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Transcript

CIRCUIT ANALYSIS METHODS

Why?

PROBLEM ?

IN ORDER TO ANALYZE A CIRCUIT WHICH HAS ne ELEMENTS, 2-ne EQUATIONS WITH 2-ne UNKNOWNS NEEDS TO BE SOLVED.

COULD IT BE POSSIBLE TO SOLVE (ANALYZE) A CIRCUIT WITH LESS THAN 2ne NUMBER OF EQUATIONS ?

YES BUT UNKNOWNS SHOULD ALSO BE REDUCED AS MANY AS EQUATIONS.

THERE ARE TWO SOLUTIONS:

MESH-CURRENT METHOD

NODE-VOLTAGE METHOD

MESH-CURRENT METHOD

MESH- CURRENT METHOD

NEW VARIABLE: MESH CURRENT

MESH CURRENT

ASSUME THAT FOR EACH MESH THERE IS A CURRENT

SUCH THAT EACH ELEMENT'S CURRENT CAN BE WRITTEN IN TERMS OF MESH CURRENTS.

ELEMENTS CURRENTS VS MESH CURRENTS

IF AN ELEMENT IS IN TWO MESHES (ELEMENT "B" IN THE FIGURE),

ELEMENT B'S CURRENT IS THE SUM OF THE MESH CURRENTS

  • WITH SIGN (+) WHEN BOTH ELEMENT CURRENT REFERENCE AND MESH CURRENT DIRECTION IS THE SAME,
  • WITH SIGN (-) OTHERWISE.

IF AN ELEMENTS IS IN ONE MESH ONLY (ELEMENT "A" IN THE FIGURE)

AND ELEMENT CURRENT REFERENCE AND MESH CURRENT DIRECTION IS THE SAME

  • THEN ELEMENT'S CURRENT IS EQUAL TO MESH CURRENT WITH (+) SIGN,
  • OTHERWISE (-)SIGN.

ELEMENTS CURRENTS VS

MESH CURRENTS

MESH_CURRENT METHOD

STEP 1

Write independent KVL equations:

number of KVL equations: (ne-nd+1)

Unknowns: ne elements voltages.

STEP 3

Substitute elements currents for mesh currents

  • (ne-nd+1) equations
  • unknowns: (ne-nd+1) mesh currents

Unique solution

In step 2,

if an element voltage cannot be substituted for its current

then

  • keep this element voltage as a new unknown(s) ( these could be voltages of independent/dependent current sources i.e Vıs ,Vcccs, Vvccs).
  • Since a new unknown (s) came in (addition to mesh-current unknowns) then an additional equation(s) is needed to write, this equation is called "supplementary equation".

Supplementary equation is written by

the definition relation of the new introduced voltage unknown which could be Vıs ,Vcccs or Vvccs.

In this definition relation all the variables should be transferred to mesh currents.

STEP 2

Using elements definition relations, substitute elements voltages for elements currents;

  • (ne-nd+1) equations:
  • unknowns: ne elements currents

STEPS

Once the linear equations are solved, all the mesh currents and new introduced unknowns (if any) are found.

This solution is unique since number of equations is equal to number of unknowns.

MESH-Currents/Elements currents

FINDING 2Ne-UNKNOWNS

  • Once the mesh currents found then each element's current calculated as it is explained before (two frames back).
  • Then elements voltages are found by using elements definition relations.

EXAMPLE

It is possible directly write the third step mesh current equations! write a rule for that.

What happens if an independent source is not DC but AC, such as VB=10sin(2t+60)?

What can yo say about elements currents and voltages depend on what?

NODE VOLTAGE METHOD

NODE-VOLTAGE METHOD

NEW VARIABLE: NODE VOLTAGE

NODE-VOLTAGES

NODE-VOLTAGE: THE VOLTAGE MEASURED BETWEEN A NODE AND REFERENCE NODE (GROUND) IS CALLED THAT NODE-VOLTAGE.

ASSUME THAT FOR EACH NODE THERE SHOULD BE A VOLTAGE (NODE-VOLTAGE) SUCH THAT

  • EACH ELEMENT'S VOLTAGE CAN BE CALCULATED
  • BY THESE NODE-VOLTAGES
  • ASSIGNED TO THE ELEMENT'S TERMINALS.

ELEMENTS VOLTAGES VS NODE-VOLTAGES

ELEMENTS VOLTAGES VS

NODE_VOLTAGES

RULE:

ELEMENT'S VOLTAGE (Va) =

=[Element's (+) related terminal node-voltage, (vd1)] -[Element's (-) related terminal node-voltage(vd2)]

NODE-VOLTAGE METHOD

STEP 1

Write the independent KCL equations

  • (nd-1) equations
  • number of unknowns: ne-element currents

In step 2,

if an element current cannot be substituted for its voltage

then keep this element current as a new unknown ( these could be currents of independent/dependent voltage sources i.e Ivs ,Icccs, Ivccs).

  • Since a new unknown came in (addition to node-voltages unknowns)
  • then an additional equation is needed to write,
  • this equation is called "supplementary equation".

Supplementary equation is written by

the definition relation of the new introduced current unknown which could be Ivs ,Icccs, Ivccs.

In this definition relation all the variables should be transferred to node-voltages.

STEP 2

Substitute elements current for element voltages by using elements definition relations

  • number of equations:(nd-1)
  • unknowns: ne element voltages

STEP 3

Write elements voltages in terms of node-voltages (as explained two frame back)

  • (nd-1) equations
  • unknowns: (nd-1) node-voltages

STEPS

FINDING EACH ELEMENT'S CURRENT AND VOLTAGES

FINDING 2ne ELEMENTS VOLTAGES AND CURRENTS

Once the node-voltages found

  • then each element's voltage calculated as it is explained before (two frames back).
  • Then elements currents are found by using elements definition relations.

EXAMPLE FOR MESH-CURRENT METHOD

EXAMPLE

Vd2=29.8V, Vd3=19.2V, Vd4=16.9V

It is possible directly write the third step node-voltage current equations! write a rule for that.

What happens if independent source not DC but AC, such as 10sin(2t+60)?

What can yo say about elements' currents and voltages depend on what?

CONCLUSION

A simple analysis of resistive circuits can be done

  • by two methods
  • in two levels.

CONCLUSIONS

Mesh current method:

Level 1:

  • (ne-nd+1)X (ne-nd+1) range linear equations and
  • supplementary equations are solved
  • to find mesh currents.

Level 2:

  • Element currents are found in terms of mesh-currents and then
  • element voltages are found by definition relations.

Node-voltage method:

Level 1:

  • (nd-1)X (nd-1) range linear equations and
  • supplementary equations are solved
  • to find node-voltages.

Level 2:

  • Element voltages are found in terms of node-voltages and then
  • element currents are found by definition relations.

Which method is chosen when?why?

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