11.3 Linear Transformers: Ideal and Non-ideal
4 min read•july 30, 2024
Linear transformers are essential components in magnetically coupled circuits, transferring electrical energy between circuits through electromagnetic induction. They operate based on Faraday's Law and can step up or step down voltage and current depending on their .
Ideal transformers have perfect energy transfer, while non-ideal transformers experience and parasitic effects. Understanding these differences is crucial for designing efficient power systems and analyzing real-world transformer behavior in various applications.
Ideal Transformer Principles
Electromagnetic Induction and Energy Transfer
Top images from around the web for Electromagnetic Induction and Energy Transfer
23.7 Transformers – College Physics View original
Is this image relevant?
Faraday’s Law of Induction: Lenz’s Law | Physics View original
Is this image relevant?
Magnetic Flux, Induction, and Faraday’s Law | Boundless Physics View original
Is this image relevant?
23.7 Transformers – College Physics View original
Is this image relevant?
Faraday’s Law of Induction: Lenz’s Law | Physics View original
Is this image relevant?
1 of 3
Top images from around the web for Electromagnetic Induction and Energy Transfer
23.7 Transformers – College Physics View original
Is this image relevant?
Faraday’s Law of Induction: Lenz’s Law | Physics View original
Is this image relevant?
Magnetic Flux, Induction, and Faraday’s Law | Boundless Physics View original
Is this image relevant?
23.7 Transformers – College Physics View original
Is this image relevant?
Faraday’s Law of Induction: Lenz’s Law | Physics View original
Is this image relevant?
1 of 3
- Ideal linear transformer transfers electrical energy between circuits through electromagnetic induction without losses
- Primary creates magnetic field when alternating current applied induces voltage in secondary winding
- Obeys principle of (power input to primary equals power output from secondary)
- Governed by Faraday's Law of Induction and Lenz's Law explaining how changing magnetic fields induce voltages in windings
- Can step up or step down voltage and current depending on turns ratio (ratio of primary to secondary winding turns)
- Core assumed to have infinite permeability and zero reluctance allowing perfect coupling between windings
Turns Ratio and Winding Relationships
- Turns ratio defined as number of primary winding turns to secondary winding turns
- Voltage ratio directly proportional to turns ratio: (V₁, N₁ primary voltage and turns; V₂, N₂ secondary voltage and turns)
- Current ratio inversely proportional to turns ratio: (I₁, I₂ primary and secondary currents)
- Power factor remains constant across transformer (phase relationship between voltage and current preserved)
- Dot convention in schematics indicates relative winding polarity and phase relationship between primary and secondary voltages
Transformer Voltage and Current Relationships
Impedance Transformation
- Impedance transformation follows square of turns ratio: (Z₁, Z₂ impedances at primary and secondary sides)
- Used for impedance matching in circuits optimizing power transfer between source and load
- Reflected impedance allows representation of secondary-side elements as equivalent components on primary side simplifying circuit analysis
- Examples:
- 10:1 turns ratio steps down 100Ω load to 1Ω when viewed from primary side
- 1:3 turns ratio steps up 50Ω source impedance to 450Ω when viewed from secondary side
Power and Efficiency Considerations
- Ideal transformer has 100% (no power losses)
- Power input equals power output:
- Voltage-current trade-off: higher voltage corresponds to lower current and vice versa
- Examples:
- 1000W input at 100V primary yields 1000W output at 1000V secondary (10:1 step-up)
- 500VA apparent power remains constant from primary to secondary sides
Non-Ideal Transformer Characteristics
Core and Copper Losses
- Core losses consist of hysteresis losses and eddy current losses reducing efficiency
- Hysteresis losses result from energy required to realign magnetic domains in
- Eddy current losses caused by circulating currents induced in core material
- Copper losses (I²R losses) occur in windings due to conductor resistance
- Examples:
- Hysteresis losses in silicon steel core approximately 2-3% of total power
- Copper losses in 1kVA transformer may be around 10-20W at full load
Parasitic Effects and Non-Linear Behavior
- Leakage inductance arises from imperfect coupling between windings (some flux doesn't link both windings)
- Winding capacitance (inter-turn and inter-winding) introduces parasitic effects significant at high frequencies
- Core saturation occurs when magnetic flux density in core material reaches maximum value leading to non-linear behavior
- Magnetizing current required to establish magnetic field in core (additional primary current draw not contributing to power transfer)
- Examples:
- Leakage inductance of 1mH in 60Hz transformer can cause 0.38V voltage drop at 1A
- Winding capacitance of 100pF becomes significant above 1MHz in RF transformers
Impact of Non-Ideal Transformers on Circuits
Efficiency and Voltage Regulation
- Efficiency reduction due to core and copper losses leads to power dissipation and potential heating
- Voltage regulation (change in secondary voltage from no-load to full-load) affected by leakage inductance and winding resistance
- Frequency response limitations arise from core losses, leakage inductance, and winding capacitance restricting useful bandwidth
- Examples:
- 98% efficient 1kVA transformer dissipates 20W as heat
- 5% voltage regulation means 100V no-load output drops to 95V at full load
Distortion and Electromagnetic Effects
- Harmonic distortion introduced by core saturation (particularly near/beyond designed flux density limits)
- Inrush current at transformer energization significantly higher than normal operating current due to core saturation
- Impedance matching accuracy compromised potentially leading to suboptimal power transfer
- EMI (Electromagnetic Interference) generation and susceptibility increase due to leakage fields and imperfect shielding
- Examples:
- Inrush current can be 10-15 times rated current for first few cycles
- Harmonic distortion in saturated transformer can reach 5-10% THD (Total Harmonic Distortion)
Key Terms to Review (16)
Conservation of Energy: Conservation of energy is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. This principle is crucial in understanding how energy moves and changes within electrical systems, particularly in the analysis of transformers and coupled circuits where energy is transferred between magnetic and electrical forms. Recognizing this concept helps to evaluate system efficiency and performance.
Core material: Core material refers to the substance used in the construction of inductors and transformers that enhances magnetic properties and efficiency. This material plays a crucial role in guiding and concentrating magnetic fields generated by electric currents, improving energy transfer and reducing losses in electromagnetic devices.
Efficiency: Efficiency is a measure of how effectively a system converts input energy into useful output energy, expressed as a percentage. In various electrical contexts, this term helps assess the performance of devices and systems, indicating how much of the total input power is utilized in doing useful work while minimizing losses due to factors like heat dissipation or magnetic leakage.
Faraday's Law of Electromagnetic Induction: Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the wire. This principle is fundamental in understanding how linear transformers operate, as they rely on the induction of voltage through magnetic flux changes. The law highlights the relationship between electricity and magnetism, forming the basis for many electrical devices, including generators and transformers.
Ideal transformer equation: The ideal transformer equation describes the relationship between the primary and secondary voltages and currents in an ideal transformer, highlighting the principle of energy conservation and magnetic coupling. In this context, it establishes that the voltage ratio between the primary and secondary coils is equal to the turns ratio, while the current ratio is inversely proportional. This relationship is fundamental for understanding how transformers transfer electrical energy between circuits while maintaining efficiency in an ideal scenario.
Losses: Losses refer to the energy that is dissipated as heat or other forms of energy in electrical components, particularly in transformers, due to non-ideal conditions. These losses can significantly affect the efficiency and performance of electrical systems, including transformers, which ideally should convert input energy to output energy without any loss. Understanding losses is essential for analyzing the behavior of transformers and optimizing their design and operation.
Ohm's Law: Ohm's Law is a fundamental principle in electrical engineering that states the relationship between voltage, current, and resistance in a circuit. It asserts that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. This relationship helps in understanding how circuits operate and facilitates the analysis of various electrical components.
P1 = p2: The expression 'p1 = p2' signifies that the power on the primary side of a transformer (p1) is equal to the power on the secondary side (p2) under ideal conditions. This concept is fundamental to understanding how energy is conserved in transformers, ensuring that the input power and output power are equivalent in a lossless scenario, while also leading to insights about voltage and current relationships.
Power Distribution: Power distribution refers to the process of delivering electrical power from a generation source to end-users through a network of electrical components. It involves transforming voltage levels and managing the flow of electricity to ensure that power reaches consumers efficiently and reliably, while minimizing losses and maintaining safety standards.
Signal isolation: Signal isolation refers to the process of separating different electrical circuits to prevent unwanted interaction or transfer of signals between them. This is crucial in various applications where maintaining signal integrity and protecting sensitive components from interference is necessary. Isolation helps to reduce noise, enhance safety, and protect against high voltages and surges.
Step-down transformer: A step-down transformer is an electrical device that reduces the voltage from a primary circuit to a lower voltage in a secondary circuit while increasing the current. This type of transformer plays a crucial role in power distribution by allowing high-voltage transmission lines to supply electricity at a more usable level for residential or industrial use. It operates on the principle of electromagnetic induction, where the ratio of turns in the primary and secondary coils determines the voltage change.
Step-up transformer: A step-up transformer is an electrical device that increases the voltage from the primary winding to the secondary winding while decreasing the current. This type of transformer operates on the principle of electromagnetic induction, where a varying current in the primary coil creates a magnetic field that induces a higher voltage in the secondary coil, allowing for efficient energy transfer over long distances.
Turns Ratio: The turns ratio is the ratio of the number of turns of wire in the primary coil to the number of turns in the secondary coil of a transformer. This important characteristic determines how voltage and current are transformed between the primary and secondary sides, influencing the overall efficiency and functionality of both ideal and non-ideal transformers.
V1/v2 = n1/n2: The equation $$\frac{v_1}{v_2} = \frac{n_1}{n_2}$$ represents the relationship between the voltages and turns ratios in an ideal transformer. In this context, $v_1$ is the primary voltage, $v_2$ is the secondary voltage, $n_1$ is the number of turns in the primary coil, and $n_2$ is the number of turns in the secondary coil. This relationship illustrates how voltage levels can be transformed based on the winding ratios of the coils, which is fundamental in designing transformers for various applications.
Voltage Transformation: Voltage transformation refers to the process of changing the voltage level of an electrical signal using a transformer. This change can either increase (step-up) or decrease (step-down) the voltage while maintaining the power transfer and allowing for efficient transmission of electricity over long distances. Understanding voltage transformation is crucial as it relates directly to the operation of both ideal and non-ideal transformers, which have varying efficiencies and characteristics in their performance.
Winding: Winding refers to the arrangement of coils of wire used in inductors and transformers to create magnetic fields when electrical current passes through them. The design and configuration of windings play a crucial role in determining the performance characteristics, such as inductance, impedance, and voltage transformation, essential for efficient energy transfer and storage in electrical devices.