Deriving v(t) from i(t) in a Series Circuit: Finding V Across Z | Homework Help

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In summary, the problem is to find the voltage (v(t)) across Z in a series circuit with a .5H inductor, .01F capacitor, .5H inductor, and 10Ω resistor. The circuit is connected to a source i(t) = 10cos(10t). Using Ohm's law (V=IR), the impedance in the s-domain is found to be s+100/s+10 and in the frequency domain (Z(jω)) to be j((ω^2-100)/ω)+10. To find v(t), the current output (I(s)) must be obtained through laplace transform, then multiplied with the impedance, and finally finding the inverse laplace of the
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Homework Statement



I've been given a series circuit with a .5H inductor .01F cap, .5H inductor and 10Ω resistor in series.

This circuit is connected to a source i(t) = 10cos(10t) and I need to find v(t) across Z

Homework Equations


From the givens I've found the Z(s) to be s+100/s+10 and Z(jω) to be j((ω^2-100)/ω)+10

The Attempt at a Solution



I believe I need to use Ohm's law V=IR to find V. So if I multiply j((ω^2-100)/ω)+10 by 10cos(10t) I should get a v(t) but I think I'm doing something wrong with this.
 
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what you need to do is make sure you are in the s-domain, so V(s) = I(s)Z(s), you need to perform laplace transform to the current output to obtain I(s), then you can multiply that with the frequency dependent impedance, then finally get your answer from the inverse laplace of the product
 

1. How do you derive v(t) from i(t) in a series circuit?

To derive v(t) from i(t) in a series circuit, you can use Ohm's Law (V=IR) and Kirchhoff's Voltage Law (KVL). First, use Ohm's Law to solve for the total resistance (R) of the circuit by multiplying the current (i) by the total circuit impedance (Z). Then, use KVL to set up an equation that relates the voltage (V) across the total impedance (Z) to the voltage across each individual component in the circuit. Finally, solve for v(t) by plugging in the values for R and i from Ohm's Law and solving the equation.

2. What is the difference between v(t) and i(t) in a series circuit?

V(t) is the voltage across a specific component in the circuit, whereas i(t) is the current flowing through that component. In a series circuit, the current is the same throughout the entire circuit, but the voltage may vary across different components depending on their individual resistances.

3. Can you use this method to find the voltage across any component in a series circuit?

Yes, this method can be used to find the voltage across any component in a series circuit. As long as you know the total impedance of the circuit and the current flowing through it, you can use Ohm's Law and KVL to solve for the voltage across any individual component.

4. Why is it important to find the voltage across a component in a series circuit?

Finding the voltage across a component in a series circuit is important because it allows you to understand how much energy is being used or dissipated by that component. This can be helpful in troubleshooting circuit issues or designing more efficient circuits.

5. Can you use this method to find the current flowing through a component in a series circuit?

No, this method is specifically for finding the voltage across a component in a series circuit. To find the current flowing through a component, you can use Ohm's Law and Kirchhoff's Current Law (KCL) to set up an equation and solve for i(t).

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