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How do you analyze circuits with dependent and independent sources using network theorems?

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Analyzing circuits with dependent and independent sources using network theorems follows similar principles to analyzing circuits with only independent sources. The main network theorems used for circuit analysis include the Superposition theorem, Thevenin's theorem, Norton's theorem, and the Maximum Power Transfer theorem. Here's how you can apply these theorems when dealing with circuits that contain both dependent and independent sources:

Superposition Theorem:
The Superposition theorem is applicable when you have a circuit with multiple independent sources. To analyze the circuit using superposition, follow these steps:

a. For each independent source, set all other independent sources to zero (replace voltage sources with short circuits and current sources with open circuits).
b. Calculate the response of the circuit to each individual source. For dependent sources, treat them as usual; their values depend on other circuit variables, not the source you are analyzing.
c. The total response of the circuit is the sum of the individual responses obtained in step b.

Thevenin's Theorem:
Thevenin's theorem is used to simplify complex circuits into an equivalent circuit with a single voltage source and a series resistance. To apply Thevenin's theorem:

a. Find the Thevenin voltage (Vth): Disconnect the load (if any) from the circuit and determine the voltage across the load terminals (AB). This is the Thevenin voltage.
b. Find the Thevenin resistance (Rth): Again, disconnect all the sources from the circuit and find the equivalent resistance looking into the load terminals (AB) with the sources turned off (set to their internal resistances if any).
c. Replace the original circuit with the Thevenin equivalent circuit consisting of a single voltage source (Vth) in series with a resistor (Rth).

Norton's Theorem:
Norton's theorem is similar to Thevenin's theorem but deals with current sources instead of voltage sources. To apply Norton's theorem:

a. Find the Norton current (In): Disconnect the load from the circuit and determine the current flowing through the load terminals (AB). This is the Norton current.
b. Find the Norton resistance (Rn): Disconnect all sources from the circuit and calculate the equivalent resistance looking into the load terminals (AB) with the sources turned off.
c. Replace the original circuit with the Norton equivalent circuit consisting of a current source (In) in parallel with a resistor (Rn).

Maximum Power Transfer Theorem:
The Maximum Power Transfer theorem is used to find the load resistance that maximizes power transfer from a network to the load. It's applicable for both dependent and independent sources. To apply this theorem:

a. Disconnect the load resistance and calculate the Thevenin or Norton equivalent circuit at the load terminals (AB).
b. Find the load resistance that matches the Thevenin or Norton resistance of the equivalent circuit found in step a.

Keep in mind that dependent sources introduce additional complexity as their values depend on other variables in the circuit. Therefore, you may need to solve simultaneous equations or use other techniques to express the dependent source values before applying network theorems.
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