What is meant by step response of RL parallel circuit?
2 Answers
The step response of an RL parallel circuit refers to how the circuit responds over time when a sudden voltage change, typically a step input, is applied. This analysis helps in understanding the transient behavior of the circuit components: the resistor (R) and the inductor (L) arranged in parallel.
### RL Parallel Circuit Basics
1. **Components**:
- **Resistor (R)**: Opposes the flow of current, causing a voltage drop proportional to the current through it.
- **Inductor (L)**: Stores energy in a magnetic field when current flows through it and opposes changes in current.
2. **Configuration**:
In a parallel RL circuit, both the resistor and the inductor are connected across the same voltage source, leading to two paths for current. When a voltage \( V \) is applied, the circuit behavior depends on both components.
### Step Input
A step input is a sudden change in voltage from 0V to a specific value \( V_0 \) at \( t = 0 \). For example, if a circuit initially at rest (no current flowing) suddenly has 10V applied, the analysis will show how the current and voltages change over time.
### Analyzing the Step Response
1. **Initial Conditions**:
- At \( t = 0 \): The current through the inductor cannot change instantaneously; hence, \( i_L(0) = 0 \).
- The voltage across the inductor \( V_L \) at \( t = 0 \) is equal to the applied voltage \( V_0 \).
2. **Governing Equations**:
- The voltage across the inductor can be described by \( V_L = L \frac{di_L}{dt} \).
- The total current \( I(t) \) flowing from the source is the sum of the currents through the resistor \( i_R \) and the inductor \( i_L \):
\[
I(t) = i_R + i_L
\]
- The current through the resistor is given by Ohm’s law:
\[
i_R = \frac{V_0}{R}
\]
3. **Differential Equation**:
Applying Kirchhoff's voltage law to the loop gives us the following differential equation:
\[
V_0 = i_R R + L \frac{di_L}{dt}
\]
Replacing \( i_R \):
\[
V_0 = \frac{V_0}{R} R + L \frac{di_L}{dt}
\]
Simplifying this leads to:
\[
L \frac{di_L}{dt} + i_L R = V_0
\]
This is a first-order linear ordinary differential equation.
4. **Solution**:
The solution involves finding the homogeneous and particular solutions. The homogeneous part describes the natural response (decay of current through the inductor), while the particular part gives the steady-state behavior.
The characteristic equation is derived from the homogeneous part:
\[
\frac{di_L}{dt} + \frac{R}{L} i_L = 0
\]
The solution to this is:
\[
i_L(t) = I_0 e^{-\frac{R}{L}t}
\]
where \( I_0 \) is the initial current.
5. **Total Current**:
Combining the solutions gives the total current in the circuit:
\[
I(t) = I_{ss} + (I_0 - I_{ss}) e^{-\frac{R}{L}t}
\]
where \( I_{ss} = \frac{V_0}{R} \) is the steady-state current.
### Final Response Characteristics
1. **Transient Response**: Initially, the current ramps up due to the inductor opposing the change in current. This is characterized by the time constant \( \tau = \frac{L}{R} \), which indicates how quickly the system responds to the change.
2. **Steady-State Response**: After a significant amount of time (about 5 time constants), the current stabilizes at \( I_{ss} = \frac{V_0}{R} \), indicating that the circuit has reached a steady state.
### Summary
The step response of an RL parallel circuit provides insights into the time-dependent behavior of the circuit when subjected to sudden voltage changes. By analyzing the transient and steady-state responses, one can understand how energy is stored and dissipated in the circuit over time. This knowledge is essential in applications where timing and response characteristics are critical, such as in filters, oscillators, and various control systems.
### RL Parallel Circuit Basics
1. **Components**:
- **Resistor (R)**: Opposes the flow of current, causing a voltage drop proportional to the current through it.
- **Inductor (L)**: Stores energy in a magnetic field when current flows through it and opposes changes in current.
2. **Configuration**:
In a parallel RL circuit, both the resistor and the inductor are connected across the same voltage source, leading to two paths for current. When a voltage \( V \) is applied, the circuit behavior depends on both components.
### Step Input
A step input is a sudden change in voltage from 0V to a specific value \( V_0 \) at \( t = 0 \). For example, if a circuit initially at rest (no current flowing) suddenly has 10V applied, the analysis will show how the current and voltages change over time.
### Analyzing the Step Response
1. **Initial Conditions**:
- At \( t = 0 \): The current through the inductor cannot change instantaneously; hence, \( i_L(0) = 0 \).
- The voltage across the inductor \( V_L \) at \( t = 0 \) is equal to the applied voltage \( V_0 \).
2. **Governing Equations**:
- The voltage across the inductor can be described by \( V_L = L \frac{di_L}{dt} \).
- The total current \( I(t) \) flowing from the source is the sum of the currents through the resistor \( i_R \) and the inductor \( i_L \):
\[
I(t) = i_R + i_L
\]
- The current through the resistor is given by Ohm’s law:
\[
i_R = \frac{V_0}{R}
\]
3. **Differential Equation**:
Applying Kirchhoff's voltage law to the loop gives us the following differential equation:
\[
V_0 = i_R R + L \frac{di_L}{dt}
\]
Replacing \( i_R \):
\[
V_0 = \frac{V_0}{R} R + L \frac{di_L}{dt}
\]
Simplifying this leads to:
\[
L \frac{di_L}{dt} + i_L R = V_0
\]
This is a first-order linear ordinary differential equation.
4. **Solution**:
The solution involves finding the homogeneous and particular solutions. The homogeneous part describes the natural response (decay of current through the inductor), while the particular part gives the steady-state behavior.
The characteristic equation is derived from the homogeneous part:
\[
\frac{di_L}{dt} + \frac{R}{L} i_L = 0
\]
The solution to this is:
\[
i_L(t) = I_0 e^{-\frac{R}{L}t}
\]
where \( I_0 \) is the initial current.
5. **Total Current**:
Combining the solutions gives the total current in the circuit:
\[
I(t) = I_{ss} + (I_0 - I_{ss}) e^{-\frac{R}{L}t}
\]
where \( I_{ss} = \frac{V_0}{R} \) is the steady-state current.
### Final Response Characteristics
1. **Transient Response**: Initially, the current ramps up due to the inductor opposing the change in current. This is characterized by the time constant \( \tau = \frac{L}{R} \), which indicates how quickly the system responds to the change.
2. **Steady-State Response**: After a significant amount of time (about 5 time constants), the current stabilizes at \( I_{ss} = \frac{V_0}{R} \), indicating that the circuit has reached a steady state.
### Summary
The step response of an RL parallel circuit provides insights into the time-dependent behavior of the circuit when subjected to sudden voltage changes. By analyzing the transient and steady-state responses, one can understand how energy is stored and dissipated in the circuit over time. This knowledge is essential in applications where timing and response characteristics are critical, such as in filters, oscillators, and various control systems.
The step response of an RL parallel circuit describes how the current or voltage in the circuit changes over time when a step input is applied. Let’s break this down:
### RL Parallel Circuit Overview
An RL parallel circuit consists of:
- **R**: A resistor
- **L**: An inductor
- **V_in**: The input voltage source (which applies a step input)
In a parallel configuration, the resistor and inductor are connected side-by-side across the same two points.
### Step Input
A step input is a sudden change in voltage from 0V to a constant voltage \( V_{in} \) at \( t = 0 \). This type of input is used to analyze how the circuit responds to abrupt changes.
### Step Response of RL Parallel Circuit
The step response involves observing how the circuit behaves after the step input is applied. Here’s how the response can be analyzed:
1. **Initial Conditions**:
- At \( t < 0 \), the circuit is in steady state with no current flowing through the inductor (since an inductor behaves like an open circuit for DC steady states). The voltage across the resistor and inductor is zero.
2. **At \( t = 0 \)**:
- The input voltage suddenly jumps to \( V_{in} \).
3. **For \( t > 0 \)**:
- **Current through the Resistor (I_R)**: The current through the resistor \( R \) can be calculated using Ohm’s law, \( I_R = \frac{V_{in}}{R} \).
- **Current through the Inductor (I_L)**: The current through the inductor doesn’t change instantaneously due to its inherent property of resisting changes in current. The voltage across the inductor is \( V_L = L \frac{dI_L}{dt} \).
The differential equation governing the current through the inductor is:
\[
V_{in} = I_R \cdot R + L \frac{dI_L}{dt}
\]
where \( I_R = \frac{V_{in}}{R} \).
Rearranging gives:
\[
L \frac{dI_L}{dt} = V_{in} - I_R \cdot R
\]
\[
L \frac{dI_L}{dt} = V_{in} - V_{in}
\]
\[
\frac{dI_L}{dt} = \frac{V_{in}}{L} - \frac{V_{in}}{L}
\]
\[
\frac{dI_L}{dt} = 0
\]
The current through the inductor approaches a steady value as time progresses. Eventually, the inductor behaves like a short circuit, and the current through it becomes constant.
### Summary of the Step Response
- **Initial Response (At \( t = 0 \))**: The current through the inductor starts from 0 and increases as the inductor opposes the sudden change.
- **Steady-State Response (Long after \( t = 0 \))**: The circuit reaches a steady state where the current through the resistor is \( \frac{V_{in}}{R} \), and the inductor’s voltage drops to zero, making it act like a short circuit.
The exact mathematical response of the inductor current over time involves solving the differential equations derived from the circuit's governing equations. This typically results in an exponential rise or decay based on the RL time constant \( \tau = \frac{L}{R} \).
### RL Parallel Circuit Overview
An RL parallel circuit consists of:
- **R**: A resistor
- **L**: An inductor
- **V_in**: The input voltage source (which applies a step input)
In a parallel configuration, the resistor and inductor are connected side-by-side across the same two points.
### Step Input
A step input is a sudden change in voltage from 0V to a constant voltage \( V_{in} \) at \( t = 0 \). This type of input is used to analyze how the circuit responds to abrupt changes.
### Step Response of RL Parallel Circuit
The step response involves observing how the circuit behaves after the step input is applied. Here’s how the response can be analyzed:
1. **Initial Conditions**:
- At \( t < 0 \), the circuit is in steady state with no current flowing through the inductor (since an inductor behaves like an open circuit for DC steady states). The voltage across the resistor and inductor is zero.
2. **At \( t = 0 \)**:
- The input voltage suddenly jumps to \( V_{in} \).
3. **For \( t > 0 \)**:
- **Current through the Resistor (I_R)**: The current through the resistor \( R \) can be calculated using Ohm’s law, \( I_R = \frac{V_{in}}{R} \).
- **Current through the Inductor (I_L)**: The current through the inductor doesn’t change instantaneously due to its inherent property of resisting changes in current. The voltage across the inductor is \( V_L = L \frac{dI_L}{dt} \).
The differential equation governing the current through the inductor is:
\[
V_{in} = I_R \cdot R + L \frac{dI_L}{dt}
\]
where \( I_R = \frac{V_{in}}{R} \).
Rearranging gives:
\[
L \frac{dI_L}{dt} = V_{in} - I_R \cdot R
\]
\[
L \frac{dI_L}{dt} = V_{in} - V_{in}
\]
\[
\frac{dI_L}{dt} = \frac{V_{in}}{L} - \frac{V_{in}}{L}
\]
\[
\frac{dI_L}{dt} = 0
\]
The current through the inductor approaches a steady value as time progresses. Eventually, the inductor behaves like a short circuit, and the current through it becomes constant.
### Summary of the Step Response
- **Initial Response (At \( t = 0 \))**: The current through the inductor starts from 0 and increases as the inductor opposes the sudden change.
- **Steady-State Response (Long after \( t = 0 \))**: The circuit reaches a steady state where the current through the resistor is \( \frac{V_{in}}{R} \), and the inductor’s voltage drops to zero, making it act like a short circuit.
The exact mathematical response of the inductor current over time involves solving the differential equations derived from the circuit's governing equations. This typically results in an exponential rise or decay based on the RL time constant \( \tau = \frac{L}{R} \).