What is the relation between applied voltage and emf induced with respect to flux wave?
2 Answers
The relationship between applied voltage, induced electromotive force (EMF), and the magnetic flux wave is a core concept in electromagnetism and electrical engineering, particularly in the operation of electric machines like transformers, motors, and generators. Let's break down this relationship in detail.
### 1. **Magnetic Flux (Φ) and Flux Wave**
Magnetic flux (denoted as Φ) represents the total magnetic field passing through a given area. It is a measure of the strength of the magnetic field over a specific region, like a coil or a conductor. Magnetic flux is given by the formula:
\[
\Phi = B \times A \times \cos(\theta)
\]
where:
- \( \Phi \) is the magnetic flux,
- \( B \) is the magnetic field (magnetic flux density),
- \( A \) is the area through which the magnetic field lines pass,
- \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
In the case of alternating current (AC) systems, the magnetic flux varies over time. This varying flux creates a "flux wave," which is the time-varying nature of magnetic flux as it oscillates, typically in a sinusoidal manner.
### 2. **Electromotive Force (EMF) Induced (Faraday's Law of Induction)**
The induced electromotive force (EMF) is directly related to the rate of change of magnetic flux according to **Faraday’s Law of Electromagnetic Induction**. It states that:
\[
\text{EMF} = - N \frac{d\Phi}{dt}
\]
where:
- \( N \) is the number of turns in the coil,
- \( \frac{d\Phi}{dt} \) is the rate of change of the magnetic flux through the coil,
- The negative sign indicates Lenz's Law, which states that the induced EMF opposes the change in flux that caused it.
This means that whenever the magnetic flux through a coil changes, an EMF is induced. The faster the flux changes, the larger the induced EMF.
### 3. **Applied Voltage and EMF**
In electrical machines like transformers and motors, the applied voltage (the input electrical energy) is related to the induced EMF. Here's how:
- **Transformers**: In a transformer, the applied voltage on the primary coil creates a magnetic field, which in turn creates a time-varying magnetic flux in the core. This varying flux induces an EMF in the secondary coil according to Faraday’s Law. The relationship between the applied voltage \( V_p \) on the primary coil and the induced EMF \( E_s \) on the secondary coil depends on the turns ratio and the rate of change of flux:
\[
V_p = N_p \frac{d\Phi}{dt}, \quad E_s = N_s \frac{d\Phi}{dt}
\]
The ratio of the applied voltage to the induced EMF in the secondary is proportional to the turns ratio \( \frac{N_p}{N_s} \).
- **AC Motors and Generators**: In motors, the applied voltage creates a rotating magnetic field (or flux wave). This rotating flux wave interacts with the rotor to induce an EMF in the rotor windings. In generators, the mechanical motion (e.g., turning of a rotor in a magnetic field) causes a change in the magnetic flux, which induces an EMF in the windings.
### 4. **Phase Relationship Between Applied Voltage and Induced EMF**
In AC systems, especially in synchronous machines and transformers, the applied voltage and the induced EMF are related in a specific phase relationship:
- The induced EMF is **90 degrees out of phase** with the flux wave in sinusoidal systems. This means that when the magnetic flux is at its maximum (where the rate of change is zero), the induced EMF is zero.
- The applied voltage, which drives the system, is typically in phase with the current but may have a different phase relationship with the induced EMF depending on the type of machine and its operation.
### 5. **Back EMF in Motors**
In motors, particularly DC motors, an important concept is **back EMF**. When a motor runs, the rotation of the rotor in the magnetic field induces an EMF in the windings, which opposes the applied voltage. This back EMF is a direct result of the rotating flux wave and is proportional to the speed of the rotor:
\[
E_{\text{back}} = k \cdot \Phi \cdot \omega
\]
where:
- \( k \) is a constant,
- \( \Phi \) is the magnetic flux,
- \( \omega \) is the angular velocity of the rotor.
The back EMF opposes the applied voltage, and the motor reaches an equilibrium where the applied voltage equals the sum of the back EMF and the voltage drop across the motor’s internal resistance.
### 6. **Practical Implications of the Relationship**
- In **transformers**, the applied voltage on the primary side creates a varying flux, which induces an EMF on the secondary side proportional to the turns ratio and the flux wave.
- In **generators**, mechanical movement creates the changing flux that induces an EMF, which can be used to drive a load.
- In **motors**, the applied voltage creates a flux wave, which induces a back EMF opposing the applied voltage, controlling the motor's speed.
### Summary
The relationship between applied voltage, induced EMF, and the magnetic flux wave is governed by the principle of electromagnetic induction. The applied voltage generates a time-varying magnetic flux, which, in turn, induces an EMF in a coil or conductor. The key factors influencing this relationship are the rate of change of magnetic flux, the number of turns in the coil, and the phase relationship between the applied voltage, the flux wave, and the induced EMF.
### 1. **Magnetic Flux (Φ) and Flux Wave**
Magnetic flux (denoted as Φ) represents the total magnetic field passing through a given area. It is a measure of the strength of the magnetic field over a specific region, like a coil or a conductor. Magnetic flux is given by the formula:
\[
\Phi = B \times A \times \cos(\theta)
\]
where:
- \( \Phi \) is the magnetic flux,
- \( B \) is the magnetic field (magnetic flux density),
- \( A \) is the area through which the magnetic field lines pass,
- \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
In the case of alternating current (AC) systems, the magnetic flux varies over time. This varying flux creates a "flux wave," which is the time-varying nature of magnetic flux as it oscillates, typically in a sinusoidal manner.
### 2. **Electromotive Force (EMF) Induced (Faraday's Law of Induction)**
The induced electromotive force (EMF) is directly related to the rate of change of magnetic flux according to **Faraday’s Law of Electromagnetic Induction**. It states that:
\[
\text{EMF} = - N \frac{d\Phi}{dt}
\]
where:
- \( N \) is the number of turns in the coil,
- \( \frac{d\Phi}{dt} \) is the rate of change of the magnetic flux through the coil,
- The negative sign indicates Lenz's Law, which states that the induced EMF opposes the change in flux that caused it.
This means that whenever the magnetic flux through a coil changes, an EMF is induced. The faster the flux changes, the larger the induced EMF.
### 3. **Applied Voltage and EMF**
In electrical machines like transformers and motors, the applied voltage (the input electrical energy) is related to the induced EMF. Here's how:
- **Transformers**: In a transformer, the applied voltage on the primary coil creates a magnetic field, which in turn creates a time-varying magnetic flux in the core. This varying flux induces an EMF in the secondary coil according to Faraday’s Law. The relationship between the applied voltage \( V_p \) on the primary coil and the induced EMF \( E_s \) on the secondary coil depends on the turns ratio and the rate of change of flux:
\[
V_p = N_p \frac{d\Phi}{dt}, \quad E_s = N_s \frac{d\Phi}{dt}
\]
The ratio of the applied voltage to the induced EMF in the secondary is proportional to the turns ratio \( \frac{N_p}{N_s} \).
- **AC Motors and Generators**: In motors, the applied voltage creates a rotating magnetic field (or flux wave). This rotating flux wave interacts with the rotor to induce an EMF in the rotor windings. In generators, the mechanical motion (e.g., turning of a rotor in a magnetic field) causes a change in the magnetic flux, which induces an EMF in the windings.
### 4. **Phase Relationship Between Applied Voltage and Induced EMF**
In AC systems, especially in synchronous machines and transformers, the applied voltage and the induced EMF are related in a specific phase relationship:
- The induced EMF is **90 degrees out of phase** with the flux wave in sinusoidal systems. This means that when the magnetic flux is at its maximum (where the rate of change is zero), the induced EMF is zero.
- The applied voltage, which drives the system, is typically in phase with the current but may have a different phase relationship with the induced EMF depending on the type of machine and its operation.
### 5. **Back EMF in Motors**
In motors, particularly DC motors, an important concept is **back EMF**. When a motor runs, the rotation of the rotor in the magnetic field induces an EMF in the windings, which opposes the applied voltage. This back EMF is a direct result of the rotating flux wave and is proportional to the speed of the rotor:
\[
E_{\text{back}} = k \cdot \Phi \cdot \omega
\]
where:
- \( k \) is a constant,
- \( \Phi \) is the magnetic flux,
- \( \omega \) is the angular velocity of the rotor.
The back EMF opposes the applied voltage, and the motor reaches an equilibrium where the applied voltage equals the sum of the back EMF and the voltage drop across the motor’s internal resistance.
### 6. **Practical Implications of the Relationship**
- In **transformers**, the applied voltage on the primary side creates a varying flux, which induces an EMF on the secondary side proportional to the turns ratio and the flux wave.
- In **generators**, mechanical movement creates the changing flux that induces an EMF, which can be used to drive a load.
- In **motors**, the applied voltage creates a flux wave, which induces a back EMF opposing the applied voltage, controlling the motor's speed.
### Summary
The relationship between applied voltage, induced EMF, and the magnetic flux wave is governed by the principle of electromagnetic induction. The applied voltage generates a time-varying magnetic flux, which, in turn, induces an EMF in a coil or conductor. The key factors influencing this relationship are the rate of change of magnetic flux, the number of turns in the coil, and the phase relationship between the applied voltage, the flux wave, and the induced EMF.
In electrical engineering, the relationship between applied voltage and the electromotive force (EMF) induced due to magnetic flux is central to understanding the operation of electromagnetic devices such as transformers, electric generators, and inductors. Here's a detailed explanation of this relationship:
### 1. **Basic Concepts**
- **Applied Voltage (V):** This is the voltage that you apply to a circuit or a component.
- **Electromotive Force (EMF):** This is the voltage generated by a device due to changes in magnetic flux. It’s often referred to as induced voltage in contexts involving electromagnetic induction.
- **Magnetic Flux (Φ):** This is the measure of the amount of magnetic field passing through a given area. It is defined as Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field and the normal to the surface.
### 2. **Faraday’s Law of Electromagnetic Induction**
Faraday's Law states that the EMF induced in a circuit is proportional to the rate of change of magnetic flux through the circuit. Mathematically, it is expressed as:
\[ \text{EMF} = -\frac{d\Phi}{dt} \]
Here, \(\frac{d\Phi}{dt}\) represents the rate of change of magnetic flux.
### 3. **Relation Between Applied Voltage and Induced EMF**
- **In a Transformer:**
In a transformer, an alternating voltage is applied to the primary coil, creating a changing magnetic flux in the core. This changing flux induces an EMF in the secondary coil. The relationship between the applied voltage \( V_p \) and the induced EMF \( V_s \) in the secondary coil is governed by the turns ratio \( N_p/N_s \), where \( N_p \) and \( N_s \) are the number of turns in the primary and secondary coils, respectively. The relationship is given by:
\[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \]
The induced EMF in the secondary coil is directly proportional to the rate of change of the applied voltage in the primary coil, assuming ideal conditions with no losses.
- **In an Inductor:**
For an inductor, when a time-varying voltage is applied across it, a changing current creates a changing magnetic flux through the inductor. The induced EMF (or voltage) across the inductor is given by:
\[ \text{EMF} = -L \frac{dI}{dt} \]
Here, \(L\) is the inductance of the inductor, and \( \frac{dI}{dt} \) is the rate of change of current. The applied voltage must overcome this induced EMF to change the current through the inductor.
### 4. **Flux Wave and Its Impact**
When considering alternating current (AC) applications, the magnetic flux can be represented as a sinusoidal wave, and the induced EMF will also be sinusoidal, but it will be out of phase with the flux wave by 90 degrees (due to the derivative relationship).
- **For AC Systems:**
The applied AC voltage creates a time-varying magnetic flux in the core of a device. The EMF induced in the system will be sinusoidal if the applied voltage is sinusoidal, and its amplitude is proportional to the frequency of the applied voltage and the magnitude of the flux change.
The relationship can be expressed as:
\[ \text{EMF}_{\text{peak}} = 2 \pi f \Phi_{\text{peak}} \]
where \(f\) is the frequency of the AC supply, and \( \Phi_{\text{peak}} \) is the peak magnetic flux.
### Summary
- The **applied voltage** creates a changing magnetic flux in electromagnetic devices.
- The **induced EMF** is proportional to the rate of change of this magnetic flux.
- In AC systems, both the applied voltage and the induced EMF follow sinusoidal variations, with the EMF being phase-shifted by 90 degrees relative to the flux wave.
Understanding these relationships is crucial for designing and analyzing systems involving electromagnetic induction.
### 1. **Basic Concepts**
- **Applied Voltage (V):** This is the voltage that you apply to a circuit or a component.
- **Electromotive Force (EMF):** This is the voltage generated by a device due to changes in magnetic flux. It’s often referred to as induced voltage in contexts involving electromagnetic induction.
- **Magnetic Flux (Φ):** This is the measure of the amount of magnetic field passing through a given area. It is defined as Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field and the normal to the surface.
### 2. **Faraday’s Law of Electromagnetic Induction**
Faraday's Law states that the EMF induced in a circuit is proportional to the rate of change of magnetic flux through the circuit. Mathematically, it is expressed as:
\[ \text{EMF} = -\frac{d\Phi}{dt} \]
Here, \(\frac{d\Phi}{dt}\) represents the rate of change of magnetic flux.
### 3. **Relation Between Applied Voltage and Induced EMF**
- **In a Transformer:**
In a transformer, an alternating voltage is applied to the primary coil, creating a changing magnetic flux in the core. This changing flux induces an EMF in the secondary coil. The relationship between the applied voltage \( V_p \) and the induced EMF \( V_s \) in the secondary coil is governed by the turns ratio \( N_p/N_s \), where \( N_p \) and \( N_s \) are the number of turns in the primary and secondary coils, respectively. The relationship is given by:
\[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \]
The induced EMF in the secondary coil is directly proportional to the rate of change of the applied voltage in the primary coil, assuming ideal conditions with no losses.
- **In an Inductor:**
For an inductor, when a time-varying voltage is applied across it, a changing current creates a changing magnetic flux through the inductor. The induced EMF (or voltage) across the inductor is given by:
\[ \text{EMF} = -L \frac{dI}{dt} \]
Here, \(L\) is the inductance of the inductor, and \( \frac{dI}{dt} \) is the rate of change of current. The applied voltage must overcome this induced EMF to change the current through the inductor.
### 4. **Flux Wave and Its Impact**
When considering alternating current (AC) applications, the magnetic flux can be represented as a sinusoidal wave, and the induced EMF will also be sinusoidal, but it will be out of phase with the flux wave by 90 degrees (due to the derivative relationship).
- **For AC Systems:**
The applied AC voltage creates a time-varying magnetic flux in the core of a device. The EMF induced in the system will be sinusoidal if the applied voltage is sinusoidal, and its amplitude is proportional to the frequency of the applied voltage and the magnitude of the flux change.
The relationship can be expressed as:
\[ \text{EMF}_{\text{peak}} = 2 \pi f \Phi_{\text{peak}} \]
where \(f\) is the frequency of the AC supply, and \( \Phi_{\text{peak}} \) is the peak magnetic flux.
### Summary
- The **applied voltage** creates a changing magnetic flux in electromagnetic devices.
- The **induced EMF** is proportional to the rate of change of this magnetic flux.
- In AC systems, both the applied voltage and the induced EMF follow sinusoidal variations, with the EMF being phase-shifted by 90 degrees relative to the flux wave.
Understanding these relationships is crucial for designing and analyzing systems involving electromagnetic induction.