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Spectrum Software has released Micro-Cap 11, the eleventh generation of our SPICE circuit simulator.

For users of previous Micro-Cap versions, check out the new features available in the latest version. For those of you who are new to Micro-Cap, take our features tour to see what Micro-Cap has to offer.

 

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Simulating a PID Controller

 

The PID (Proportional-Integral-Derivative) controller is a common feedback mechanism used within closed loop control systems. The controller is used to automatically adjust a variable to keep a specified measurement at a set point. It can be found in applications that need a form of temperature control, flow control, or position control.

The PID controller operates by comparing the set point versus the output measurement and calculating the error between the two. The controller performs three calculations on the error value. The proportional segment calculates the reaction to the current error value. The integral segment calculates the reaction based on the history of the error value. The derivative segment calculates the reaction based on how quickly the error value is changing. The sum of these calculations is used to adjust the control system towards the set point.

Each of the proportional, integral, and derivative calculations has a scale factor that provides a weighting to the importance of that calculation in the controller output. An increase in the proportional scale value will decrease the rise time and reduce the steady state error, but too high of a value will lead to an unstable system. An increase in the integral scale value will eliminate the steady state error but can lead to an overshoot in the transient response. An increase in the derivative scale value reduces overshoot and increases the stability of the system. Potential requirements of the control system such as limiting overshoot also need to be taken into account when determining the scale factors.

Simulating a PID controller is simple in Micro-Cap due to the behavioral macros that come with the program. An example circuit that uses a PID controller to set the position of a mass connected to a spring is displayed in the figure below.

PID controller example circuit

The PID controller is modeled using five macros. The Sub macro (X1) calculates the difference between the set point value and the output measurement to produce the error value. This error value is then fed into the proportional, integral, and derivative inputs. The Amp macro (X2) calculates the proportional response. Its Gain parameter has been defined with the Kp symbolic variable. The Int macro (X3) calculates the integral response. Its Scale parameter has been defined with the Ki symbolic variable. The Dif macro (X4) calculates the derivative response. Its Scale parameter has been defined with the Kd symbolic variable. The three symbolic variables have their values set using the following define statements:

.define Kp 20
.define Ki 20
.define Kd .1

Modifying the values in the define statements will tune the PID controller for the specific application. The outputs of the Amp, Int, and Dif macros are then input into a Sum3 macro (X5) which produces the PID controller output with which the control system is adjusted.

In this example, the PID controller is used to position a mass that is connected to a spring by adjusting the force that is applied to the system. There are also a couple of friction elements present in the system. The mechanical diagram of this control system is displayed in the figure below.

Mechanical equivalent of the plant circuitry

In order to simulate this mechanical system in Micro-Cap, the elements need to be converted into their electrical analogs. The force applied to the system is represented by a current. Since this current needs to be adjusted by the PID controller, an IofV dependent source (G1) is used where the voltage input to the source is the output of the controller. The two friction elements, B1 and B2, are represented by resistors R1 and R2. These resistors actually model inverse friction so the resistance values need to be defined as 1/friction. The analog of the spring, K, is the inductor L1. The inductor models the compliance of the spring so the inductance needs to be defined as 1/spring constant. The analog of the mass, M, is the capacitor C1 connected to ground. The capacitance is defined with the mass value. The voltage of the node between the capacitor and the inductor is equivalent to the position of the mass in the system.

The sensor that performs the measurement of the mass position in this case is just a simple VofV dependent source. For many control systems, the output measurement would need to be converted into an appropriate unit to compare versus the set point. However, since the output measurement and the set point value are on a direct one to one ratio, the gain has been set to one for the dependent source. An RC time delay was added to the output of the source to model a transport delay from the sensor output to the feedback input of the PID controller.

This schematic is then simulated in transient analysis. The V1 battery which defines the set point value has been stepped from .25V to 1V in .25V increments, and the simulation time is 5s. The resultant transient simulation is displayed below.

Transient response to the stepped set point

This simulation has the scale factors defined as Kp=20, Ki=20, and Kd=.1. Note that the position of the mass will be adjusted so that it is equivalent to the set point value in each branch of the simulation. There is a slight overshoot due to the integral which can be eliminated by setting the Kd parameter to 1. Modifying these scale factors will have a large effect on the transient response of the system so the tuning of these values is very dependent on the control system that is being adjusted.

 
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