# Temperature Sensing and a Current Ratio of 17

Mathematics is the part of physics where experiments are cheap.

## Introduction

Figure 1: My Idea of a Thermometer.

One of the most common diagnostic functions requested for an electronic system is to measure its own temperature. We want to know the hardware temperature when a problem occurs because many system characteristics are affected by temperature and temperature may give us a clue to the problem's root cause.

Today, I was looking at the specification for a temperature measurement integrated circuit (TMP75B) and I noticed that it has a very common block diagram – two switched current sources of different drive levels feeding a transistor's base-emitter junction.

As I looked at this block diagram, I recalled reading that the current ratio of the current sources is normally set to 17. You don't often see the number 17 randomly appear in electronics, so there must be a reason. Let's do a little mathematical experimentation to see if we can figure out why.

## Background

### Block Diagram

Figure 2 shows a block diagram of the TMP75B. Notice the two current sources that drive the base-emitter junction of a diode-connected, bipolar transistor. This is a very common architecture for an integrated circuit-based, temperature sensor.

Figure 2: Block Diagram of TI's TMP75B.

### Ratio Quote

Here is a quote from an ON Semiconductor specification that mentions that the current ratio is typically set at 17.

Setting Ic1 as a fixed multiple, N, of Ic2 gives:

 Eq. 1 $\displaystyle \Delta {{V}_{{BE}}}=\frac{{{{k}_{b}}\cdot T}}{q}\cdot \ln \left( N \right)$

This is the equation used internally in 2−current ON Semiconductor devices to calculate temperature based on the difference in Vbe measurements. The typical value used for N is 17.

I have seen 17 used as the current ratio for other switched, dual-current source temperature sensor circuits, but I do not recall where.

### Definitions

Before we proceed to the Analysis section, we need to define a few terms.

An ADC is a device that converts a continuous physical quantity (usually voltage) to a digital number that represents the quantity's amplitude (Wikipedia).
The number of discrete binary numbers (sometimes called codes) available to represent a range of analog input, like 0 V to 1 V. The ADC count number is normally a power of 2 (Wikipedia).
LSB Voltage (δVBE)
The minimum change in voltage required to guarantee a change in the output code level is called the Least Significant Bit (LSB) voltage (Wikipedia).

## Analysis

### Approach

First, I will derive Equation 1. We can use Equation 1 to derive Equation 2, which is very useful for designers because it relates the temperature resolution to the current ratio and ADC resolution.

 Eq. 2 $\displaystyle \ln \left( N \right)\cdot {{N}_{{ADC}}}=\frac{{{{V}_\text{Ref}}\cdot q}}{{{{k}_{b}}\cdot \delta T}}$

where

• kb is Boltzmann's constant.
• δT is temperature resolution, which is smallest temperature change the system can measure.
• q is elementary charge.
• VRef is the ADC reference voltage. This represents the high-end analog voltage that the ADC will convert properly.

### Setup

Figure 3 shows the how I defined the critical constants in the analysis to follow. I also show how we can compute δT knowing the positive temperature range and maximum positive ADC value.

Figure 3: Analysis Setup and Temperature Sensor Requirements.

### Derivation of Equation 1

We can derive Equation 1 as shown in Figure 4.

Figure 4: Derivation of Equation 1.

### Derive Equation 2

Figure 5 shows how we can derive Equation 2. This is the key equation needed to explain the appearance of N= 17.

Figure 5: Relationship Between ADC Resolution and Current Ratio.

### Numerical Example

In Figure 6, I assume that the voltage used for the ADC reference voltage as 1 V (a guess). I can then evaluate the right-hand side of Equation 2 to determine N for various ADC count values.

Figure 6: Evaluation of Equation 2.

As you can see in Figure 6, a current ratio of 17 appears when we use an ADC with 16 bits. The other ratios are large, which would be difficult to control accurately. So we may have found where ratio of 17 comes from.

### Dynamic Range

One issue remains – the TMP75B only reports back 12 bits of temperature information, yet I am assuming a 16-bit ADC. What limits the amount of information that can be returned?

Figure 7 investigates that issue and, like many engineering issues, the problem has to do with dynamic range. While we need a 16-bit converter to get the temperature measurement resolution we want, the input voltage does not vary enough to use all 16-bits of the ADC.

Figure 7: Dynamic Range of Temperature Reading.

The analysis in Figure 7 shows that there is only 12 bits of information in the data returned from the ADC because ΔVBE is a small voltage and will only use a fraction of the 16-bit ADC's dynamic range.

## Conclusion

I came up with a theory as to where the current ratio of 17 came from. It may not be totally correct for each part, but it is a reasonable working theory given my lack of internal part details.

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