Kuroda’s identity is an important method when designing filters or generally converting structures that are difficult to implement in practice to more practical ones. In this post only two of the Kuroda’s identities are going to be discussed. There are two more, but we are only discuss two of them because they are the most useful ones. The derivation is not going to be given in this blog post.
Kurodo’s identities that we are going to discuss convert a series inductance/stub into shunt capacitor/stub and vice versa. If you noticed we used the terms stubs, inductors and capacitors interchangeably because 1/8 wavelength long short stubs are equivalent to inductors and 1/8 wavelength long open stubs are equivalent to capacitors according to Richard’s transform. In the following figure Kurodo’s identities are shown.
The bold lines in figure.1 represents transmission lines and the characteristic impedance of them is also shown in figure.1 between bold lines. In the figure inductor and transmission line that are connected in series is converted to a transmission line and capacitor that are connected in parallel in the first identity and in the second one a capacitor and transmission line that are in parallel converted to an inductor and transmission line that are connected in series. It’s important to note that the elements’ locations are reversed as a result of this conversion and all of the transmission lines have to be 1/8 wavelength long.
How it’s going to be useful in filter design? As it’s mentioned earlier in this post series stubs are not particularly useful and practical when it comes to implementation of the filter on a PCB. For that reason, we need to convert series structures to parallel one. Now We have general idea of what Kurodo’s identity is all about let us see how it is actually going to be useful.
Consider the following circuit.
Kurodo’s identity is generally used to get rid off series stubs by converting them to parallel stubs because as it’s mentioned before it’s not straightforward to implement series stubs on a PCB. Let’s see how the circuit in figure.2 is converted to a more useful one.
In figure.3 series stub is converted to parallel stub which is much more practical to realize. The second kurodo’s identity is quite similar to the one we discussed, and therefore it’s not going to be explained in details.
Kurodo’s identity is a very useful method to convert filters that not physically realizable, it converts these types of filters to realizable ones. For higher order filters it’s necessary to use both identities. In the future I will create a blog post including a higher order filter and show how both identities are used.
Comments