dv dt + 5v = 10B(t), v(0) : = 0 em we are just going to look at the equation for the battery. d battery when connected to the circuit at time zero have a voltage source equation B(t) = 10e-0.5t attery is off until one second (t = 1) then connects for one second (until t =2), then for one second and recharges fully and then is turned on for one second, then it 1 off for good. %3! the function for the battery voltage in the circuit using the Heaviside function and =ponential function.
dv dt + 5v = 10B(t), v(0) : = 0 em we are just going to look at the equation for the battery. d battery when connected to the circuit at time zero have a voltage source equation B(t) = 10e-0.5t attery is off until one second (t = 1) then connects for one second (until t =2), then for one second and recharges fully and then is turned on for one second, then it 1 off for good. %3! the function for the battery voltage in the circuit using the Heaviside function and =ponential function.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Battery Voltage Source Equation and Heaviside Function Application
In this educational module, we will examine the equation governing the behavior of a battery in an electric circuit.
#### Differential Equation for the Battery Voltage
The differential equation given for the voltage \( v(t) \) across the battery in the circuit is:
\[ \frac{dv}{dt} + 5v = 10B(t), \quad v(0) = 0 \]
where:
- \( \frac{dv}{dt} \) is the derivative of the voltage with respect to time.
- \( v \) represents the voltage.
- \( B(t) \) is the voltage source function.
#### Voltage Source Equation
A fully charged battery connected to the circuit at time zero has a voltage source equation described as:
\[ B(t) = 10e^{-0.5t} \]
#### Problem Scenario
**Scenario (a)**:
- The battery remains off until one second ( \( t = 1 \) ).
- At \( t = 1 \) second, it connects for one second (until \( t = 2 \) ).
- It is turned off again for one second.
- Then, it recharges fully and turns on for one more second.
- Finally, it is turned off for good.
#### Task
Develop the function \( B_1(t) \) for the battery voltage in the circuit using the Heaviside function and the exponential function.
**Note**: The Heaviside function, \( u(t) \), is defined as:
\[ u(t) =
\begin{cases}
0, & t < 0 \\
1, & t \geq 0
\end{cases}
\]
Using the Heaviside function, you can model the various on-off states of the battery.
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Transcribed Image Text:### Battery Voltage Source Equation and Heaviside Function Application
In this educational module, we will examine the equation governing the behavior of a battery in an electric circuit.
#### Differential Equation for the Battery Voltage
The differential equation given for the voltage \( v(t) \) across the battery in the circuit is:
\[ \frac{dv}{dt} + 5v = 10B(t), \quad v(0) = 0 \]
where:
- \( \frac{dv}{dt} \) is the derivative of the voltage with respect to time.
- \( v \) represents the voltage.
- \( B(t) \) is the voltage source function.
#### Voltage Source Equation
A fully charged battery connected to the circuit at time zero has a voltage source equation described as:
\[ B(t) = 10e^{-0.5t} \]
#### Problem Scenario
**Scenario (a)**:
- The battery remains off until one second ( \( t = 1 \) ).
- At \( t = 1 \) second, it connects for one second (until \( t = 2 \) ).
- It is turned off again for one second.
- Then, it recharges fully and turns on for one more second.
- Finally, it is turned off for good.
#### Task
Develop the function \( B_1(t) \) for the battery voltage in the circuit using the Heaviside function and the exponential function.
**Note**: The Heaviside function, \( u(t) \), is defined as:
\[ u(t) =
\begin{cases}
0, & t < 0 \\
1, & t \geq 0
\end{cases}
\]
Using the Heaviside function, you can model the various on-off states of the battery.
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