# Normalized Determinant Function (NDF)

The normalized determinant function was first described by Platzker and Struble in a classic IEEE MTT-S paper and GaAs IC symposium paper back in 1993. NDF stands for normalized determinant function. Many people refer to the NDF and the NDF function, even though the "F" in NDF stands for function. If you are a serious power amplifier designer, you will need to dive into this topic, to at least understand the difference between stable and potentially unstable networks by examining origin encirclements of the NDF plotted from  -∞ to + ∞.

Below is a description of what NDF is all about, contributed by one of the original authors on this important topic, Wayne Struble. Thanks, Wayne!

All linear systems can be described with a characteristic equation whereby the natural and forced time response of that system is governed by this equation.  This characteristic equation can be written as a ratio of two polynomials in terms of s as E=P(s)/Q(s).  Stability of the system is governed by the roots of E (i.e. poles and zeroes of the characteristic equation).  If any poles of the system fall in the right half plane (RHP), the system’s time response will grow exponentially over time (i.e. the system is unstable).  Conversely, if there are no RHP poles, the system’s time response will not grow exponentially over time (i.e. the system is stable).  The NDF function is simply a way to determine if there are any RHP poles in the characteristic equation of the system.  It does this by counting the number of RHP zeroes in the full network determinant (which is the same as counting the number of RHP poles of the characteristic equation of the system).  We do this by using the “Principle of the Argument Theorem” from complex theory.  Basically, one plots the complex NDF function (magnitude and phase) versus frequency on a polar plot and counts the number of clockwise encirclements of the origin (0,0) as frequency increases from –infinity to +infinity.  The number of encirclements is equal to the number of RHP poles in the system’s characteristic equation.  If there are any RHP poles present, the system is unstable.

This technique is mathematically rigorous for all linear systems.  At the bottom of the page are some good references, and here are some presentations you can download from Microwaves101 (contributed by Wayne Struble, thanks again!)

Stability Analysis for RF and Microwave Circuit Design, Wayne Struble and Aryeh Platzker.

Appendix 1: NDF Stability Analysis of Linear Networks from Return Ratios, Wayne Struble and Aryeh Platzker.

  E. Routh, “Dynamics of a System of Rigid Bodies”, 3rd Ed., Macmillan, London, 1877

  H. Nyquist, “Regeneration Theory”, Bell System Technical Journal, Vol. 11, pp. 126-147, Jan. 1932

  A. Platzker, W. Struble, and K. Hetzler, “Instabilities Diagnosis and the Role of K in Microwave Circuits”, IEEE MTT-S Digest, vol. 3, pp. 1185-1188, Jun. 1993

  W. Struble and A. Platzker, “A Rigorous Yet Simple Method For Determining Stability of Linear N-port Networks”, 15th Annual GaAs IC Symposium Digest, pp. 251-254, Oct. 1993

  A. Platzker and W. Struble, “Rigorous Determination of The Stability of Linear N-node Circuits From Network Determinants and The Appropriate Role of The Stability Factor K of Their Reduced Two-Ports”, 3rd International Workshop on Integrated Nonlinear Microwave and Millimeterwave Circuits, pp. 93-107, Oct. 1994

Below are two PDF files on NDF, make sure you grab them both.