9. Kirchhoff's voltage law states that the sum of the voltage drops across a resistor, R, an inductor, L, and a capacitor, C, in an electrical circuit must be the same as the voltage source, E (t), applied to that RLC circuit. Applying the additional fact that the current, I, is related to the charge, q, by the relationship I = a, the resulting ODE model for the charge dt' in a circuit is: d²q dq 1 +R dt² dt +=q= E (t).
9. Kirchhoff's voltage law states that the sum of the voltage drops across a resistor, R, an inductor, L, and a capacitor, C, in an electrical circuit must be the same as the voltage source, E (t), applied to that RLC circuit. Applying the additional fact that the current, I, is related to the charge, q, by the relationship I = a, the resulting ODE model for the charge dt' in a circuit is: d²q dq 1 +R dt² dt +=q= E (t).
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![### Kirchhoff's Voltage Law in RLC Circuits
Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor (R), an inductor (L), and a capacitor (C) in an electrical circuit must be equal to the voltage source \( E(t) \) applied to that RLC circuit. By incorporating that the current \( I \) is related to the charge \( q \) by the relationship \( I = \frac{dq}{dt} \), the resulting ordinary differential equation (ODE) model for the charge in a circuit is given by:
\[ L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t). \]
#### Example Problem
If a \( 100 \sin(60t) \) source is connected to an RLC circuit with:
- An inductor \( L \) of \( \frac{1}{20} \) henry,
- A resistor \( R \) of 1 ohm, and
- A capacitor \( C \) of \( \frac{1}{130} \) farad,
Find the charge \( q(t) \), given that \( q(0) = 0 \) and \( I(0) = q'(0) = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b491fec-b8bd-45cc-a06e-6176bd563445%2F92ebc110-9391-40c6-8d4c-d60354bb104f%2Focq9mm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Kirchhoff's Voltage Law in RLC Circuits
Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor (R), an inductor (L), and a capacitor (C) in an electrical circuit must be equal to the voltage source \( E(t) \) applied to that RLC circuit. By incorporating that the current \( I \) is related to the charge \( q \) by the relationship \( I = \frac{dq}{dt} \), the resulting ordinary differential equation (ODE) model for the charge in a circuit is given by:
\[ L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t). \]
#### Example Problem
If a \( 100 \sin(60t) \) source is connected to an RLC circuit with:
- An inductor \( L \) of \( \frac{1}{20} \) henry,
- A resistor \( R \) of 1 ohm, and
- A capacitor \( C \) of \( \frac{1}{130} \) farad,
Find the charge \( q(t) \), given that \( q(0) = 0 \) and \( I(0) = q'(0) = 0 \).
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