6.1 Block Diagram Representation

Block diagrams are visual tools that represent dynamic systems, showing how components interact and signals flow. They use blocks for system parts, arrows for signals, and junctions for combining inputs. This representation helps us understand how a system processes information and achieves its goals.

By constructing and analyzing block diagrams, we can simplify complex systems and determine their overall behavior. We can combine blocks, identify feedback loops, and calculate transfer functions. This approach allows us to study system stability, performance, and response to different inputs.

Block diagrams for dynamic systems

Components and structure

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• Block diagrams consist of blocks representing system components, arrows representing signals, summing junctions, and branch points
  • Blocks depict transfer functions that mathematically describe the input-output relationships of system components
  • Arrows indicate the direction of signal flow and establish interconnections between components (e.g., from a sensor to a controller)
  • Summing junctions add or subtract incoming signals before processing by a component (e.g., combining a reference signal and a feedback signal)
  • Branch points allow a signal to be distributed to multiple blocks (e.g., a control signal sent to multiple actuators)
• The overall structure of the block diagram illustrates the sequence and relationships among the components of the dynamic system
  • The arrangement of blocks and interconnections visually represents the flow of signals and the dependencies between components
  • The structure helps in understanding how the system processes inputs, generates outputs, and achieves its desired behavior

Signal flow and relationships

• Each block represents a transfer function that mathematically describes the relationship between its input and output signals
  • Transfer functions can be represented using mathematical expressions, such as Laplace transforms or differential equations
  • The transfer function of a block determines how the input signal is transformed or processed to produce the output signal
• Arrows indicate the direction of signal flow and establish the interconnections between system components
  • Signals flow from the output of one block to the input of another, representing the propagation of information or energy
  • The interconnections define the dependencies and interactions between the components of the dynamic system
• The block diagram representation allows for a clear visualization of the signal flow and the relationships between the system components
  • It helps in understanding how the system operates, identifying feedback loops, and analyzing the propagation of signals through the system

Constructing block diagrams

Representing system components

• Identify the key components of the dynamic system and represent each component as a block with its corresponding transfer function
  • Each component, such as a sensor, controller, actuator, or plant, is depicted as a separate block in the diagram
  • The transfer function of each block mathematically describes the input-output relationship of the corresponding component
• Determine the input and output signals for each component and represent them as arrows entering and leaving the respective blocks
  • Input signals are represented by arrows pointing towards the block, indicating the information or energy flowing into the component
  • Output signals are represented by arrows pointing away from the block, indicating the processed or transformed signal leaving the component
• Arrange the blocks and interconnections to accurately represent the flow of signals and the relationships between the system components
  • The placement of blocks should follow the logical sequence of signal processing and the dependencies between the components
  • Interconnections should be drawn to establish the proper signal flow and capture the interactions between the blocks

Simplifying block diagrams

• Use summing junctions to combine signals when multiple inputs are added or subtracted before being processed by a component
  • Summing junctions are represented by a circle with a plus (+) or minus (-) sign, indicating the operation performed on the incoming signals
  • Multiple input signals can be connected to a summing junction, and the output of the junction is the sum or difference of the inputs
• Employ branch points to distribute a signal to multiple components when necessary
  • Branch points are represented by a dot or a small circle, indicating that the incoming signal is split and sent to multiple blocks
  • The output signal from a branch point is identical to the input signal, allowing the same signal to be used by multiple components
• Simplify the block diagram by combining blocks in series, parallel, or feedback configurations when appropriate
  • Series-connected blocks can be combined into a single block with an equivalent transfer function obtained by multiplying the individual transfer functions
  • Parallel-connected blocks can be combined into a single block with an equivalent transfer function obtained by adding the individual transfer functions
  • Feedback loops can be simplified by determining the transfer function of the forward path and the feedback path separately and combining them using the appropriate formula

Input-output relationships in block diagrams

Analyzing signal paths

• Determine the individual transfer functions for each block in the system
  • Each block represents a specific component or subsystem, and its transfer function mathematically describes the relationship between its input and output signals
  • Transfer functions can be expressed using mathematical expressions, such as Laplace transforms or differential equations
• Identify the paths that signals take from the input to the output of the system
  • Trace the signal flow from the input block to the output block, following the arrows and considering the effects of summing junctions and branch points
  • Determine the sequence of blocks that the signal passes through and the operations performed on the signal along each path

Equivalent transfer functions

• Calculate the equivalent transfer function for series-connected blocks by multiplying their individual transfer functions
  • When blocks are connected in series, the output of one block becomes the input of the next block
  • The equivalent transfer function for series-connected blocks is obtained by multiplying the transfer functions of the individual blocks in the order they appear in the signal path
• Compute the equivalent transfer function for parallel-connected blocks by adding their individual transfer functions
  • When blocks are connected in parallel, the input signal is distributed to multiple blocks, and their outputs are summed
  • The equivalent transfer function for parallel-connected blocks is obtained by adding the transfer functions of the individual blocks

Feedback effects

• Analyze the effects of feedback loops on the system's input-output relationship
  • Feedback loops occur when the output of a system is fed back to influence its input, creating a closed loop
  • Feedback can be positive or negative, depending on whether the fed-back signal reinforces or opposes the input signal
• Positive feedback reinforces the input signal, potentially leading to instability
  • In positive feedback, the fed-back signal is in phase with the input signal, amplifying its effect
  • Positive feedback can cause the system to become unstable, leading to oscillations or unbounded growth of the output
• Negative feedback attenuates the input signal, promoting stability and reducing sensitivity to disturbances
  • In negative feedback, the fed-back signal is out of phase with the input signal, counteracting its effect
  • Negative feedback helps stabilize the system by reducing the impact of disturbances and maintaining the output close to the desired value

Transfer function from block diagrams

Simplifying block diagrams

• Simplify the block diagram by combining blocks connected in series, parallel, or feedback configurations
  • Series-connected blocks can be combined into a single block with an equivalent transfer function obtained by multiplying the individual transfer functions
  • Parallel-connected blocks can be combined into a single block with an equivalent transfer function obtained by adding the individual transfer functions
  • Feedback loops can be simplified by determining the transfer function of the forward path and the feedback path separately
• For series-connected blocks, multiply the individual transfer functions to obtain the equivalent transfer function
  • The equivalent transfer function for series-connected blocks is the product of the individual transfer functions in the order they appear in the signal path
  • For example, if blocks with transfer functions $G_1(s)$ and $G_2(s)$ are connected in series, the equivalent transfer function is $G_{eq}(s) = G_1(s) \times G_2(s)$
• For parallel-connected blocks, add the individual transfer functions to obtain the equivalent transfer function
  • The equivalent transfer function for parallel-connected blocks is the sum of the individual transfer functions
  • For example, if blocks with transfer functions $G_1(s)$ and $G_2(s)$ are connected in parallel, the equivalent transfer function is $G_{eq}(s) = G_1(s) + G_2(s)$

Closed-loop transfer function

• When dealing with feedback loops, determine the transfer function of the forward path and the feedback path separately
  • The forward path is the path from the input to the output without considering the feedback loop
  • The feedback path is the path from the output back to the input, representing the signal that is fed back
• Calculate the closed-loop transfer function using the formula: $\frac{C(s)}{R(s)} = \frac{G(s)}{1 \pm G(s)H(s)}$
  • $C(s)$ represents the output signal, $R(s)$ represents the input signal, $G(s)$ is the forward path transfer function, and $H(s)$ is the feedback path transfer function
  • The sign in the denominator depends on whether the feedback is positive (+ sign) or negative (- sign)
  • The closed-loop transfer function relates the input signal to the output signal, taking into account the effects of the feedback loop
• Simplify the resulting expression to obtain the overall transfer function of the system
  • Perform algebraic manipulations to simplify the closed-loop transfer function
  • Express the overall transfer function in a simplified form, such as a ratio of polynomials in the Laplace domain
• The overall transfer function provides a mathematical representation of the input-output relationship of the entire system, considering the effects of all the components and their interconnections