SOiz92 brought up Microchip AN710 appnote and based on it prepared a xls to calculate inductance of several printed antennas. I happened to see an article about NFC antenna design online. The article uses HFSS to design and simulate NFC antenna.
HFSS environment is as below.
The article investigated three types of antennas. The first one is N-turn rectangular antenna as below.
This one is similar to the N-turn rectangular multilayer coil discussed in Microchip AN710.
Let’s see how the simple equation in AN710 works compared with HFSS simulation result. The HSFF article doesn’t give the dimension of antenna so let’s make a reasonable guess based on the drawing shape and the fact that it is for NFC application. For AN710 equation, we set x to be 2cm and y to be 3.5cm. HFSS article has parameters for trace width and distance. AN710 rect coil assumes no gap among coils. Here we use the sum of trace width and distance as width of cross section in AN710. Another difference is HFSS article has a parameter for height (coil build up) of cross section but HSFF case is printed trace so we just set height of cross section to be zero in calculation.
With b=0.4cm, AN710 calculation is as below.
With trace width=trace distance=0.2mm, HSFF simulation is as below.
So at low frequency, the result matches well, 5.92uH for AN710 and 5.91uH for HFSS. However, HSFF curve clearly shows the inductance is a function of operating frequency and inductance increases as freq increases. This effect is not shown in AN710 equation. In other words AN710 equation is sort of for low freq case estimation.
Let’s change N (number of turns) next. For AN710, when N=3 L is 0.533uH and when N=6 L is 2.13uH. Below diagram shows HFSS result. Again the results match well at low freq. At high freq, HFSS shows no freq effect when N=3 or N=6. This is unexpected since freq effect is very clear in N=10 case.
The HFSS article also investigates the effect of trace width and trace distance on inductance. Below is for the case that trace width stays at 0.2mm and trace distance changes from 0.2mm to 0.5mm. So inductance increases from 5.9uH to 7.8uH as trace distance increases.
Below is for the case that trace distance stays at 0.2mm and trace width changes from 0.2mm to 0.5mm. So at low freq, trace width has no effect on inductance.
Here is AN710 equation result (x=2cm, y=3.5cm, h=0, N=10)
b(cm) 0.4 0.5 0.6 0.7 0.8 0.9
L(uH) 5.92 5.57 5.25 4.97 4.71 4.49
Inductance decreases with the increase of cross section width. Unfortunately two method results do not match.