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Understanding RHPZ

27 Aug 2013  | Sergio Franco

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During my academic career I have observed that one of the most challenging subjects to teach as well as to learn is systems theory. All those poles and zeros may make sense in class, but once the student tries to relate them to a physical circuit in the lab, there seems to be a chasm between the blackboard and the bench.

My contention is that the student should be taught first to investigate the circuit intuitively, using as much physical insight as possible, and then to use mathematical tools for a more systematic validation. I will try to illustrate using the much dreaded right half-plane zero (RHPZ) as a vehicle.

Let us start out with the basic circuit of figure 1a, consisting of an RC network and an op amp buffer, giving

As we know, the value of s for which the denominator vanishes represents a pole. This value is

Since it is negative, the pole lies in the left half of the s plane (LHP). The circuit's response to a 0-to-1 V step, depicted in figure 1b, is the familiar exponential transient. (Note that the time scale has been chosen slow enough that we can ignore typical op amp dynamic limitations such as slew rate limiting.)

Figure 1: Single-pole circuit and its response to a 1-V input step.

Suppose we now modify the circuit as in figure 2a. What is its step response going to be? Using physical insight we observe that initially, with C still discharged, we have Vp = 0, so the op amp will momentarily act as an inverting amplifier with a gain of –R2/R1 (= –0.5 V/V with the component values shown.) Consequently, in response to the +1-V input step, Vo will jump to –0.5 V, as shown in figure 2b. However, as C charges up, Vp will again produce the exponential transient of figure 1b, so we eventually have VpVi, in turn implying VoVi. Note, incidentally, that the higher the R2/R1 ratio, the bigger the initial negative jump of Vo.

Figure 2: Circuit with a RHPZ and its response to a 1-V input step.

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