7.3 Non-ideal transformer characteristics and equivalent circuits

Transformers aren't perfect. They've got quirks that affect how well they work. We'll look at the losses that eat up power and the pesky currents that don't do any useful work. These issues make real transformers less efficient than their ideal counterparts.

We'll also check out how transformers behave when they're not quite ideal. Things like magnetic flux that doesn't go where it should and resistance in the windings can mess with the voltage. We'll see how engineers model these issues to predict how transformers will perform in the real world.

Transformer Losses

Core and Copper Losses

• Core losses occur in the transformer's magnetic core due to hysteresis and eddy currents
  • Hysteresis losses result from the energy required to repeatedly magnetize and demagnetize the core material
  • Eddy current losses arise from induced currents circulating within the core, generating heat
• Copper losses stem from the resistance of the transformer windings
  • Caused by the current flowing through the primary and secondary coils
  • Increases with the square of the current ($P_{loss} = I^2R$)
• Both core and copper losses contribute to reduced transformer efficiency
  • Core losses remain relatively constant regardless of load
  • Copper losses vary with load, increasing as the current increases

Magnetizing Current and Efficiency

• Magnetizing current flows in the primary winding to establish the magnetic field in the core
  • Present even when the secondary winding is open-circuited
  • Typically small in magnitude (1-5% of rated current) but contributes to core losses
• Transformer efficiency measures the ratio of output power to input power
  • Calculated as $\eta = \frac{P_{out}}{P_{in}} \times 100\%$
  • Affected by core losses, copper losses, and magnetizing current
  • Modern power transformers can achieve efficiencies over 99% at full load
• Efficiency varies with load, typically peaking at 50-75% of full load
  • Decreases at very low loads due to fixed core losses
  • Decreases at very high loads due to increased copper losses

Transformer Non-Idealities

Leakage Inductance and Its Effects

• Leakage inductance results from magnetic flux that does not link both primary and secondary windings
  • Caused by imperfect coupling between windings
  • Modeled as separate inductances in series with each winding
• Effects of leakage inductance include:
  • Voltage drop in the windings, reducing the overall voltage transfer
  • Phase shift between primary and secondary voltages
  • Limits the rate of change of current, affecting transformer response to transients
• Leakage inductance can be minimized through proper winding design and core geometry
  • Interleaving primary and secondary windings
  • Using toroidal cores to reduce flux leakage paths

Winding Resistance and Voltage Regulation

• Winding resistance represents the ohmic resistance of the copper wire used in the windings
  • Causes voltage drop and power loss in the transformer
  • Increases with wire length and decreases with wire cross-sectional area
• Voltage regulation quantifies the change in secondary voltage from no-load to full-load conditions
  • Calculated as $VR = \frac{V_{nl} - V_{fl}}{V_{fl}} \times 100\%$
  • Affected by winding resistance, leakage inductance, and load power factor
  • Lower voltage regulation indicates better performance (closer to ideal transformer behavior)
• Strategies to improve voltage regulation:
  • Using larger wire gauge to reduce winding resistance
  • Minimizing leakage inductance through improved winding design
  • Employing tap changers for voltage adjustment under varying load conditions

Transformer Modeling

Equivalent Circuit Models and Analysis

• Equivalent circuit models represent transformer behavior using lumped circuit elements
  • Simplify analysis by reducing the complex electromagnetic system to a circuit representation
  • Enable calculation of voltages, currents, and power flow in the transformer
• Common equivalent circuit models include:
  • T-model: Represents leakage inductances and magnetizing branch separately
  • Π-model: Alternative representation with magnetizing branch split between primary and secondary
• Key components of equivalent circuit models:
  • Primary and secondary winding resistances ($R_1$ and $R_2$)
  • Leakage inductances ($L_1$ and $L_2$)
  • Magnetizing branch (parallel combination of $R_m$ and $L_m$)
• Analysis techniques using equivalent circuits:
  • Referred quantities: Secondary parameters referred to the primary side for simplified analysis
  • Phasor analysis: Used to solve for voltages and currents in AC circuits
  • Power calculations: Determine input power, output power, and losses

Applications and Limitations of Equivalent Circuits

• Applications of transformer equivalent circuits:
  • Predicting transformer performance under various load conditions
  • Analyzing voltage regulation and efficiency
  • Designing transformer protection systems
  • Studying transformer behavior in power system simulations
• Limitations and considerations:
  • Neglect of frequency-dependent effects (skin effect, proximity effect)
  • Assumption of linear magnetic core behavior (may not hold for high currents or frequencies)
  • Simplified representation of complex 3D electromagnetic fields
• Advanced modeling techniques for specific applications:
  • Finite element analysis for detailed magnetic field simulations
  • Frequency-dependent models for transient analysis
  • Thermal models for temperature rise predictions