Summing points

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

Now if = 0 and the summing point is replaced by —1, then the response C (.v) to input i 2(.v) acting alone is given by Figure 4.12. The choice as to whether the summing point is replaced by +1 or —1 depends upon the sign at the summing point. 

To consider pH as a controlled variable, we use a pH electrode to measure its value and, with a transmitter, send the signal to a controller, which can be a little black box or a computer. The controller takes in the pH value and compares it with the desired pH, what we call the set point or reference. If the values are not the same, there is an error, and the controller makes proper adjustments by manipulating the acid or the base pump—the actuator.2 The adjustment is based on calculations using a control algorithm, also called the control law. The error is calculated at the summing point where we take the desired pH minus the measured pH. Because of how we calculate the error, this is a negative feedback mechanism. 

We identify two locations after the summing points with lower case e and a to help us.2 We can write at the summing point below H ... 

How do we decide the proper locations We do not know for sure, but what should help is after a summing point where information has changed. We may also use the location before a branch off point, helping to trace where the information is routed. 

As before, we can write down the algebraic equations about the summing point (the comparator) for X3, and the two... 

Figure 5.2. Information around the summing point in a negative feedback system.

For the rest of the control loop, Gc is obviously the controller transfer function. The measuring device (or transducer) function is Gm. While it is not shown in the block diagram, the steady state gain of Gm is Km. The key is that the summing point can only compare quantities with the same units. Hence we need to introduce Km on the reference signal, which should have the same units as C. The use of Km, in a way, performs unit conversion between what we dial in and what the controller actually uses in comparative tests. 2... 

The next order of business is to derive the closed-loop transfer functions. For better readability, we ll write the Laplace transforms without the 5 dependence explicitly. Around the summing point, we observe that... 

The time delay effect is canceled out, and this equation at the summing point is equivalent to a system without dead time (where the forward path is C = GCGE). With simple block diagram algebra, we can also show that the closed-loop characteristic polynomial with the Smith predictor... 

Although blocks are used to identify many types of mathematical operations, operations of addition and subtraction are represented by a circle, called a summing point. As shown in Figure 6, a summing point may have one or several inputs. Each input has its own appropriate plus or minus sign. A summing point has only one output and is equal to the algebraic sum of the inputs. 

A takeoff point is used to allow a signal to be used by more than one block or summing point (Figure 7). 

The reference point is an external signal applied to the summing point of the control system to cause the plant to produce a specified action. This signal represents the desired value of a controlled variable and is also called the "setpoint."... 

The feedback signal is a function of the output signal. It is sent to the summing point and algebraically added to the reference input signal to obtain the actuating signal. 

To carry out amperometric or voltammetric experiments simultaneously at different electrodes in the same solution is not difficult. In principle, any number of working electrodes could be studied however, it is unlikely that more than two or three would ever be widely used in practice. The bulk of the solution can have only one controlled potential at a time (if there are significant iR drops, there will be severe control problems with multiple-electrode devices). It is necessary to use a single reference electrode to monitor the difference between this inner solution potential and the inner potential of W1 at the summing point of an operational amplifier current-to-voltage converter (this is the potential of the circuit common see OA-2 in Fig. 6.17). The potential difference between... 

Sum Points for All Risk Markets 3. Look up Risk Corresponding to Point Total... 

For operations of addition and subtraction, the block is replaced by a small circle, called a summing point, with the appropriate plus or minus sign associated with the... 

In this problem (see initial Figure 38), the output from the summing point, S, is... 

Consider the circuit shown in Figure 15.2.1. The resistor, Rf, is the feedback element, through which there is a feedback current, if. The input is a current, ijn, which might be from a working electrode or a photomultiplier tube. From the conservation of charge (Kirchhoff s law), the sum of all the currents into the summing point, S, must be zero, and since a negligibly small current passes between the inputs. 

The adder potentiostat shown in Figure 15.4.4 remedies both drawbacks of the control circuit considered above, and is by far the most widely used design. Since the currents into the summing point S must add to zero. 

The facility for addition of input signals allows the straightforward synthesis of complex waveforms, and each input signal is individually referred to circuit ground. Any reasonable number of signals can be added at the input. One simply requires a resistor into the summing point for each of them. 

The input network can be expanded by adding resistors into the summing point to create a system that will provide a cell current equal to the sum of input currents in the fashion of an adder. Each input voltage source must be capable of supplying its contribution to the cell current, as one can see from Figure 15.5.1. This requirement can create problems in systems intended for applying high currents. 

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