Determine the stability of the system whose characteristics equation is:

.

# Solution

All coefficients are positive and non-zero; therefore, the necessary condition for stability is satisfied. Let's write the Routh array:

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Determine the stability of the system whose characteristics equation is:

.

All coefficients are positive and non-zero; therefore, the necessary condition for stability is satisfied. Let's write the Routh array:

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Three resistors are in series and their equivalent, , is parallel with the voltage source. So, according to the Ohm's law: . The negative sign comes from the direction .

Applying KCL at the bottom node:

.

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Determine the driving-point impedance of the network at a frequency of kHz:

Lets first find impedance of elements one by one:

The resistor impedance is purely real and independent of frequency.

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Find and :

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Find using superposition rule:

The superposition theorem states that the response (voltage or current) in any branch of a linear circuit which has more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are turned off (made zero).

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Find Thevenin's and Norton's Equivalent Circuits:

Suppose that , and .

The circuit has both independent and dependent sources. In these cases, we need to find open circuit voltage and short circuit current to determine Norton's (and also Thevenin's) equivalent circuits.

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Find current of resistors, use the current division rule.

Suppose that , , , and

Solution:

and are parallel. The current of is passing through them and it is actually divided between them. The branch with lower resistance has higher current because electrons can pass through that easier than the other branch. Using the current division rule, we get

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Solve the circuit and find the power of sources:

, , , , , .

Solution:

There are three meshes in the circuit. So, we need to assign three mesh currents. It is better to have all the mesh currents loop in the same direction (usually clockwise) to prevent errors when writing out the equations.

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Find the voltage across the current source and the current passing through the voltage source.

Assume that , , , , , ,

Solution

is in series with the current source; therefore, the same current passing through it as the current source:

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Determine voltage across and using voltage division rule.

Assume that

, , , and

Solution:

Please note that the voltage division rule cannot be directly applied. This is to say that:

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