For designs that require accurate current measurements a shunt or sense resistor is often used. An unknown current flows through the sense resistor and the resulting measured voltage drop across the known resistance is converted to a current measurement using: V / R = I. When determining a shunt resistor's effect on the accuracy of a current measurement, often circuit designers focus more on the tolerance or accuracy specification of the shunt resistor. But the Temperature Coefficient of Resistance or TCR specification is also important to consider. And there is the commonly overlooked aspect of self heating from the flow of current through the shunt resistor that needs to be considered. In this tutorial we will look at all three of these factors as we walk through a detailed uncertainty calculation of two example shunt resistors. 

Example shunt resistor #1 specifications:

• 0.05 Ω
• 0.25% tolerance / accuracy
• 50 PPM/°C TCR (Temperature Coefficient of Resistance)
• 0.5W max rating

Example shunt resistor #2 specifications:

• 0.05 Ω
• 0.5% tolerance
• 10 PPM/°C TCR
• 1 W max rating

25 C is almost always the starting point of TCR, this is defined by standards like IEC 60115 and MIL-PRF

Operating conditions for our two example sense resistors:

• Max current through the shunt 2.5 A
• Shunt will be used in environments with an ambient temperatures up to 50°C
• The design that is implementing the shunt resistor will not have active cooling
  

We have a +25°C ambient difference from test conditions.

Estimate self heating of the sense resistor

Self-heating is often a misunderstood contributor to shunt error and it needs to be part of the overall TCR error calculation. A lot of shunt manufacturers do not make calculating self heating effects on the shunt resistor's accuracy straightforward. But one way to do it is to first calculate the max power dissipated by the shunt and then look at its power derating chart in its datasheet:

Derating chart from example shunt resistor #1 datasheet

Derating chart from shunt resistor #2 datasheet (blue line):

For shunt #1, if derating shows 0.5 W capacity until 70°C and zero at 125°C, then we know 0.5 W causes ~55°C rise (125°C - 70°C)

Approximate thermal resistance:

55°C / 0.5W ≈ 110°C/W

Calculate max power dissipated by shunt:

2.5A^2 x 0.05Ohms = 0.3125W

Now calculate heat rise at 0.3125 W:

0.3125W × 110°C/W ≈ 34.4°C

Shunt #1: ~59.4°C above its calibration temperature (25°C ambient + 34.4°C from self heating).

Let’s do the same calculations for shunt #2:

Approximate thermal resistance:

55°C / 1W ≈ 55°C/W

Calculate max power dissipated by shunt:

2.5A^2 x 0.05Ohms = 0.3125W

Now calculate heat rise at 0.3125 W:

0.3125W × 55°C/W ≈ 17.2°C

Shunt resistor #2 is now ~42.2°C above its calibration temperature (25°C ambient + 17.2°C from self heating).

Convert temperature drift Into resistance uncertainty

Apply TCR to shunt #1:

ΔR% = TCR × ΔT

50ppm/°C × 59.4°C = 2970ppm = 0.297%

Apply TCR to shunt #2:

ΔR% = TCR × ΔT

10ppm/°C × 42.2°C = 422ppm = 0.0422%

Combine with the tolerance specification to get total uncertainty

Shunt resistor #1 worst case +/- uncertainty:

0.25% tolerance + 0.297% temperature drift = ~0.547%

Convert to resistance:

50mOhm × 0.00547 ≈ ±0.27mOhm

Shunt resistor #2 worst case +/- uncertainty:

0.5% tolerance + 0.0422% temperature drift = ~0.542%

Convert to resistance:

50mOhm × 0.00542 ≈ ±0.27mOhm

As you can see from the resulting uncertainty calculations that even though one of the sense resistors had a tolerance specification that was twice that of the other their uncertainty is essentially the same. This result is due to shunt resistor #2 having a lower TCR specification and a higher power rating so it was less susceptible to self heating. When you are using a shunt resistor in an area where the environmental temperature can vary a lot from 25 C a shunt resistor’s TCR can become a big contributor to its uncertainty. Then there is also self heating to consider which further contributes to TCR based uncertainty. Note that physics tells us when a resistor heats up its resistance increases and does not decrease so the +/- uncertainty could be changed with that in mind. You could also apply some statistics to this uncertainty calculation, such as the Root Sum of Squares (RSS), to further tighten up the calculated uncertainty. 

Keep in mind, in this exercise we just considered the shunt resistor’s uncertainty contribution to measuring current. There is also the measurement uncertainty to contributors of the measurement device or circuit itself. For more information on designing a circuit to measure a current shunt resistor check out the Anabit video at the end of this tutorial.     

Tips to Reduce Uncertainty and Error when using a Shunt Resistor to Measure Current

• To reduce uncertainty from self heating, pick a sense resistor with a lot of margin in its power rating. As you can see from our example calculations the higher the power rating the lower the self heating.
• Use a high accuracy resistance measurement device to calibrate your shunt resistor. Measure the actual resistance of the shunt resistor and save that information in software as a correction factor. This will essentially eliminate the uncertainty caused by the tolerance specification.
• Put a temperature sensor on your shunt resistor. At the same time you make a current measurement, measure the sense resistor’s temperature. Use this temperature measurement to correct for TCR uncertainty contributors.
• Make sure there are not any “hot” circuits around the shunt resistor in your design, such as a linear voltage regulator. This could heat up the shunt resistor and lead to further TCR uncertainty that you did not account for. 

For a deeper dive into the TCR specification check out "Interpretation of TCR Specifications
for Precision Resistors" from VPG