Norton, PI, and T Transformations

When LC filters contain an L, PI, T, or M configuration of like elements, as shown in Figure 1 below, it is possible to alter the impedance level of either side of the L, PI, T, or M without changing the frequency response shape of the filter. This is advantageous when specific element values are desired to be used one side of the L, PI, or T, or when the termination resistance of the filter needs to be changed.

Directly Nortonable LC resonators require a PI or L of parallel resonators, or a T or L of series resonators. A T of parallel LC resonators or a PI of series LC resonators require two Norton transformations, one on each side, and consequently two selectable values are presented in the pop-up Norton control panel when this selection is made.

 

Figure 1: L, Pi, T, and M Examples

 

To perform a Norton Transformation:

To do a Norton transformation, right click on the desired L, PI, T, or M combination of like elements, as shown below. The change control panel will appear with a list of desired Norton effects, and a selection of which side (source or load) to alter the impedance of. Make the desired selection and hit "OK" or "Apply".

Pass Cursor Over PI, Tee, L, or M elements to Select Them for Norton Transformation

 

To change a left L to a right L, select "Set Left Element Value" in the Norton selection menu, and enter "0" or "Inf". The left element will be removed, and a new right element will be installed, having the effect of changing the left L to a right L. The same hold for right L to left L.

M inductor combination includes any cross coupled resonator set in a cross coupled resonator filters. One Norton is permitted for the entire cross coupled set of elements.

 

Negative Element Values

Norton transformations work in part by installing negative element values in parallel with existing positive elements. If the existing positive element value is too small of a capacitor or too large of an inductor, new negative element values are created in the filter. When this happens, the specific Norton transformation values cannot be used in a real filter.

 

Pi to T and T to PI Transformation

All PI combinations of elements have a unique equivalent T combination, and all T combinations have a unique and equivalent PI combination. To convert between PI and T combinations of element, right click on the desired PI or T elements, and select PI to T or T to PI in the pop-up Change control Panel. Only single element types may be converted. LC resonators do not have T to PI or PI to T transformation.

 

Norton Theory

The theory behind Norton transformation is very simple and well documented. It is based on the equivalence of the following two circuits:

 

 

Figure 2: Norton Admittance Equivalence

 

Admittance is used in the above illustration because Nortons are usually performed on capacitors. It is easily seen above why Norton transformations are sometimes referred to as "Norton Transformers" or "Capacitor Transformers" in that they may substitute for an ideal coupled inductor transformer.

 

Impedance Matching

Norton transformations may be used to match band pass filters with unequal resistors when it is desirable to achieve a matched reflection response with unequal resistors. To do this, design an equally terminated band pass filter, then Norton the load or the source to the desired value. The result will be a filter with unequal terminations and a matched reflection response.

The following example depicts a 50 ohm source and 10 Ohm load matched with the use of a Norton transformation. Note that the reflection coefficient is the same as that for a matched load:

 

 

Norton Transformation to Match Unequal Loads

 

Note that this function may be performed automatically with the use of the "Match Impedance" Checkbox in the Lumped Control Panel.

 

LC Transformations

Using the right mouse key to click on outer reactive element of a band pass filters that is of differing type and orientation than the next outer element (an "L" of LC") permits the swapping of the L and C, and applying a transform that restores an approximate frequency response.  In the case of coupled resonator filters, the new approximate frequency response is generally more symmetrical than the original frequency response, and applying this transformation is useful in cases where a more symmetrical frequency response is desired than what is obtained from direct coupled resonator or tubular filters.

The LC transformation will change one of the termination values.  However, performing the same transformation at both ends of the band pass filter will restore the termination value in most cases.  When it does not, a Norton transform will almost always restore the value.

The below example illustrates the process of selecting an outer capacitor with the right mouse key, selecting, "Series C" to transform it to a series capacitance, and the adjacent inductor to a shunt inductance, and applying the subsequent impedance scale to the source resistance.

 

Selecting an LC Transform of outer LC Elements in an "L" configuration

 

 

Resulting Elements and Termination Designed for a More Symmetrical Frequency Response

 

 

Selecting and Joining the Series Capacitors to Make One Capacitor