Transfer Function and Block Diagrams Questions and Answers

Answer: B

The block diagram is (B).

Call the voltage across the shunt R, V2, the current in the first series R, I1, and the current in the second series R, I2. Writing the network as cause and effect, with G = 1/R and Z = 1/(sC):

\[ \begin{aligned} I_1 &= G\,(V_1 - V_2) \\ V_2 &= R\,(I_1 - I_2) \\ I_2 &= G\,(V_2 - V_3) \\ V_3 &= Z\,I_2 \end{aligned} \]

That needs three summing points in a chain, with blocks G, R, G and Z in that order, and three feedback paths: V2 (after R) back to the first junction, I2 (after the second G) back to the second junction, and V3 (after Z) back to the third. Diagram B is exactly this.

A has only two summing points and no feedback of V3. C puts Z before G, so it applies an impedance to a voltage difference, and its feedback paths come from the wrong places. D starts with R instead of G, which would multiply a voltage by a resistance. The printed diagrams carry no plus or minus signs; the signs above come from the network equations.

Answer: D

The transfer function is G₂(s) / [1 + G₁(s)G₂(s)H(s)], which is option D reading the G(s) printed in the options as G₂(s), the only block the disturbance passes through.

Set R(s) = 0 and follow the loop. The disturbance Td enters just before G₂, and the output is fed back with a minus sign through H to G₁:

\[ \begin{aligned} y &= G_2\,(Td - G_1 H y) \\ y\,(1 + G_1 G_2 H) &= G_2\,Td \\ \frac{y}{Td} &= \frac{G_2}{1 + G_1 G_2 H} \end{aligned} \]

Option C has a minus in the denominator, which would be positive feedback, and the figure shows negative feedback. Option B (zero) would hold only if the disturbance had no path to the output, and option A is not the form of any loop gain here. The options do not define G(s); this explanation takes it to mean G₂(s), the reading under which the key is right.

Answer: C

The value that does not define a zero is -2.

The zeros of F(s) are the values of s that make the numerator zero:

\[ s(s - 1 + j2)(s - 1 - j2) = 0 \]

so the zeros are s = 0, s = 1 - j2 and s = 1 + j2. The value -2 makes the factor (s + 2) in the denominator zero, so it is a pole, as are \(\pm j1\). The other three options are all zeros.

Answer: B

An externally introduced signal affecting the controlled output is called a stimulus.

In control-system terminology the stimulus is the independent input applied to a system from outside, the command or disturbance that sets it responding; the system's reaction is the response. A reference input that commands a new output, and a disturbance that upsets it, are both stimuli in this sense.

Feedback is a signal derived from the output and returned inside the loop, so it is not externally introduced. 'Signal' is too general a word to define anything, and gain control is an adjustment of amplification, not an input to the system.

Answer: A

The output of the controller is given to the final control element.

In a feedback loop the sensor measures the controlled variable, the comparator forms the error between it and the set point, and the controller processes that error into a corrective signal. That signal drives the final control element, for example a control valve, a motor drive or a heater, which acts directly on the process to change the controlled variable.

The comparator and sensor come before the controller in the signal path, so its output is not fed to them. An amplifier may be part of the controller or of the actuator, but the block the controller's output is defined as feeding is the final control element.

Answer: B

The statement that is not correct is that an open-loop system is more expensive.

An open-loop system has no sensor or feedback path, so it is simpler and less expensive than a closed-loop system. That makes 'less expensive' correct and 'more expensive' the incorrect statement.

The other statements are true of open-loop systems. The input command alone determines the control action, since the output is not measured. Because there is no feedback to correct errors, non-linearities, parameter changes and disturbances cause the output to deviate, that is, malfunction.

Answer: B

A closed-loop system is distinguished by feedback.

In a closed-loop system the output is measured and fed back to be compared with the reference input; the error between them drives the controller. An open-loop system acts only on the input command and never checks the output. Feedback is what lets a closed-loop system correct for disturbances and parameter changes.

A servomechanism is one kind of closed-loop system (a feedback system controlling position or speed), not the distinguishing feature. Both kinds of system have inputs and outputs, so the input or output pattern does not distinguish them.

Answer: D

The diagram is equivalent to \((G_{1}+G_{2})G_{3}/(1+G_{3}H)\). The input R reaches the summing point through the two parallel blocks \(G_{1}\) and \(G_{2}\), both added, so the forward path before the junction is \(G_{1} + G_{2}\). Then \(G_{3}\) follows in cascade, and the output C is fed back through H with a negative sign:

\[ C = G_{3}\left[(G_{1}+G_{2})R - HC\right] \]

\[ \frac{C}{R} = \frac{(G_{1}+G_{2})G_{3}}{1 + G_{3}H} \]

Option (C) treats \(G_{1}\) and \(G_{2}\) as cascaded instead of parallel. Options (A) and (B) combine the wrong blocks, and neither is a cascade or a parallel pair as drawn.

Answer: B

The transfer function is needed to calculate the output for a given input, the second option. Since \(C(s) = G(s) R(s)\), knowing G(s) and the input R(s) gives the output C(s) and so the response.

The time constant, the steady state gain and the order of the system can all be read from the transfer function, but each is only a property of the system, while the output for a particular input is the main purpose for which the transfer function is determined.

Answer: A

The transfer function is defined for linear and time-invariant systems, statements (i) and (iii), the first option. It is the ratio of the Laplace transform of the output to that of the input with zero initial conditions, and this ratio exists as a function of s only when the system is linear and its parameters do not change with time.

Nonlinear systems and time-variant systems do not have a transfer function in this sense, so every option containing (ii) or (iv) is wrong.