Stability using Routh Stability Criterion

Determine the stability of the system whose characteristics equation is:

a(s) = 2s^5 + 3s^4 + 2s^3 + s^2 + 2s + 2 .

Solution

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

\begin{array}{l | c c c} s^5 & 2 & 2 & 2 \\ s^4 &  3 & 1 & 2 \\ s^3 & & &  \\  s^2 & & & \\ s^1 & & & \\ s^0 & & &\end{array}
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Winner of Electrical Circuits Contest #1

Problem

Find I_x and I_y :
Electrical Circuit Contest #1

Solution

Three resistors are in series and their equivalent, 6\Omega , is parallel with the voltage source. So, according to the Ohm's law: I_y=-\frac{6V}{6 \Omega}=-1 A . The negative sign comes from the direction I_y .
Applying KCL at the bottom node:
-(-2A)+I_x+I_y=0 \rightarrow I_x=-1 A .
The lucky winner of the Electrical Circuits Contest #1 is Kunal Marwaha from UC Berkeley. I would like to say thank you to all participants and I am thinking of holding contest #2 soon. Kunal, congratulations and soon you will receive the prize by Paypal.

Find Equivalent Impedance - AC Steady State Analysis


Determine the driving-point impedance of the network at a frequency of 2 kHz:

Determine Impedance

Solution

Lets first find impedance of elements one by one:

Resistor R

The resistor impedance is purely real and independent of frequency.

Z_R=R=20 \Omega

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Electrical Circuit Contest - Win $10!

What's up? We are going to have fun!
This is our first contest for electrical circuits. Solve the problem, submit your answers and cross your fingers to be the lucky winner!

Problem

Find I_x and I_y :
Electrical Circuit Contest #1

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Superposition method - Circuit with two sources

Find I_x using superposition rule:
Main cuircuit to be analyzed using superposition method

Solution

Superposition

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

Find Thevenin's and Norton's Equivalent Circuits:
1254-1
Suppose that R_1=5\Omega , R_2=3\Omega and I_S=2 A .

Solution

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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Solve Using Current Division Rule

Find current of resistors, use the current division rule.
Problem 1246 (1)
Suppose that R_1=2 \Omega , R_2=4 \Omega , R_3=1 \Omega , I_S=5 A and V_S=4 V

Solution:
R_2 and R_3 are parallel. The current of I_S 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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Mesh Analysis - Supermesh

Solve the circuit and find the power of sources:
Problem 1226 - 1
V_S=10V , I_S=4 A , R_1=2 \Omega , R_2=6 \Omega , R_3=1 \Omega , R_4=2 \Omega .

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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