dynamic response, that is, it has higher G0 and higher accuracy.

The amplitude and phase of frequency response are as follows:

For ideal fast operational amplifier（ τ T = 0s) and when CF = CI（ τ f = τ i) When the frequency FG (or ω g) There is a response at:

If fi = 10 x FG, the amplitude error is about 0.5%. Because fi = 10 x FG, the phase error is about 6O. Phase error is more sensitive to frequency effect than amplitude error. This is very important in the circuit design of impedance meter, and sometimes in the photoelectric detection amplifier, because the photoelectric detection waveform has to be synchronized with some other waveforms.

Circuit avoiding large feedback resistance

For some z-meter (ZM) designs with transimpedance amplifiers, the RR should be large enough, that is, 10m Ω Or more. When RR becomes very large, shunt CF must be very small and shunt parasitic capacitance may be too large to obtain the desired damping. To avoid this problem, the following circuit can be used instead.

Figure 3: use this circuit to avoid excessive resistance shunt parasitic capacitance.

To make the operational amplifier a high gain monopole operational amplifier, G ≈– 1 / s X τ T (see the derivation of G in the first part of this series). The transfer function of the feedback divider is as follows:

And τ f = RR x Cf。 When the circuit is solved by RP = R1 | R2:

Ideal operational amplifier（ τ ZM of T = 0s) is reduced to:

For RP = 0 Ω， The span resistance is further reduced to:

If you insert a fast filter between the output and RR and CF × 1 buffer amplifier, the output resistance of R1 and R2 voltage divider need not be too small (RP < RR). So if RP = 0 Ω And the operational amplifier has τ At t:

There are two differences between this circuit and the case without output divider: RR and τ T increased 1 / hdiv effectively.

epilogue

It can be seen from the two parts of this paper that even a simple circuit with only two devices may involve complex dynamic derivation. Designers sometimes avoid using these derivations to reduce the trouble of mathematical calculation, but using these formulas can better understand the performance of a given circuit under various conditions. The analysis of transimpedance amplifier introduced in this paper can provide a template for such circuit design, and provide a guiding example of how to analyze the dynamic characteristics of the amplifier.

Don’t refuse to use S-field algebra to solve circuit dynamic problems because of cubic or higher polynomial. In this example, we encounter a cubic term, but it is not necessary to solve it, because the polynomial can be reduced to a quadratic equation by simplification, which is convenient for later analysis and calculation. This is common because circuits are often modularized at the design stage, either isolated from each other or interacting with each other through controlled port impedance. Template scheme can be used in design, but it is usually limited to quadratic equation in s domain.

Editor in charge: GT