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Laplace Circuit Analysis

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Please give a full and simplified answer to all parts of question 6 to aid me with revision.

When a force F is applied across the face of the crystal, the current is generated. The interconnecting cable can be represented by a lumped capacitance CC. The input resistance of the amplifier, RL, acts as a load to the
crystal. The output of the amplifier drives a display (a moving coil voltmeter that is calibrated in units of acceleration.

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This solution provides calculations regarding Laplace circuit analysis.

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Circuit analysis branch

1. A circuit has a branch with three resistors in series of 2, 3, and 8 ohms respectively, connected in series. What is the equivalent resistance of the branch? 13 ohms
2. A circuit has two resistors in parallel, 4 ohms each. Can they be replaced by a single resistor? If so, what would be the equivalent value? 2 ohms
3. What is the energy stored in a 10 F capacitor at t = 4 seconds if v(t) = 2 + 3t ?
4. Using the definition of the Laplace transform, find the Laplace transform of f(at).
5. What is the inverse Laplace transform of 1/s^3?
6. Find the Laplace transform of e^(-3t)cos4t.
7. Determine the inverse Laplace Transform of H(s) = s/(s^2 - 1).
8. The transfer function for a circuit is H(s) = 1/(s+4). What is the natural response of the system?
9. The transfer function for a circuit is H(s) = (s+2)/((s+2)^2 + 4)). What is the natural response of the system?
10. A 21 V source is in series with a 3 ohm and a 4 ohm resistor. What is the total power absorbed by the 4 ohm resistor?

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