Divide (48.625)$_{10}$ by (5)$_{10}$ using A/D converter.

To divide a decimal number by another using an Analog-to-Digital (A/D) converter, we need to understand that A/D converters are designed to convert analog signals to digital signals, not to perform arithmetic operations directly. However, we can simulate the division process by using a digital-to-analog converter (DAC) in conjunction with an A/D converter in a feedback loop. This process is iterative and is known as successive approximation.

Here's a step-by-step approach to perform the division of 48.625 by 5 using an A/D converter through successive approximation:

Step 1: Set Up the DAC and A/D Converter

First, we need a DAC that can convert digital values to analog voltages and an A/D converter that can convert analog voltages back to digital values. The A/D converter should have a reference voltage set to the divisor, which is 5 in this case.

Step 2: Initialize Variables

We will initialize a digital counter (which will hold our approximation of the quotient) to 0. We will also need a variable to hold the current DAC output, which will start at 0.

Step 3: Successive Approximation Loop

We will iteratively increase the digital counter, convert this value to an analog voltage using the DAC, and then compare this voltage with the dividend (48.625) using the A/D converter. The goal is to find the digital counter value that causes the DAC output to be just less than the dividend when converted back to a digital value by the A/D converter.

Step 4: Perform Iteration

1. Increment the digital counter by 1.
2. Convert the digital counter value to an analog voltage using the DAC.
3. Use the A/D converter to convert this analog voltage back to a digital value, which will be scaled according to the reference voltage (5).
4. If the A/D converter output is less than the dividend (48.625), continue to the next iteration. If it is equal to or greater than the dividend, decrement the counter by 1 to get the last value that was less than the dividend.

Step 5: Determine Quotient and Remainder

Once the loop ends, the digital counter will hold the quotient of the division. To find the remainder, we can subtract the product of the quotient and the divisor from the original dividend.

Example

Let's assume we have an 8-bit A/D converter and DAC, and the reference voltage for the A/D converter is set to 5 volts. The dividend (48.625) is represented by an analog voltage input to the A/D converter.

Here's a simplified example of how the process might look:

Iteration	Digital Counter	DAC Output (V)	A/D Conversion	A/D Output (Digital)	Dividend (Digital)	Action
1	1	0.0390625	0.0390625 * 51.2	2	48.625 * 10.24 = 498	Continue
2	2	0.078125	0.078125 * 51.2	4	498	Continue
...	...	...	...	...	...	...
9	9	0.3515625	0.3515625 * 51.2	18	498	Continue
10	10	0.390625	0.390625 * 51.2	20	498	Stop (>= 48.625)

In this example, the digital counter value of 9 is the last value before the DAC output exceeds the dividend when converted back to digital. Therefore, the quotient is 9.

To find the remainder:

$$ \text{Remainder} = \text{Dividend} - (\text{Quotient} \times \text{Divisor}) $$

$$ \text{Remainder} = 48.625 - (9 \times 5) = 48.625 - 45 = 3.625 $$

So, the quotient is 9 and the remainder is 3.625.

Please note that this is a conceptual example. In practice, the resolution of the DAC and A/D converter, as well as the reference voltage, will affect the precision of the result. The actual implementation would require more detailed consideration of the hardware specifications and the scaling factors used for the conversion between digital and analog values.