System Response Time & Servo Bandwidth

Overview

The analog response time is the time it takes from when an analog input changes to when a responding analog output can change. How fast this response is sets how fast the Quarto can respond to a signal change, and thus the servo bandwidth of the Quarto. This total response time can be broken down into several parts as an analog input goes into the Quarto and makes it way to an analog output:

1. DAC Update Latency
2. Analog Output Filter
3. Analog Input Filter
4. ADC Measurement Latency
5. Interrupt Latency
6. MCU Processing Time

DAC Update Time

The DAC Update Latency Time is 1µs¹.

Analog Output Filter

The output filter between the DAC and the Quarto's Analog Output has a bandwidth of 180 kHz¹. While this filter isn't a simple delay circuit, as the delay varies for different frequency components, for frequencies at 100 kHz or below, the delay is 1.2µs.

Analog Input Filter

The input filter in front of the Analog Input has a bandwidth of 400 kHz. While this filter isn't a simple delay circuit, as the delay varies for different frequency components, for frequencies at 100 kHz or below, the delay is 550ns.

ADC Measurement Time

The ADC Measurement Time is 800ns².

Interrupt Latency

Please see Measuring Interrupt Latency App Note note for details, but the worst-case interrupt latency is 210ns.

MCU Processing Time

Clearly this depends on the function that that is executing. In the Measuring Interrupt Latency App Note the processing time was only 160ns. More complex functions will take longer to run. If unsure how long something takes to execute, as done in the app note, can you use a trigger line to measure the execution time with an O-scope. The code would be:

triggerWrite(1,HIGH); // Set Trigger 1 high at start on function
functiontoTime();
triggerWrite(1,LOW); // Set Trigger 1 low after function completes

But for straight-forward arithmetic functions (no loops), 500ns is a safe estimate.

Closed Loop Bandwidth / Delay

In the case of a servo where the Quarto's analog output is connected to its analog input, then we can measure the delay through the entire closed loop. If at t=0, we look at a signal at the Quarto's analog input, then that signal gets to the ADC at time 800ns. The interrupt latency adds 210ns, so the MCU gets the data at time 1.01µs. Assuming the MCU takes 290ns to calculate the DAC update value, then at t=1.3µs the MCU updates the DAC. That update hits the DAC output at t=2.3µs and gets through the output filter at 3.5µs and through the ADC input filter at 4.05us. This means that the closed-loop signal delay is 4.05µs. However, this ignores the sampling time of the ADC. If the ADC is sampling every 1µs, then some information is delayed by nearly 1µs before the ADC samples it and other information is barely delayed at all. A reasonable approximation is to assume that all signals are delayed by, on average, half the sampling time. When sampling every 1µs, this adds an additional delay of 500ns. If sampling at 2µs, then the extra delay would be 1µs.

Putting this all together, we have a total delay of 4.05µs plus the time between a voltage change event and when the ADC samples that change. This delay is (on average) approximately half the sampling rate. So if you have a 4-channel PID servo, where each channel channel is sampling at 4µs, then that adds 2µs of delay, for a total of 6.05µs. The closed loop oscillation frequency is where the servo has 180° phase shift, which is when the delay is equal to half the period of oscillation. At 100kHz, one period is 10µs, so oscillations at 100kHz occurs when the delay is 5µs. The table below shows the delay for a few different ADC sampling rates along with the expected servo loop oscillation frequency.

ADC Sample Rate (µs)	Max # of PID Loops	Total Delay (µs)	Expected Oscillation Freq (kHz)
1	1	4.55	110
2	2	5.05	100
3	3	5.55	90
4	4	6.05	82

Footnotes

1. See DAC section of Specifications Page. ↩ ↩²

2. See ADC section of Specifications Page. ↩