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Advanced Circuit Analysis: Exploring Theorems and Methods in DC and AC Circuits

a. Applying KVL to each loop in the circuit in figure one below we obtained the expression for voltage as:

Figure 1: Question 1a

Loop with I1:

Loop with I2:

Loop with I3:

Therefore the matrix formed using equations i, ii and iii is given by:

...............i

QUESTION 2

The generalized electrical equations of the DC circuit as shown in figures Q2 using:

a. Maxwell’s current theorem by following the direction of currents flowing in the loops as shown in figure Q2a.

Figure Q2a

Applying KVL to each loop the expression for electrical circuit can be obtained as:

Loop with I1:

Loop with I2:

Loop with I3:

Therefore the matrix formed using equations i, ii and iii is given by:

b. Millman’s theorem to find equivalent voltage and equivalent resistance across electrical point AB as shown in figure Q2b.

Figure 3: Q2b

Solution:

VOLTAGE:

RESISTANCE:

c. Obtain Thevenin’s voltage and Thevenin’s resistance across AB in figure Q2c and prove that they are same as the equivalent voltage and equivalent resistance obtained from Millamn’s Theorem in 2(b).

Figure 4: Q2c

Solution:

RESISTANCE:

THEVENIN VOLTAGE

We assume the voltage at node A as V and apply KCL to this node as follows:

It can be observed that equations 2II and 2cii are equal and therefore the same.

d. Use above and apply Thevenin’s theorem to solve for current flowing in R4.

Solution

The circuit in figure Q2C can be redrawn as:

Therefore current I can be calculated as:

e. Show that current I flowing in figure Q2 in 2(a) is same as in 2(d) for a given values of R1=10; R2=5; R3=15; R4=20; E1=50; E2=10; E3=5 and E4=10. All units are in SI. Use elimination method of solving algebraic equations.

Solution:

Substituting the values of resistances and voltage in matrix ii current I will be equal to current I3.

I1= A   , I2= ¼ A , I3=¼ A

Therefore I=¼ A

Substituting for values of resistance and voltages in equation di I can be obtained as:

QUESTION 3

For the AC network shown below in figures Q3:

a.  Determine the equivalent impedance and equivalent voltage across A-B both in polar and rectangular co-ordinates by using Millman’s theorem for the circuit shown in figure Q3a. Given Z1=2+j5; Z2=3-j4; Z3=4+j5; Z4=3-j5; E1=10+j0 and E2=20-j5.

Figure 5: Q3a

Solution:

VOLTAGE:

Susbstituting for values of voltage and resistance we obtain the voltage EAB as:

RESISTANCE:

Substituting for values of z in equation 3ii above we obtain Z as:

b. Use this equivalent circuit to find out the current flowing in branch A-B in figure Q3b both in polar and rectangular co-ordinates by applying Thevenin’s theorem. Draw an equivalent Thevenin’s circuit.

Figure 6: Q3b

The equivalent circuit:

Solution:

c. Use Maxwell’s current law to find loop currents in figure Q3c and to verify that the current flowing in branch AB is same as in 3(b). Use elimination method of solving algebraic equations.

Figure 7: Q3c

Applying KVL to each loop we have the following equations:

Loop with I1:

Loop with I2:

Loop with I3:

Therefore the matrix formed using equations i, ii and iii is given by:

...............i

Substituting for values of Z and E and solving the equation we have the currents as:

The two currents are equal as obtained from the previous section.