Gy's sampling theory

Template:Redirects here A signal-flow graph (SFG) is a special type of block diagram—and directed graph—consisting of nodes and branches. Its nodes are the variables of a set of linear algebraic relations. An SFG can only represent multiplications and additions. Multiplications are represented by the weights of the branches; additions are represented by multiple branches going into one node. A signal-flow graph has a one-to-one relationship with a system of linear equations. In addition to this, it can also be used to represent the signal flow in a physical system; i.e., it can represent relations of cause and effect.

Example 1: Simple amplifier

File:Sfg simple amplifier.svg

Figure 1: SFG of a simple amplifier

The amplification of a signal V₁ by an amplifier with gain a₁₂ is described mathematically by

$V_{2}=a_{12}V_{1}\,.$

This relationship represented by the signal-flow graph of Figure 1. is that V₂ is dependent on V₁ but it implies no dependency of V₁ on V₂. See Kou page 57.

Example 2: Two-port network

File:Sfg ymatrix.svg

Figure 2: SFG describing two-port admittance matrix equations

In electrical circuits, two equations describing a two-port network in admittance matrix form,

${\begin{bmatrix}I_{1}\\I_{2}\end{bmatrix}}={\begin{bmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{bmatrix}}{\begin{bmatrix}V_{1}\\V_{2}\end{bmatrix}}$

can be drawn as the SFG shown in Figure 2. This signal-flow graph describes causality properly if and only if V₁ and V₂ are designated as inputs. If any other designation of inputs is made then the SFG is either incomplete or invalid.

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Example 2a: Circuit containing a two-port network

Signal flow graph of a circuit containing a two port. The forward path from input to output is shown in a different color.

The figure to the right depicts a circuit that contains a two-port network. V_in is the input of the circuit and V₂ is the output. The two-port equations impose a set of linear constraints between its port voltages and currents. The terminal equations impose other constraints. All these constraints are represented in the SFG (Signal Flow Graph) below the circuit. There is only one path from input to output which is shown in a different color and has a gain of -R_Ly₂₁. There are also three loops: -R_iny₁₁, -R_Ly₂₂, R_inR_Ly₁₂y₂₁. Sometimes a loop indicates intentional feedback but it can also indicate a constraint on the relationship of two variables. For example, the equation that describes a resistor says that the ratio of the voltage across the resistor to the current through the resistor is a constant which is called the resistance. This can be interpreted as the voltage is the input and the current is the output, or the current is the input and the voltage is the output, or merely that the voltage and current have a linear relationship. Virtually all passive two terminal devices in a circuit will show up in the SFG as a loop.

The SFG and the schematic depict the same circuit, but the schematic also suggests the circuit's purpose. Compared to the schematic, the SFG is awkward but it does have the advantage that the input to output gain can be written down by inspection using Mason's rule.

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Example 3 Asymptotic gain formula

Figure 3: A possible signal-flow graph for the asymptotic gain model

Figure 4: A different signal-flow graph for the asymptotic gain model

A possible SFG for the asymptotic gain model for a negative feedback amplifier is shown in Figure 3, and leads to the equation for the gain of this amplifier as

$G={\frac {y_{2}}{x_{1}}}$ $=G_{\infty }\left({\frac {T}{T+1}}\right)+G_{0}\left({\frac {1}{T+1}}\right)\ .$

The interpretation of the parameters is as follows: T = return ratio, G_∞ = direct amplifier gain, G₀ = feedforward (indicating the possible bilateral nature of the feedback, possibly deliberate as in the case of feedforward compensation). Figure 3 has the interesting aspect that it resembles Figure 2 for the two-port network with the addition of the extra feedback relation x₂ = T y₁.

From this gain expression an interpretation of the parameters G₀ and G_∞ is evident, namely:

$G_{\infty }=\lim _{T\to \infty }G\ ;\ G_{0}=\lim _{T\to 0}G\ .$

There are many possible SFG's associated with any particular gain relation. Figure 4 shows another SFG for the asymptotic gain model that can be easier to interpret in terms of a circuit. In this graph, parameter β is interpreted as a feedback factor and A as a "control parameter", possibly related to a dependent source in the circuit. Using this graph, the gain is

$G={\frac {y_{2}}{x_{1}}}$ $=G_{0}+{\frac {A}{1-\beta A}}\ .$

To connect to the asymptotic gain model, parameters A and β cannot be arbitrary circuit parameters, but must relate to the return ratio T by:

$T=-\beta A\ ,$

and to the asymptotic gain as:

$G_{\infty }=\lim _{T\to \infty }G=G_{0}-{\frac {1}{\beta }}\ .$

Substituting these results into the gain expression,

$G=G_{0}+{\frac {1}{\beta }}{\frac {-T}{1+T}}$

$=G_{0}+(G_{0}-G_{\infty }){\frac {-T}{1+T}}$

$=G_{\infty }{\frac {T}{1+T}}+G_{0}{\frac {1}{1+T}}\ ,$

which is the formula of the asymptotic gain model.

Signal flow graphs are used in many different subject areas besides control and network theory, for example, stochastic signal processing.

Example 4: Position servo with multi-loop feedback

Angular position servo and signal flow graph. θ_C = desired angle command, θ_L = actual load angle, K_P = position loop gain, V_ωC = velocity command, V_ωM = motor velocity sense voltage, K_V = velocity loop gain, V_IC = current command, V_IM = current sense voltage, K_C = current loop gain, V_A = power amplifier output voltage, L_M = motor inductance, V_M = voltage across motor inductance, I_M = motor current, R_M = motor resistance, R_S = current sense resistance, K_M = motor torque constant (Nm/amp), T = torque, M = moment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, G_M = motor back EMF constant, G_T = tachometer conversion gain constant,. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring. Constants are shown in black and variables in purple.

This example is representative of a SFG (signal-flow graph) used to represent a servo control system and illustrates several features of SFGs. Some of the loops (loop 3, loop 4 and loop 5) are extrinsic intentionally designed feedback loops. These are shown with dotted lines. There are also intrinsic loops (loop 0, loop1, loop2) that are not intentional feedback loops, although they can be analyzed as though they were. These loops are shown with solid lines. Loop 3 and loop 4 are also known as minor loops because they are inside a larger loop.

• The forward path begins with $\theta _{C}\,$, the desired position command. This is multiplied by K_P which could be a constant or a function of frequency. K_P incorporates the conversion gain of the DAC and any filtering on the DAC output. The output of K_P is the velocity command $V_{\omega C}\,$ which is multiplied by K_V which can be a constant or a function of frequency. The output of K_V is the current command, V_IC which is multiplied by K_C which can be a constant or a function of frequency. The output of K_C the amplifier output voltage, V_A. The current, I_M, though the motor winding is the integral of the voltage applied to the inductance. The motor produces a torque, T, proportional to I_M. Permanent magnet motors tend to have a linear current to torque function. The conversion constant of current to torque is K_M. The torque, T, divided by the load moment of inertia, M, is the acceleration, $\alpha \,$, which is integrated to give the load velocity $\omega \,$ which is integrated to produce the load position, $\theta _{L}\,$.

• The forward path of loop 0 asserts that acceleration is proportional to torque and the velocity is the time integral of acceleration. The backward path says that as the speed increases there is a friction or drag that counteracts the torque. Torque on the load decreases proportionately to the load velocity until the point is reached that all the torque is used to overcome friction and the acceleration drops to zero. Loop 0 is intrinsic.

• Loop1 represents the interaction of an inductor's current with its internal and external series resistance. The current through an inductance is the time integral of the voltage across the inductance. When a voltage is first applied, all of it appears across the inductor. This is shown by the forward path through ${\frac {1}{sL_{M}}}\,$. As the current increases, voltage is dropped across the inductor internal resistance R_M and the external resistance R_S. This reduces the voltage across the inductor and is represented by the feedback path -(R_M + R_S). The current continues to increase but at a steadily decreasing rate until the current reaches the point at which all the voltage is dropped across (R_M + R_S). Loop 1 is intrinsic.

• Loop2 expresses the effect of the motor back EMF. Whenever a permanent magnet motor rotates, it acts like a generator and produces a voltage in its windings. It does not matter whether the rotation is caused by a torque applied to the drive shaft or by current applied to the windings. This voltage is referred to as back EMF. The conversion gain of rotational velocity to back EMF is G_M. The polarity of the back EMF is such that it diminishes the voltage across the winding inductance. Loop 2 is intrinsic.

• Loop 3 is extrinsic. The current in the motor winding passes through a sense resister. The voltage, V_IM, developed across the sense resister is fed back to the negative terminal of the power amplifier K_C. This feedback causes the voltage amplifier to act like a voltage controlled current source. Since the motor torque is proportional to motor current, the sub-system V_IC to the output torque acts like a voltage controlled torque source. This sub-system may be referred to as the "current loop" or "torque loop". Loop 3 effectively diminishes the effects of loop 1 and loop 2.

• Loop 4 is extrinsic. A tachometer (actually a low power dc generator) produces an output voltage $V_{\omega M}\,$ that is proportional to is angular velocity. This voltage is fed to the negative input of K_V. This feedback causes the sub-system from $V_{\omega C}\,$ to the load angular velocity to act like a voltage to velocity source. This sub-system may be referred to as the "velocity loop". Loop 4 effectively diminishes the effects of loop 0 and loop 3.

• Loop 5 is extrinsic. This is the overall position feedback loop. The feedback comes from an angle encoder that produces a digital output. The output position is subtracted from the desired position by digital hardware which drives a DAC which drives K_P. In the SFG, the conversion gain of the DAC is incorporated into K_P.

see Mason's rule for development of Mason's Gain Formula for this example.

Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com

See also

• Control Systems/Signal Flow Diagrams in the Control Systems Wikibook
• Mason's rule
• Minor loop feedback
• Noncommutative signal-flow graph

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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External links

• Khoman Phang: An overview of signal-flow graphs This chapter is from Dr. Phang's Ph. D. Thesis © Copyright by Khoman Phang 2001
• M. L. Edwards: S-parameters, signal flow graphs, and other matrix representations All Rights Reserved