Network Theorems

Step 1: Write the given current function.

Step 2: Differentiate with respect to time.

Step 3: Set the derivative equal to zero for maximum value.

Since ,

Step 4: Solve for .

Step 5: Conclusion.
The current attains its maximum value at

Step 1: Identify the resistive network.
The circuit consists of two resistors of value connected in parallel.
The equivalent resistance of two equal resistors in parallel is

Step 2: Identify the capacitive network.
The circuit also consists of two capacitors of value connected in parallel.
The equivalent capacitance of two capacitors in parallel is

Step 3: Write the expression for time constant.
The time constant of an RC network is given by

Step 4: Substitute the equivalent values.


Step 5: Conclusion.
Hence, the time constant of the given network is

Step 1: Identify the load terminals A–B.
The load is the capacitive reactance connected between terminals A and B. To apply Norton’s theorem, this load is first removed.
Step 2: Calculate Norton current .
Norton current is the short-circuit current between terminals A and B. Hence, terminals A and B are shorted.
With A–B shorted, the series resistor is directly connected to ground, making the circuit equivalent to a source feeding a parallel combination.
Impedances seen from the source are:
Inductive reactance:
Parallel branch:
Total impedance:

Source current is:



Since the current divides equally between the two parallel branches,

Step 3: Calculate Norton impedance .
Deactivate the source by replacing the voltage source with a short circuit.
Seen from terminals A–B:



Step 4: Reconnect the load and find .
Load impedance:

Using current division:


Magnitude and angle:

Hence,


Step 5: Conclusion.
The load current is

Step 1: Remove the load resistance to find Thevenin voltage.
To determine the Thevenin equivalent as seen from terminals A–B, the load resistance of is removed. The circuit is now open at terminals A–B.
Step 2: Calculate the Thevenin voltage .}
With the load open, no current flows through the right-side resistor and the source.
The left part of the circuit consists of a source feeding two resistors, one in series and one to ground.
Using voltage division, the voltage at the junction node is

The source raises the voltage further at terminal A, hence

Step 3: Find the Thevenin resistance .}
Deactivate all independent sources:
- Replace the source by a short circuit
- Replace the source by a short circuit
Now, from terminal A, the resistor is in series with two resistors connected in parallel to ground.

Therefore,

Step 4: Reconnect the load resistance and calculate .
The total resistance seen by the Thevenin voltage source is

Hence, the load current is

Step 5: Conclusion.
The current flowing through the load resistance is