Some of the workhorse analog circuits in many systems, such as active filters and amplifiers, may include poles in the transfer function. Even in passive filters, reactive components create poles in a transfer function. This may lead to an undesired transient response when the circuit is driven with a signal and eventually settles into its steady state. If the transient is small enough, we typically don't worry about them. But if transient behavior is undesirable, how can we cancel out a pole and thereby eliminate the transient behavior in the circuit?
A designer will have to add something to a circuit in order to cancel the pole and thereby compensate for the transient response. The specific method used to do this depends on exactly what the circuit needs to do. The method used in a feedback loop will be different from the method used in a passive filter. While we can't address every example in this article, we'll look at a few examples that illustrate the concepts involved in pole cancellation or suppression.
Two Simple Pole Cancellation Methods
Techniques that modify a transfer function without affecting core functionality can be difficult and can force an entire circuit redesign. Fortunately, if you've already created a circuit, there are two potential methods that can help solve problems relating to undesired poles in a transfer function or loop gain spectrum, but without requiring major changes to the circuit. These are:
Add damping with a discrete resistor
Add a zero that matches the pole frequency
The first approach is a more specific case where you have a transient response while driving with a fast edge rate signal. In other instances, this might not be desirable, such as when you want to avoid power loss on the discrete resistor. A more elaborate approach is to add a zero that matches the pole frequency. This can also involve adding components or changing one part of the circuit such that a new zero perfectly counterbalances a pole.
Consider the case of an RLC circuit, where the component values are selected such that the transient response is an underdamped oscillation. In the limiting case where there is no resistor, the response is a very high Q bandpass response right at the resonant frequency. In the other extreme, the response is damped very quickly, which would occur when there was very little inductance or very large resistance.
If the transient response is oscillatory and you simply need to suppress the pole, you could add a resistor as shown below. The previous point discusses this in terms of an RLC circuit but technically any circuit that exhibits an RLC-like response can have its transient response suppressed with additional resistance.
R2 (boxed in red) suppresses the pole created by L1 and C2 by adding some damping.
The example circuit above has small resistances intentionally added into the circuit to provide damping. If the damping rate and the response frequency are known, it is possible to calculate the exact amount of resistance required for critical damping. This value would be a lower limit on the additional resistance in order to fully suppress the pole.
There are three downsides to the approach of pole suppression with additional damping:
Some power is lost over the added resistance
This only works in circuits with higher order passive filter behavior
The slower transient response might not be appropriate for all applications
Finally, this approach doesn't exactly remove a pole, it just suppresses its transient response to the point that the pole is very weak. The other approach actually eliminates a pole from the transfer function.
Adding a Zero at the Pole Frequency
To add a zero in a transfer function, a simple approach is to place a circuit element in parallel with the circuit creating the pole. This usually works because, typically when reactive components are placed in parallel, it is highly likely that a zero will arise somewhere in a transfer function. This means that your job is to determine that frequency where the zero should be added.
To see how this works, consider the RC circuit below. In this circuit, the output is measured across the capacitor, and we have R1 = R2.
The pole in the transfer function arises at a value of f = πRC. If an identical capacitor is placed across R1, the pole has been perfectly eliminated. This can be seen if we just look at Kirchoff’s voltage law around the loop created by the capacitors, as shown below.
This voltage loop illustrates how a transient response is compensated by bypassing the resistor R1
Around this loop, there is no resistance, so the time constant for this loop is theoretically zero and the pole frequency has been moved to infinity. The result is now a flat transfer function with no poles or zeros.
Verify Your Pole Cancellation in Simulation
Pole cancellation is verified by simulating the transfer function for a circuit. In some cases, the value of a required additional circuit element might be difficult to determine by hand due to the complexity of a circuit. When this is the case, you can iterate through component values using a parameter sweep and verify pole cancellation visually. SPICE simulators can do the hard work for you and give you a quick way to verify your transfer function has been properly compensated.
Whenever you want to build and analyze your analog circuits, make sure you simulate your designs with the complete set of tools in PSpice from Cadence. PSpice users can access a powerful SPICE simulator as well as specialty design capabilities like model creation, graphing and analysis tools, and much more.