7.2 Ideal transformer model and analysis

Transformers are essential for power systems, allowing voltage levels to be changed efficiently. The ideal transformer model simplifies analysis by assuming perfect energy transfer between windings. This section explores the key principles and relationships that govern ideal transformer behavior.

Understanding ideal transformers is crucial for grasping real-world applications. We'll dive into voltage and current ratios, impedance transformation, and analysis techniques. These concepts form the foundation for working with more complex transformer circuits and power systems.

Ideal Transformer Model

Assumptions and Basic Principles

• Ideal transformer operates with perfect efficiency, no energy losses
• Core permeability approaches infinity, resulting in negligible magnetizing current
• Zero winding resistance eliminates copper losses
• No flux leakage ensures all magnetic flux links both primary and secondary windings
• Operates on principle of electromagnetic induction discovered by Faraday
• Consists of two or more coils wound around a common ferromagnetic core
• Primary coil connected to AC source induces voltage in secondary coil
• Magnetic flux in core transfers energy between primary and secondary circuits

Voltage and Current Relationships

• Turns ratio (N) defines relationship between primary and secondary windings
• Voltage ratio directly proportional to turns ratio: $\frac{V_p}{V_s} = \frac{N_p}{N_s} = N$
• Current ratio inversely proportional to turns ratio: $\frac{I_p}{I_s} = \frac{N_s}{N_p} = \frac{1}{N}$
• Step-up transformer increases voltage (N > 1), step-down decreases voltage (N < 1)
• Ampere-turns balance maintains equilibrium between primary and secondary sides
• Dot convention indicates relative polarity of windings (determines phase relationships)

Power Conservation Principle

• Ideal transformer conserves power between primary and secondary sides
• Input power equals output power: $P_p = P_s$
• Power conservation expressed as: $V_p I_p = V_s I_s$
• Apparent power (VA) remains constant across transformer
• Power factor unaffected by ideal transformer
• Reactive power (VAR) transfers between primary and secondary without losses
• Efficiency of ideal transformer always 100% (real transformers approach 98-99%)

Transformer Analysis

Impedance Transformation

• Transformer modifies impedance seen by source or load
• Impedance transformation ratio proportional to square of turns ratio
• Reflected impedance formula: $Z_r = N^2 Z_L$
• Allows impedance matching between source and load for maximum power transfer
• Useful in audio systems, RF circuits, and power distribution networks
• Impedance transformation preserves complex nature of load (resistive and reactive components)
• Can be used to step up or step down impedance based on application requirements

Analyzing Reflected Impedance

• Reflected impedance represents load impedance as seen from primary side
• Simplifies analysis by referring all circuit elements to one side of transformer
• Enables use of series and parallel circuit analysis techniques
• Maintains power relationships between primary and secondary circuits
• Reflected current calculated using turns ratio: $I_r = \frac{I_s}{N}$
• Reflected voltage determined by: $V_r = N V_s$
• Useful for analyzing transformer circuits with complex loads or multiple windings

Phasor Diagram Representation

• Graphical tool for visualizing voltage and current relationships in transformers
• Represents magnitude and phase angle of electrical quantities as vectors
• Primary and secondary voltages typically drawn in phase (ideal transformer)
• Current phasors may lead or lag voltage phasors depending on load characteristics
• Dot convention determines relative orientation of voltage phasors
• Useful for analyzing power factor and phase relationships in transformer circuits
• Helps visualize effects of non-ideal transformer characteristics (winding resistance, leakage inductance)

No-Load Condition Analysis

• Represents transformer behavior when secondary winding open-circuited
• Primary current consists only of magnetizing current (negligible in ideal case)
• Secondary voltage induced but no current flows in secondary winding
• Voltage ratio maintains turns ratio relationship: $V_s = \frac{V_p}{N}$
• Core losses (hysteresis and eddy currents) become significant in real transformers
• No-load test used to determine core losses and magnetizing current in practical transformers
• Provides insight into transformer's magnetic circuit characteristics