Lecture 09

Mesh Current Method and Loop Current Method

Previously, Branch current method�

• Procedure of branch current method

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• Select the associated reference direction of current and voltage of each element (component) and/or voltage/current source (if any) in each branch.

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• For each of arbitrarily selected n-1 nodes, listing the KCL equations.

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• Selecting b-(n-1) independent loops and listing the KVL equations for each loop based on VCL.

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• Note
• advantage: the branch current method is convenient and intuitive.
• disadvantage: it can involve quite a lot equations when the circuit graph has many branches.

Mesh current method�

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A method by taking the imaginary currents flowing along the b-(n-1) meshes as the unknowns, by which to list the circuit equations and to solve these equation.

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Motivation: to use less unknowns in solving the equation.

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• In contrast, the number of equations in the branch current method is b (the number of branches).

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• Supposing there is a virtual mesh current in each mesh;

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• The current of each branch is a combination of two virtual mesh currents.

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Mesh current method�

• b-(n-1) (b=3, n=2) independent loops (Single-link loops or Mesh loops)

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• Imagine two virtual mesh currents i_M1 and i_M2.

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• The branch current can be represented by the mesh current.

• Note
• In the mesh current method, a mesh current flows in a single closed loop. Therefore, KCL is satisfied (because there is only one mesh current in a mesh).
• Thus, only KVL equation is enough for each mesh. The number of equations equal the number of meshes.

Mesh current method�

• Let R₂₁ = R₁₂ = -R₂ (or mutual-resistance of mesh 1 and mesh 2)

• Let u_M1 = u_s1 –u_s2 be the total voltage of all voltage sources of mesh 1.

• Let u_M2 = u_s2 be the total voltage of all voltage sources of mesh 2.

Mesh current method�

• Note
• The self-resistance of each mesh is positive.
• When two mesh currents on the same branch are in the same direction, the mutual resistance is positive; Otherwise, it is negative.
• When the voltage direction of the voltage source is consistent with the mesh current direction, the voltage is negative; Otherwise, it is positive.

If the circuit has two meshes, the standard form of mesh current equation is as follows.

Mesh current method�

More generally, if the circuit has K meshes, the standard form of mesh current equation is as follows.

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Note:

• The self-resistance of mesh i, R_ii (i=1,…,K), is positive.
• The mutual-resistance two neighboring mesh p and mesh q, R_pq (p=1,…,K, q=1,…,K), is positive when the directions of the two mesh currents on R_pq are consistent. Otherwise, R_pq is negative.
• If mesh p and mesh q are not neighbors, R_pq and R_qp are zero.

Mesh current method�

• Example: to solve i using the mesh current method.

i = i_M2 – i_M3

Note:

• The coefficient matrix corresponding to the linear circuit network without dependent source is symmetric, i.e., R_pq = R_qp.
• When all the mesh current takes the clockwise (or counterclockwise) reference direction, R_pq is negative.

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Loop current method�

Motivation: to use less unknowns in solving the equation. Previously, the mesh current method can only be applied to planar circuit. Loop current method is a more general analysis method and can be applied to non-planar loops. Mesh is a special loop.

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• The “loop” here only refers to the single-link loops, which are independent loops.

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• Supposing there is a virtual current in each single-link loop. The current of each branch is a combination of the virtual loop currents.

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• A method by taking the imaginary current flowing along each single-link loop as the unknowns, by which to list b-(n-1) KVL equations and to solve these equation.

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• In contrast, the number of equations in the branch current method is b (the number of branches).

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Loop current method�

• Example: to solve i using loop current method.

select a tree

How?

three single-link loops

- R₁*i_L1 – (R₁+R₄)*i_L2 + (R_s+ R₁+ R₄)*i_L3 = u_s (3)

(R₁+R₂ +R₅)*i_L1 + (R₁+R₂)*i_L2 - R₁*i_L3 = 0 (1)

(R₁+R₂)*i_L1 + (R₁+R₂+R₃+R₄)*i_L2 - (R₁+R₄)*i_L3 = 0 (2)

Loop current method�

More generally, if the circuit has K = b-(n-1) single-link loops, the standard form of loop current equation is as follows.

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Note:

• The self-resistance of loop i, R_ii (i=1,…,K), is positive.
• The mutual-resistance of two neighboring loop p and loop q, R_pq (p=1,…,K, q=1,…,K), is positive when the directions of the two loop currents on R_pq are consistent. Otherwise, R_pq is negative.
• If loop p and loop q are not neighbors, R_pq and R_qp are zero.

Loop current method�

Procedure:

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• Select K = b-(n-1) single-link loops, and select the reference direction of loop current.

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• List the KVL equation for each loop with respect to the unknown loop current.

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• Solve the K KVL equations to obtain the K loop current.

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• Solve the branch current based on the obtained loop current.

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• Solve the other required quantities if any.

Loop current method�

• Special cases: the existence of ideal current source

Solution I: Introducing voltage for the current source, then adding the constraint between loop current and the current of current-source.

i_L1

i_L2

i_L3

(R_s+ R₁+ R₄)*i_L1 - R₁*i_l2 – R₄*i_l3 = u_s (1)

KVL equations for each loop:

-R₁*i_L1 + (R₁+ R₂)*i_L2 = u (2)

-R₄*i_L1 + (R₃+ R₄)*i_L3 = -u (3)

Extra constraint:

i_L2 – i_L3 = i_s

Note:

We have four unknowns. To easily solve the above equations, it would be better to rearrange the above equations as follows.

Loop current method�

• Special cases: the existence of ideal current source

Solution I: Introducing voltage for the current source, then adding the constraint between loop current and the current of current-source.

(R_s+ R₁+ R₄)*i_L1 - R₁*i_l2 – R₄*i_l3 = u_s (1)

Rearranged equations:

-R₁*i_L1 + (R₁+ R₂)*i_L2 - u = 0 (2)

-R₄*i_L1 + (R₃+ R₄)*i_L3 + u = 0 (3)

i_L2 – i_L3 = i_s (4)

Loop current method�

• Special cases: the existence of ideal current source

Solution II: Selecting independent loops (single-link loops), such that the ideal current branch belongs to only one loop.

select a tree

(R_s+ R₁+ R₄)*i_L1 - R₁*i_l2 – (R₁+R₄)*i_l3 = u_s (1)

KVL equations for each loop:

i_L2 = i_s (2)

-(R₁ + R₄) *i_L1 + (R₁ + R₂) *i_L2 + (R₁+ R₂ +R₃+ R₄)*i_L3 = 0 (3)

Loop current method�

• Special cases: the existence of dependent voltage or current source

5u

+

-

i_L1

i_L2

i_L3

Solution: Viewing each dependent source as an independent source, then applying the loop current method.

+

-

u

KVL equations for each loop:

R₃*i_L3 = u (4)

Extra constraint:

Loop current method�

• Special cases: the existence of dependent voltage or current source

i_L1

i_L2

i_L3

i_L4

KVL equations for each loop:

-R₅*i_L3 + R₅*i_L4 = u₃ -μu₁ (4)

Extra constraints:

R₁*i_L1 = -u₁ (5)

i_L1 – i_L2 = i_s (6)

-i_L2 + i_L4 = gu₁ (7)