In Part 1, we reviewed the low frequency behavior of the HCNR200 circuit, and obtained the stability conditions by analyzing its open-loop transfer function. In this session, we still study the system’s closed-loop transfer function and use that to design our system.

To analyze the system’s performance, we first calculate the closed-loop transfer function of the system. We still model the Op Amp as a single pole system with a DC gain of AV, and its transfer function is

The closed-loop transfer function is

because C1 loads the output of the Op Amp, and therefore shouldn’t be much larger (close to Av times) than COUT.

Our actual system’s output, PD2’s output current, is a function of VOUT:

where K0 is the low frequency gain of K , and ωK = 2π9MHz. ωK is the Op Amp’s crossover angular frequency.

Therefore the system’s overall transfer function is,

Now, we have the system’s transfer function. In this session, we will discuss how to optimize the system for specific performance requirements. Before getting into more details, we first take a look at some general constrains from the system, and thus roughly in what ranges ωc , R3, R1, K, and C1 are.

First, ωc is 2π times GBW of an Op Amp. This is usually a large number about from tens to hundreds of Mega Rad/s.

Second, the LED’s current has a range: the typical value is 10mA, and maximum, 20mA [1]. This current flows through R3, which is driven by the Op Amp. If R3 is too large, the Op Amp will saturate. That means if the Op Amp’s power rail is +/-5V, and the LED is also biased by 5V, then R3 has to be smaller than 840 Ω to generate a maximum 10mA LED current. (LED’s forward voltage is about 1.6V [1]) Of course, Op Amp’s power rail can be higher and the LED can have a smaller maximum current. Nerveless, it is reasonable to assume R3 is not larger than several KΩ.

Is there any limit on how small R3 can be? In fact, the external resistor in series with the LED can be as small as 0, because the LED has an Rb = 3.469 Ω [1]. Therefore, even the external resistor is 0, the system can still be stable. In the later part of this article, we will refer R3 as the sum of external resistor and Rb for simplicity. I.e. our R3 has a lower limit of 3.469 Ω.

Third, we want to find out in what range R1 should be. R1 directly translates VIN into photodiode current, which has a typical value of 10mA×0.5%=50µA. Most instruments generate reliably a voltage in the order of a couple Volts. If that’s where VIN lays, R1 will be about from tens to hundreds KΩ. (We will see that a relatively large R1 helps increase bandwidth, and lower THD and noise in later sessions.)

Forth, K is 0.5% at low frequency, and decrease as frequency goes higher.

Last but not least, C1 loads the output of the Op Amp. It shouldn’t be a huge number. It can’t be 0 either because of the stability constrains. In real system, its lower limit can be from the parasitic capacitance from the Op Amp and PCB.

Now, we know the rough ranges of the parameters we can adjust in the system. Next, we will discuss how to fine tune those parameters for specific dynamic performances.

First we study how to design for high bandwidth. From the transfer function:

we can see that the system’s bandwidth is only determined by how fast the denominator increases with frequency, since there is no zeros in the system. This is because the pole from β2 (transfer function from VIN to V-) cancels the zero from β1 (transfer function from VOUT to V-). At the frequency that the denominator’s magnitude reaches 1%, the system reaches its 3dB bandwidth.

That mean, to have a high bandwidth system, we need to have small R3, C1, and large ωc and R1. To start a high bandwidth design, we can first set R3 to it low limit 3.469 Ω. Then we make R1 the largest possible value. There is no upper limit for R1 if only the transfer function is concerned. However, this value can be limited by other factors such as the highest voltage the proceeding device can generate, the parasitic capacitance of the high value resistors, noise immunity, and etc. Last, if we have an Op Amp already, we can back calculate a C1 that gives the desirable bandwidth while keeping the system stable. Or we can make the C1 as small as the PCB parasitic capacitor, then pick an Op Amp whose GBW is high enough.

i.e., when R1 and Op Amp GBW are both large, C1 will have the major impact to the dominator, because the transfer function can be approximate by

This also means, keep increasing the Op Amp’s bandwidth won’t change the system’s bandwidth much after a certain point, because the 2nd order term in the denominator will dominant the polynomial at the -3dB frequency. That indicates a well-chosen Op Amp’s bandwidth should make the 2nd order and 3rd order terms yield similar values at the system -3dB frequency in a high bandwidth design.

Fig. 7 plots the simulation results of IOUT when C1 = 1, 5, and 10pF, and R1 = 100KΩ, ωc =2π500 MRad/s, and R3 = 3.469Ω. Ideal Op Amp was used.

As we can see from the plot, when C1 = 1pF, the 3dB bandwidth was about 30MHz, while C1 = 10pF yields only a 9MHz bandwidth.

This also shows one advantage of this circuit: the system can have a much higher bandwidth that the optocoupler’s bandwidth, 9MHz. This is because the feedback extended the bandwidth.

Readers probably already noticed that the “C1 = 1pF” plot has more peaking than others. This translates to ringing in step response in time domain. Why does a high C1 suppress the ringing?

As we have stated before, a large C1 make the 2nd order term dominant in the polynomial, if Op Amp’s bandwidth is high. Imaging an extreme case that the Op Amp’s GBW product is very high, then at frequencies up to the -3dB bandwidth of the system, the transfer function can be simplified as

That means a larger C1 is translated into a larger damping coefficient. This also shows that there is a tradeoff between bandwidth and step response’s ringing. A larger damping coefficient requires larger R3 and C1, while high bandwidth need both to be small. If we desire to damp the system with less sacrifice of bandwidth, we can increase C1 but keep R3 at its minimum value, because R3 scales the transfer function denominator’s both 3rd order and 2nd order terms by the same factor, while C1 affects the 3rd order term less.  Another way to damp the system is to increase the bandwidth of the Op Amp. This will suppress the 3rd order term. However, the bandwidth has to be much larger to achieve a similar effect. 

Fig. 8a shows the impact on peaking from R3, C1, and Op Amp’s bandwidth. R1 was kept at 100kΩ all the time. Fig. 8b shows their corresponding the step responses.

As we can see from the plots, by increasing C1 by 2pf, the peaking is largely suppressed. To achieve a similar damping of the system, R3 is increased by 15Ω. However, the bandwidth reduction due to R3’s increase is much more than due to C1’s. Increasing Op Amp’s bandwidth to 1GHz shows a similar damping to the system while maintaining the bandwidth. However, a 500MHz bandwidth increase may be less economical.   

Fig8a. Peaking at different conditions

Fig8b. step response at different conditions

—Yong Liao is a Sr. Test/HW Engineer at Broadcom with an MSEE.

—Xi Cheng is an R&D Test Engineer at Broadcom, with an MSEE and a physics PhD.

You must Sign in or Register to post a comment.