{"id":365,"date":"2019-05-21T21:05:31","date_gmt":"2019-05-21T21:05:31","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=365"},"modified":"2019-11-18T17:53:20","modified_gmt":"2019-11-18T17:53:20","slug":"1-6-basic-block-diagrams","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/1-6-basic-block-diagrams\/","title":{"raw":"1.6 Basic Block Diagrams","rendered":"1.6 Basic Block Diagrams"},"content":{"raw":"

1.6.1 Blocks in Series<\/strong><\/h3>\r\n

Consider two cascade blocks as shown in Figure 1\u201117. What is the transfer function relating signals [latex]Y(s)[\/latex] and [latex]R(s)[\/latex]?<\/p>\r\n\r\n\r\n[caption id=\"attachment_369\" align=\"aligncenter\" width=\"373\"]\"Figure Figure 1-17 Blocks in Series[\/caption]\r\n\r\n[latex]X(s) = R(s) \\cdot G_{1}(s)[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]Y(s) = R(s) \\cdot G_{1}(s) \\cdot G_{2}(s)[\/latex]\r\n\r\n[latex]Y(s) = X(s) \\cdot G_{2}(s)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]G(s) = \\frac{Y(s)}{R(s)} = G_{1}(s)\\cdot G_{2}(s)[\/latex]\r\n

1.6.2 Blocks in Parallel<\/strong><\/h3>\r\nConsider two blocks in parallel as shown in Figure 1\u201118. What is the transfer function relating signals [latex]Y(s)[\/latex] and [latex]R(s)[\/latex]?\r\n\r\n[latex]X_{1}(s) = R(s) \\cdot G_{1}(s)[\/latex]\r\n\r\n[latex]X_{2}(s) = R(s) \\cdot G_{2}(s)[\/latex]\r\n\r\n[latex]Y(s) = X_{1}(s) + X_{2}(s)[\/latex]\r\n\r\n[latex]G(s) = \\frac{Y(s)}{R(s)} = \\frac{R(s) \\cdot G_{1}(s) + R(s) \\cdot G_{2}(s)}{R(s)} = G_{1}(s) + G_{2}(s)[\/latex]\r\n\r\n[caption id=\"attachment_375\" align=\"aligncenter\" width=\"360\"]\"Figure Figure 1-18 Blocks in Parallel[\/caption]\r\n

1.6.3 Basic Closed Loop Revisited\u00a0<\/strong><\/h3>\r\n

Consider the basic closed loop system identical to the one shown in Chapter 1.2.2, but this time with the static block gains replaced by their dynamic equivalents, i.e. their transfer functions describing a relationship of time domain signals as represented in Laplace Transform domain. Let us derive the closed loop transfer function relating system output to the reference.<\/p>\r\n[latex]Y(s) = E(s) \\cdot G(s)[\/latex]\r\n\r\n[latex]E(s) = R(s) - B(s)[\/latex]\r\n\r\n[latex]B(s) = Y(s) \\cdot H(s)[\/latex]\r\n\r\n[latex]G_{cl}(s) = \\frac{Y(s)}{R(s)} = ?[\/latex]\r\n\r\n \r\n\r\n[caption id=\"attachment_376\" align=\"aligncenter\" width=\"354\"]\"Figure Figure 1-19 Basic Closed Loop[\/caption]\r\n\r\n\r\n\r\n
[latex]G_{cl}(s) = \\frac{Y(s)}{R(s)} = \\frac{Y(s)}{E(s) + B(s)} = \\frac{E(s) \\cdot G(s)}{E(s) + Y(s) \\cdot H(s)}[\/latex]\r\n\r\n[latex]G_{cl} = \\frac{E(s) \\cdot G(s)}{E(s) + E(s) \\cdot G(s) \\cdot H(s)} = \\frac{E(s) \\cdot G(s)}{E(s) \\cdot (1+G(s) \\cdot H(s))}[\/latex]\r\n\r\n[latex]G_{cl}(s) = \\frac{G(s)}{1+G(s) \\cdot H(s)}[\/latex]<\/td>\r\nEquation 1\u201120<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nClosed loop characteristic equation is then described as follows:\r\n\r\n\r\n\r\n
[latex]1 + G(s)H(s) = 0[\/latex]<\/td>\r\nEquation 1\u201121<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n ","rendered":"

1.6.1 Blocks in Series<\/strong><\/h3>\n

Consider two cascade blocks as shown in Figure 1\u201117. What is the transfer function relating signals [latex]Y(s)[\/latex] and [latex]R(s)[\/latex]?<\/p>\n

\"Figure
Figure 1-17 Blocks in Series<\/figcaption><\/figure>\n

[latex]X(s) = R(s) \\cdot G_{1}(s)[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]Y(s) = R(s) \\cdot G_{1}(s) \\cdot G_{2}(s)[\/latex]<\/p>\n

[latex]Y(s) = X(s) \\cdot G_{2}(s)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]G(s) = \\frac{Y(s)}{R(s)} = G_{1}(s)\\cdot G_{2}(s)[\/latex]<\/p>\n

1.6.2 Blocks in Parallel<\/strong><\/h3>\n

Consider two blocks in parallel as shown in Figure 1\u201118. What is the transfer function relating signals [latex]Y(s)[\/latex] and [latex]R(s)[\/latex]?<\/p>\n

[latex]X_{1}(s) = R(s) \\cdot G_{1}(s)[\/latex]<\/p>\n

[latex]X_{2}(s) = R(s) \\cdot G_{2}(s)[\/latex]<\/p>\n

[latex]Y(s) = X_{1}(s) + X_{2}(s)[\/latex]<\/p>\n

[latex]G(s) = \\frac{Y(s)}{R(s)} = \\frac{R(s) \\cdot G_{1}(s) + R(s) \\cdot G_{2}(s)}{R(s)} = G_{1}(s) + G_{2}(s)[\/latex]<\/p>\n

\"Figure
Figure 1-18 Blocks in Parallel<\/figcaption><\/figure>\n

1.6.3 Basic Closed Loop Revisited\u00a0<\/strong><\/h3>\n

Consider the basic closed loop system identical to the one shown in Chapter 1.2.2, but this time with the static block gains replaced by their dynamic equivalents, i.e. their transfer functions describing a relationship of time domain signals as represented in Laplace Transform domain. Let us derive the closed loop transfer function relating system output to the reference.<\/p>\n

[latex]Y(s) = E(s) \\cdot G(s)[\/latex]<\/p>\n

[latex]E(s) = R(s) - B(s)[\/latex]<\/p>\n

[latex]B(s) = Y(s) \\cdot H(s)[\/latex]<\/p>\n

[latex]G_{cl}(s) = \\frac{Y(s)}{R(s)} = ?[\/latex]<\/p>\n

 <\/p>\n

\"Figure
Figure 1-19 Basic Closed Loop<\/figcaption><\/figure>\n\n\n\n
[latex]G_{cl}(s) = \\frac{Y(s)}{R(s)} = \\frac{Y(s)}{E(s) + B(s)} = \\frac{E(s) \\cdot G(s)}{E(s) + Y(s) \\cdot H(s)}[\/latex]<\/p>\n

[latex]G_{cl} = \\frac{E(s) \\cdot G(s)}{E(s) + E(s) \\cdot G(s) \\cdot H(s)} = \\frac{E(s) \\cdot G(s)}{E(s) \\cdot (1+G(s) \\cdot H(s))}[\/latex]<\/p>\n

[latex]G_{cl}(s) = \\frac{G(s)}{1+G(s) \\cdot H(s)}[\/latex]<\/td>\n

Equation 1\u201120<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Closed loop characteristic equation is then described as follows:<\/p>\n\n\n\n
[latex]1 + G(s)H(s) = 0[\/latex]<\/td>\nEquation 1\u201121<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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