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2.4: BLOCK DIAGRAMS

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    59330
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    A block diagram is a graphical method of representing the relationships among variables in a system. The symbols used to form a block diagram were introduced in Section 2.2. Advantages of this representation include the insight into system operation that it often provides, its clear indication of various feedback loops, and the simplification it affords to determining the transfer functions that relate input and.output variables of the system. The discussion in this section is limited to linear, time-invariant systems, with the enumeration of certain techniques useful for the analysis of non­linear systems reserved for Chapter 6.

    Forming the Block Diagram

    Just as there are many complete sets of equations that can be written to describe the relationships among variables in a system, so there are many possible block diagrams that can be used to represent a particular system. The choice of block diagram should be made on the basis of the insight it lends to operation and the ease with which required transfer functions can be evaluated. The following systematic method is useful for circuits where all variables of interest are node voltages.

    1. Determine the node voltages of interest. The selected number of voltages does not have to be equal to the total number of nodes in the circuit, but it must be possible to write a complete, independent set of equations using the selected voltages. One line (which may split into two or more branches in the final block diagram) will represent each of these variables, and these lines may be drawn as isolated segments.
    2. Determine each of the selected node voltages as a weighted sum of the other selected voltages and any inputs or disturbances that may be applied to the circuit. This determination requires a set of equations of the form
      \[V_j = \sum_{n \ne j} a_{nj} V_n + \sum_m b_{mj} E_m\label{eq2.4.1} \]
      where \(V_k\) is the \(k\)th node voltage and \(E_k\) is the \(k\)th input or disturbance.
    3. The variable \(V_j\) is generated as the output of a summing point in the block diagram. The inputs to the summing point come from all other vari­ables, inputs, and disturbances via blocks with transmissions that are the \(a\)'s and \(b\)'s in Equation \(\ref{eq2.4.1}\). Some of the blocks may have transmissions of zero, and these blocks and corresponding summing-point inputs can be elimi­nated.

    The set of equations required in Step 2 can be determined by writing node equations for the complete circuit and solving the equation written about the \(j\)th node for \(V_j\) in terms of all other variables. If a certain node voltage \(V_k\) is not required in the final block diagram, the equation relating \(V_k\) to other system voltages is used to eliminate \(V_k\) from all other members of the set of equations. While this degree of formality is often unnecessary, it always yields a correct block diagram, and should be used if the desired diagram cannot easily be obtained by other methods.

    As an example of block diagram construction by this formal approach, consider the common-emitter amplifier shown in Figure 2.7\(a\). (Elements used for bias have been eliminated for simplicity.) The corresponding small-signal equivalent circuit is obtained by substituting a hybrid-pi(The hybrid-pi model will be used exclusively for the analysis of bipolar transistors operating in the linear region. The reader who is unfamiliar with the development or use of this model is referred to P. E. Gray and C. L. Searle, Electronic Principles: Physics, Models, and Circuits,Wiley, New York, 1969. ) model for the transistor and is shown in Figure 2.7\(b\). Node equations are(G's and R's (or g's and r's) are used to identify corresponding conductances and resistances, while Y's and Z's (or y's and z's) are used to identify corresponding admittances and impedances. Thus for example, \(G_A = 1/R_A\) and \(z_b = 1/y_b\).)

    \[\begin{array} {rcl} {G_S V_i} & = & {(G_S + g_x) V_a - g_x V_b} \\ {0} & = & {-g_x V_a + [(g_x + g_{\pi}) + (C_{\mu} + C_{\pi} ) s] V_b - C_{\mu} s V_o} \\ {0} & = & {(g_m - C_{\mu} s) V_b + (G_L + C_{\mu} s) V_o} \end{array}\label{eq2.4.2} \]

    If the desired block diagram includes all three node voltages, Equation \(\ref{eq2.4.2}\) is arranged so that each member of the set is solved for the voltage at the node about which the member was written. Thus,

    \[\begin{array} {rcl} {V_a} & = & {\dfrac{g_x}{g_a} V_b + \dfrac{G_s}{g_a} V_i} \\ {V_b} & = & {\dfrac{g_x}{y_b} V_a + \dfrac{C_{\mu} s}{y_b} V_o} \\ {V_o} & = & {\dfrac{C_{\mu} s - g_m}{y_o} V_b} \end{array} \nonumber \]

    where

    \[\begin{array} {rcl} {g_a} & = & {G_S + g_x} \\ {y_b} & = & {[(g_x + g_{\pi}) + (C_{\mu} + C_{\pi}) s]} \\ {y_o} & = & {G_L + C_{\mu} s} \end{array}\nonumber \]

    The block diagram shown in Figure 2.7\(c\) follows directly from this set of equations.

    Figure 2.8 is the basis for an example that is more typical of our intended use of block diagrams. A simple operational-amplifier medel is shown con­nected as a noninverting amplifier. It is assumed that the variables of interest are the voltages \(V_b\) and \(V_o\). The voltage \(V_o\) can be related to the other selected voltage, \(V_b\), and the input voltage, \(V_i\), by superposition.

    with \(V_i = 0\),

    \[V_o = -a V_b\label{eq2.4.4} \]

    while with \(V_b = 0\),

    \[V_o = aV_i\label{eq2.4.5} \]

    The equation relating \(V_o\) to other selected voltages and inputs is simply the superposition of the responses represented by Equations \(\ref{eq2.4.4}\) and \(\ref{eq2.4.5}\), or

    \[V_o = aV_i - aV_b\label{eq2.4.6} \]

    The voltage \(V_b\) is independent of \(V_i\) and is related to \(V_o\) as

    \[V_b = \dfrac{Z_1}{Z_1 + Z_2} V_o\label{eq2.4.7} \]

    截屏2021-08-05 下午3.14.00.png
    Figure 2.7 Common-emitter amplifier. (\(a\)) Circuit. (\(b\)) Incremental equivalent circuit. (\(c\)) Block diagram.

    Equations \(\ref{eq2.4.6}\) and \(\ref{eq2.4.7}\) are readily combined to form the block diagram shown in Figure 2.8\(b\).

    It is possible to form a block diagram that provides somewhat greater insight into the operation of the circuit by replacing Equation \(\ref{eq2.4.6}\) by the pair of equations

    \[V_a = V_i - V_b\label{eq2.4.8} \]

    and

    \[V_o = aV_a \label{eq2.4.9} \]

    Note that the original set of equations were not written including \(V_a\), since \(V_a\), \(V_b\), and \(V_i\) form a Kirchhoff loop and thus cannot all be included in an independent set of equations.

    The alternate block diagram shown in Figure 2.8\(c\) is obtained from Equations \(\ref{eq2.4.7}\), \(\ref{eq2.4.8}\), and \(\ref{eq2.4.9}\). In this block diagram it is clear that the summing point models the function provided by the differential input of the operational amplifier. This same block diagram would have evolved had V0 and V, been initially selected as the amplifier voltages of interest.

    The loop transmission for any system represented as a block diagram can always be determined by setting all inputs and disturbances to zero, break­ing the block diagram at any point inside the loop, and finding the signal returned by the loop in response to an applied test signal. One possible point to break the loop is illustrated in Figure 2.8c. With \(V_i = 0\), it is evident that

    \[\dfrac{V_o}{V_t} = \dfrac{-aZ_1}{Z_1 + Z_2} \nonumber \]

    The same result is obtained for the loop transmission if the loop in Figure 2.8\(c\) is broken elsewhere, or if the loop in Figure 2.8\(b\) is broken at any point.

    Figure 2.9 is the basis for a slightly more involved example. Here a-fairly detailed operational-amplifier model, which includes input and output im­pedances, is shown connected as an inverting amplifier. A disturbing current generator is included, and this generator can be used to determine the closed-loop output impedance of the amplifier \(V_o/I_d\).

    It is assumed that the amplifier voltages of interest are \(V_a\) and \(V_o\). The equation relating \(V_a\) to the other voltage of interest \(V_o\), the input \(V_i\), and the disturbance \(I_d\), is obtained by superposition (allowing all other signals to be nonzero one at a time and superposing results) as in the preceding example. The reader should verify the results

    \[V_a = \dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} V_i + \dfrac{Z_i || Z_1}{Z_2 + Z_i || Z_1} V_o \label{eq2.4.11} \]

    and

    \[V_o = \dfrac{-aZ_2 + Z_o}{Z_2 + Z_o} V_a + (Z_o || Z_2) I_d \label{eq2.4.12} \]

    The block diagram of Figure 2.9\(b\) follows directly from Equations \(\ref{eq2.4.11}\) and \(\ref{eq2.4.12}\).

    截屏2021-08-05 下午3.26.25.png
    Figure 2.8 Noninverting amplifier. (\(a\)) Circuit. (\(b\)) Block diagram. (\(c\)) Alternative block diagram.
    截屏2021-08-05 下午3.29.54.png
    Figure 2.9 Inverting amplifier. (\(a\)) Circuit. (\(b\)) Block diagram.

    Block-Diagram Manipulations

    There are a number of ways that block diagrams can be restructured or reordered while maintaining the correct gain expression between an input or disturbance and an output. These modified block diagrams could be obtained directly by rearranging the equations used to form the block diagram or by using other system variables in the equations. Equivalences that can be used to modify block diagrams are shown in Figure 2.10.

    It is necessary to be able to find the transfer functions relating outputs to inputs and disturbances or the relations among other system variables from the block diagram of the system. These transfer functions can always be found by appropriately applying various equivalences of Figure 2.10 until a single-loop system is obtained. The transfer function can then be deter­mined by loop reduction (Figure 2.10\(h\)). Alternatively, once the block diagram has been reduced to a single loop, important system quantities are evident. The loop transmission as well as the closed-loop gain approached for large loop-transmission magnitude can both be found by inspection.

    Figure 2.11 illustrates the use of equivalences to reduce the block diagram of the common-emitter amplifier previously shown as Figure 2.7\(c\). Figure 2.11\(a\) is identical to Figure 2.7\(c\), with the exceptions that a line has been replaced with a unity-gain block (see Figure 2. 10\(a\)) and an intermediate variable \(V_c\) has been defined. These changes clarify the transformation from Figure 2.11\(a\) to 2.11\(b\), which is made as follows. The transfer function from \(V_c\) to \(V_b\) is determined using the equivalance of Figure 2.10\(h\), recognizing that the feed­back path for this loop is the product of the transfer functions of blocks 1 and 2. The transfer function \(V_b/ V_c\) is included in the remaining loop, and the transfer function of block 1 links \(V_o\) to \(V_b\).

    The equivalences of Figs. 2.10\(b\) and 2.10\(h\) using the identification of transfer functions shown in Figure 2.11b (unfortunately, as a diagram is re­duced, the complexities of the transfer functions of residual blocks increase) are used to determine the overall transfer function indicated in Figure 2.11\(c\).

    The inverting-amplifier connection (Figure 2.9) is used as another example of block-diagram reduction. The transfer function relating \(V_o\) to \(V_i\) in Figure 2.9\(b\) can be reduced to single-loop form by absorbing the left-hand block in this diagram (equivalence in Figure 2.10\(d\)). Figure 2.12 shows the result of this absorption after simplifying the feedback path algebraically, eliminating the disturbing input, and using the equivalence of Figure 2.10\(e\) to introduce an inversion at the summing point. The gain of this system ap­proaches the reciprocal of the feedback path for large loop transmission; thus the ideal closed-loop gain is

    \[\dfrac{V_o}{V_i} = -\dfrac{Z_2}{Z_1} \nonumber \]

    The forward gain for this system is

    \[\begin{array} {rcl} {\dfrac{V_o}{V_e}} & = & {\left [\dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} \right ] \left [\dfrac{-aZ_2 + Z_o}{Z_2 + Z_o} \right ] } \\ {} & = & {\left [\dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} \right ] \left [\dfrac{-aZ_2}{Z_2 + Z_o} \right ] + \left [\dfrac{Z_i || Z_2}{Z1 + Z_i || Z_2} \right ] \left [\dfrac{Z_o}{Z_2 + Z_o} \right ] } \end{array}\label{eq2.4.14} \]

    截屏2021-08-05 下午3.41.35.png

    截屏2021-08-05 下午3.42.09.png
    Figure 2.10 Block-diagram equivalences. (\(a\)) Unity gain of line. (\(b\)) Cascading. (\(c\)) Summation. (\(d\)) Absorption. (\(e\)) Negation. (\(f\)) Branching. (\(g\)) Factoring. (\(h\)) Loop reduction.

    The final term on the right-hand side of Equation \(\ref{eq2.4.14}\) reflects the fact that some fraction of the input signal is coupled directly to the output via the feedback network, even if the amplifier voltage gain \(a\) is zero. Since the impedances included in this term are generally resistive or capacitive, the magnitude of this coupling term will be less than one at all frequencies. Similarly, the component of loop transmission attributable to this direct path, determined by setting \(a = 0\) and opening the loop is

    \[\begin{array} {rcl} {\dfrac{V_f}{V_e} |_{a = 0}} & = & {\left [\dfrac{Z_1}{Z_2} \right ] \left [\dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} \right ] \left [\dfrac{Z_o}{Z_2 + Z_o} \right ]} \\ {} & = & {\left [\dfrac{Z_i Z_1}{Z_i Z_1 + Z_i Z_2 + Z_1 Z_2} \right ] \left [\dfrac{Z_o}{Z_2 + Z_o} \right ]} \end{array}\label{eq2.4.15} \]

    and will be less than one in magnitude at all frequencies when the im­pedances involved are resistive or capacitive.

    截屏2021-08-05 下午3.48.24.png
    Figure 2.11 Simplification of common-emitter block diagram. (\(a\)) Original block diagram. (\(b\)) After eliminating loop generating \(V_b\). (\(c\)) Reduction to single block.
    截屏2021-08-05 下午3.49.43.png
    Figure 2.12 Reduced diagram for inverting amplifier.

    If the loop-transmission magnitude of the operational-amplifier connec­tion is large compared to one, the component attributable to direct coupling through the feedback network (Equation \(\ref{eq2.4.15}\)) must be insignificant. Conse­quently, the forward-path gain of the system can be approximated as

    \[\dfrac{V_o}{V_e} \simeq \left [\dfrac{-aZ_2}{Z_2 + Z_o} \right ] \left [\dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} \right ] \nonumber \]

    in this case. The corresponding loop transmission becomes

    \[\dfrac{V_f}{V_e} \simeq \left [\dfrac{-aZ_1}{Z_2 + Z_o} \right ] \left [\dfrac{Z_i || Z_2}{Z_1 + Z_i || Z_2} \right ] \nonumber \]

    It is frequently found that the loop-transmission term involving direct coupling through the feedback network can be neglected in practical operational-amplifier connections, reflecting the reasonable hypothesis that the dominant gain mechanism is the amplifier rather than the passive network. While this approximation normally yields excellent results at frequencies where the amplifier gain is large, there are systems where sta­bility calculations are incorrect when the approximation is used. The reason is that stability depends largely on the behavior of the loop transmission at frequencies where its magnitude is close to one, and the gain of the amplifier may not dominate at these frequencies.

    Closed-Loop Gain

    It is always possible to determine the gain that relates any signal in a block diagram to an input or a disturbance by manipulating the block diagram until a single path connects the two quantities of interest. Alternatively, it is possible to use a method developed by Mason(S. J. Mason and H. J. Zimmermann, Electronic Circuits, Signals, and Systems, Wiley, New York, 1960, Chapter 4, "Linear Signal-Flow Graphs.") to calculate gains directly from an unreduced block diagram.

    In order to determine the gain between an input or disturbance and any other points in the diagram, it is necessary to identify two topological features of a block diagram. A path is a continuous succession of blocks, lines, and summation points that connect the input and signal of interest and along which no element is encountered more than once. Lines may be traversed only in the direction of information flow (with the arrow). It is possible in general to have more than one path connecting an input to an output or other signal of interest. The path gain is a product of the gains of all elements in a path. A loop is a closed succession of blocks, lines, and

    summation points traversed with the arrows, along which no element is encountered more than once per cycle. The loop gain is the product of gains of all elements in a loop. It is necessary to include the inversions indicated by negative signs at summation points when calculating path or loop gains. The general expression for the gain or transmission of a block diagram is

    \[T = \dfrac{\displaystyle \sum_a P_a \left(1 - \displaystyle \sum_b L_b + \displaystyle \sum_{c,d} L_c L_d - \displaystyle \sum_{e,f,g} L_e L_f L_g + \cdots -\right)}{1 - \displaystyle \sum_h L_h + \sum_{i, j} L_i L_j - \displaystyle \sum_{k, l, m} L_k L_l L_m + \cdots -} \label{eq2.4.18} \]

    The numerator of the gain expression is the sum of the gains of all paths connecting the input and the signal of interest, with each path gain scaled by a cofactor. The first sum in a cofactor includes the gains of all loops that do not touch (share a common block or summation point with) the path; the second sum includes all possible products of loop gains for loops that do not touch the path or each other taken two at a time; the third sum includes all possible triple products of loop gains for loops that do not touch the path or each other; etc.

    The denominator of the gain expression is called the determinant or characteristic equation of the block diagram, and is identically equal to one minus the loop transmission of the complete block diagram. The first sum in the characteristic equation includes all loop gains; the second all possible products of the gains of nontouching loops taken two at a time; etc. Two examples will serve to clarify the evaluation of the gain expression. Figure 2.13 provides the first example. In order to apply Mason's gain formula for the transmission \(V_o/ V_i\), the paths and loops are identified and

    their gains are evaluated. The results are:

    \[\begin{array} {rcl} {P_1} & = & {ace} \\ {P_2} & = & {ag} \\ {P_3} & = & {-h} \\ {L_1} & = & {-ab} \\ {L_2} & = & {cd} \\ {L_3} & = & {-ef} \\ {L_4} & = & {-acei} \end{array}\nonumber \]

    The topology of Figure 2.13 shows that path \(P_1\) shares common blocks with and therefore touches all loops. Path \(P_2\) does not touch loops \(L_2\) or \(L_3\), while path \(P_3\) does not touch any loops. Similarly, loops \(L_1\), \(L_2\), and \(L_3\) do not touch each other, but all touch loop \(L_4\). Equation \(\ref{eq2.4.18}\) evaluated for this system becomes

    \[\dfrac{V_o}{V_i} = \dfrac{P_1 + P_2 (1 - L_2 - L_3 + L_2 L_3) + P_3 (1 - L_1 - L_2 - L_3 - L_4 + L_1 L_2 + L_2 L_3 + L_1 L_3 - L_1 L_2 L_3)}{1 - L_1 - L_2 - L_3 - L_4 + L_1 L_2 + L_2 L_3 + L_1 L_3 - L_1 L_2 L_3} \nonumber \]

    A second example of block-diagram reduction and some reinforcement of the techniques used to describe a system in block-diagram form is pro­vided by the set of algebraic equations

    \[\begin{array} {rcl} {X + Y + Z} & = & {6} \\ {X + Y - Z} & = & {0} \\ {2X + 3Y + Z} & = & {11} \end{array}\label{eq2.4.20} \]

    截屏2021-08-05 下午4.47.54.png
    Figure 2.13 Block diagram for gain-expression example.

    In order to represent this set of equations in block-diagram form, the three equations are rewritten

    \[\begin{array} {rcl} {X} & = & {-Y - Z + 6} \\ {Y} & = & {-X + Z} \\ {Z} & = & {-2X - 3Y + 11} \end{array} \nonumber \]

    This set of equations is shown in block-diagram form in Figure 2.14. If we use the identification of loops in this figure, loop gains are

    \[\begin{array} {rcl} {L_1} & = & {1} \\ {L_2} & = & {-3} \\ {L_3} & = & {-3} \\ {L_4} & = & {2} \\ {L_5} & = & {2} \end{array} \nonumber \]

    Since all loops touch, the determinant of any gain expression for this sys­tem is

    \[1 - L_1 - L_2 - L_3 - L_4 - L_5 = 2 \nonumber \]

    (This value is of course identically equal to the determinant of the coeffi­cients of Equation \(\ref{eq2.4.20}\).)

    Assume that the value of \(X\) is required. The block diagram shows one path with a transmission of +1 connecting the excitation with a value of 6 to \(X\). This path does not touch \(L_2\). There are also two paths (roughly paralleling \(L_3\) and \(L_5\)) with transmissions of - 1 connecting the excitation with a value of 11 to \(X\). These paths touch all loops. Linearity allows us to combine the \(X\) responses related to the two excitations, with the result that

    \[X = \dfrac{6[1 - (-3)] - 11 -11}{2} = 1 \nonumber \]

    The reader should verify that this method yields the values \(Y = 2\) and \(Z = 3\) for the other two dependent variables.


    This page titled 2.4: BLOCK DIAGRAMS is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James K. Roberge (MIT OpenCourseWare) .