Mesh Analysis And Nodal Analysis

In the previous chapter, we've seen that the use of Kirchhoff's laws for AC circuit analysis not only results in many equations (as too with DC circuits), but also (due to the use of complex numbers) doubles the number of unknowns. To reduce the number of equations and unknowns there are two other methods we can use: the node potential and the mesh (loop) current methods. The only difference from DC circuits is that in the AC case, we have to work with complex impedances (or admittances) for the passive elements and complex peak or effective (rms) values for the voltages and currents.

In this chapter we will demonstrate these methods by two examples.

Let's first demonstrate the use of the node potentials method.

Example 1

Find the amplitude and phase angle of the current i(t) if R = 5 ohm; L = 2 mH; C₁ = 10 mF; C₂ = 20 mF; f = 1 kHz; v_S(t) = 10 cos wt V and i_S(t) = cos wt A

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

Here we have only one independent node, N₁ with an unknown potential: j = v_R = v_L = v_C2 = v_IS . The best method is the node potential method.

The node equation:

Express j_M from the equation:

 

Now we can calculate I_M (the complex amplitude of the current i(t)):

 A

The time function of the current:

i(t) = 0.3038 cos (wt + 86.3^°) A

Using

om:=2000*pi; V:=10; Is:=1; Sys fi (fi-V)*j*om*C1+fi*j*om*C2+fi/j/om/L+fi/R1-Is=0 end; I:=(V-fi)*j*om*C1; abs(I)=[303.7892m] radtodeg(arc(I))=[86.1709]

Now an example of the mesh current method

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

Example 2

Find the current of the voltage generator V = 10 V, f = 1 kHz, R = 4 kohm, R₂ = 2 kohm, C = 250 nF, L = 0.5 H, I = 10 mA, v_S(t) = V cosw t, i_S(t) = I sinw t

Although we could again use the method of node potential with only one unknown, we will demonstrate the solution with the mesh current method.

Let's first calculate the equivalent impedances of R₂,L (Z₁) and R,C (Z₂) to simplify the work: and

We have two independent meshes (loops).The first is: v_S, Z₁ and Z₂ and the second: i_S and Z₂. The direction of the mesh currents are: I₁ clockwise, I₂ counterclockwise.

The two mesh equations are: V_S = J₁*(Z₁ + Z₂) + J₂*Z₂ J₂ = I_s

You must use complex values for all the impedances, voltages and currents.

The two sources are: V_S = 10 V; I_S = -j*0.01 A.

We calculate the voltage in volts and the impedance in kohm so we get the current in mA.

Hence:

j₁(t) = 10.5 cos (w×t -7.1°) mA

Solution

{Solution}  Vs:=10; Is:=-j*0.01; om:=2000*pi; Z1:=R2*j*om*L/(R2+j*om*L); Z2:=R/(1+j*om*R*C); Sys I Vs=I*(Z1+Z2)+Is*Z2 end; I=[10.406m-1.3003m*j] abs(I)=[10.487m] radtodeg(arc(I))=[-7.1224]

Finally, let's check the results using