Two coils can exchange energy without sharing a conducting wire. The magnetic field connects their changing currents, but it does not offer free power. Careful winding signs and a complete energy balance keep the explanation honest.
We build the coupled-coil equations from declared winding directions, test their coefficients with energy, and then derive the transformer’s voltage, current and impedance relations.
What you will learn to do
- Distinguish single-turn flux from winding linkage.
- Use dotted terminal references consistently in induced voltages.
- Derive and apply the passive mutual-inductance bound.
- Infer mutual coupling from series-aiding/opposing measurements.
- Derive voltage and current ratios for an ideal transformer.
- Account for reflected resistance, power loss and finite core limits.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning Learning goals: - Distinguish single-turn flux from winding linkage. - Use dotted terminal references consistently in induced voltages. - Derive and apply the passive mutual-inductance bound. - Infer mutual coupling from series-aiding/opposing measurements. - Derive voltage and current ratios for an ideal transformer. - Account for reflected resistance, power loss and finite core limits. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Can an open coil have an induced voltage while delivering no sustained load power?
Compare your reasoning
Yes. A changing linked flux can induce a terminal voltage even when the open circuit carries no sustained conduction current. Voltage alone does not establish power delivery.
Revisit the explanation →02 / Coupled coils and transformer reasoning
State terminal polarity before deciding a sign
A dot convention is a way to record the relative winding sense without repeatedly redrawing every turn. An increasing current entering one dot makes the mutual contribution to the other winding’s terminal voltage positive at its dot. It does not say that every actual current must flow into a dot.
Define v₁ and v₂ as dot-to-undotted terminal voltage drops, and i₁ and i₂ as currents entering the dots. The passive terminal equations then contain positive mutual derivative terms. Faraday’s negative circulation sign has not disappeared: the passive voltage drop and an oriented induced emf around the wire are different signed quantities.
If one current reference is reversed, transform both that variable and the associated coefficients consistently. Guessing “minus because of Lenz” while leaving the rest of the references unchanged can reverse the physical prediction twice.
These equations describe ideal winding inductive drops; winding resistances add their own Ri terms. Dots specify the relative sign of mutual contributions, not a universal sign for a measured voltage.
WORKED EXAMPLE
Read a negative derivative without redrawing the coil
An open secondary shares M = 8 mH with a primary. Primary current is referenced into its dot and decreases at 150 A/s. Find the secondary dot-to-undotted voltage.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning Focus: State terminal polarity before deciding a sign Learning goals: - Distinguish single-turn flux from winding linkage. - Use dotted terminal references consistently in induced voltages. - Derive and apply the passive mutual-inductance bound. - Infer mutual coupling from series-aiding/opposing measurements. - Derive voltage and current ratios for an ideal transformer. - Account for reflected resistance, power loss and finite core limits. Equations in this section (LaTeX): v_1=L_1\dot i_1+M\dot i_2,\qquad v_2=M\dot i_1+L_2\dot i_2 Meaning: These equations describe ideal winding inductive drops; winding resistances add their own Ri terms. Dots specify the relative sign of mutual contributions, not a universal sign for a measured voltage. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Coupled coils and transformer reasoning
Energy restricts how strong mutual coupling can be
Each source may do work against the voltage induced by the other changing current. Add the terminal powers v₁i₁ + v₂i₂ using the same passive references. For fixed reciprocal coefficients this is the derivative of a single stored-energy expression.
The cross term can be negative for opposing current directions, but the total magnetic energy of passive ideal coils cannot be negative for any pair of currents. Completing the square gives a bound on |M|. This is a physical constraint on an inductance matrix, not a rule about whichever current happens to be used in one experiment.
Define the coupling coefficient κ = |M|/√(L₁L₂). It lies between zero and one for this passive model. Values below one describe incomplete shared linkage. κ = 1 is ideal perfect coupling; real leakage flux usually makes κ smaller.
The sign of Mi₁i₂ follows the declared current references. Changing both the reference and its associated sign leaves the physical energy unchanged.
Since the square can be made zero by an allowed current choice, the remaining coefficient must be nonnegative. This derives the mutual-inductance bound directly.
WORKED EXAMPLE
Check an opposing-current state without dropping the cross term
L₁ = 20 mH, L₂ = 80 mH and M = 12 mH. Currents into the dotted terminals are i₁ = +2 A and i₂ = −1 A. Find κ and stored energy.
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STOP & THINK
A proposed passive linear pair has L₁ = 20 mH, L₂ = 80 mH and |M| = 60 mH. What should you conclude?
- The claimed coefficients are inconsistent with this passive model
- Negative total energy is acceptable because flux can oppose
- Coupling greater than one improves efficiency
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning
Focus: Energy restricts how strong mutual coupling can be
Learning goals:
- Distinguish single-turn flux from winding linkage.
- Use dotted terminal references consistently in induced voltages.
- Derive and apply the passive mutual-inductance bound.
- Infer mutual coupling from series-aiding/opposing measurements.
- Derive voltage and current ratios for an ideal transformer.
- Account for reflected resistance, power loss and finite core limits.
Equations in this section (LaTeX):
W=\frac12L_1i_1^2+\frac12L_2i_2^2+Mi_1i_2\ge0,\qquad |M|\le\sqrt{L_1L_2}
Meaning: The sign of Mi₁i₂ follows the declared current references. Changing both the reference and its associated sign leaves the physical energy unchanged.
W=\frac12L_2\left(i_2+\frac{M}{L_2}i_1\right)^2+\frac12\left(L_1-\frac{M^2}{L_2}\right)i_1^2
Meaning: Since the square can be made zero by an allowed current choice, the remaining coefficient must be nonnegative. This derives the mutual-inductance bound directly.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Coupled coils and transformer reasoning
Adding two coils can increase or decrease the effective inductance
Connect the windings in series so the same current magnitude passes through each. The sign of the mutual contribution depends on whether the linked fluxes reinforce or oppose. For reinforcing linkage the effective inductance is L₁ + L₂ + 2M; for opposing linkage it is L₁ + L₂ − 2M.
The factor two comes from the two mutual voltage contributions. It also follows from the cross term in the stored energy: set i₁ = i₂ = i for one orientation, or i₂ = −i₁ for the other, and match the result to ½Leff i².
Series-aiding and series-opposing measurements can determine both the self-inductance sum and M. They do not determine L₁ and L₂ separately without another measurement. This distinction prevents extracting more unknown information than the data contain.
WORKED EXAMPLE
Infer coupling from two wiring measurements
The same fixed pair reads 124 mH in one series orientation and 76 mH after reversing one winding. Determine M and L₁ + L₂.
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STOP & THINK
Series-aiding and series-opposing measurements give 90 mH and 50 mH. What is M in mH?
Answer in mH.
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning Focus: Adding two coils can increase or decrease the effective inductance Learning goals: - Distinguish single-turn flux from winding linkage. - Use dotted terminal references consistently in induced voltages. - Derive and apply the passive mutual-inductance bound. - Infer mutual coupling from series-aiding/opposing measurements. - Derive voltage and current ratios for an ideal transformer. - Account for reflected resistance, power loss and finite core limits. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Coupled coils and transformer reasoning
A common changing flux fixes a voltage ratio
An ideal transformer has two windings linking the same time-varying core flux. If leakage and winding resistance are neglected, the instantaneous induced terminal voltages are proportional to their turn counts, with polarity set by the dots. The voltage ratio is a flux statement, not an energy argument by itself.
For sinusoidal flux Φ = Φmax sin(ωt), the voltage peak is NωΦmax and the RMS voltage is NωΦmax/√2. The familiar numerical factor about 4.44 multiplies fNΦmax only for a sine wave. A square or distorted waveform needs its own flux integral.
A finite core cannot support a continually increasing flux. Applying a constant nonzero winding voltage would require a continuously ramping flux in the lossless equation, eventually leaving the linear core regime. A real DC-fed winding is then limited by resistance and other nonideal effects; the steady sinusoidal transformer model is not a DC voltage converter.
Use the exact 2π/√2 factor when precision matters. The signed instantaneous ratio assumes both voltages are measured from dotted to undotted terminals.
WORKED EXAMPLE
Determine a flux limit from the applied sine-wave voltage
A 400-turn winding is driven by 24 V RMS at 50 Hz. In the ideal sinusoidal common-flux model, what is Φmax?
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning
Focus: A common changing flux fixes a voltage ratio
Learning goals:
- Distinguish single-turn flux from winding linkage.
- Use dotted terminal references consistently in induced voltages.
- Derive and apply the passive mutual-inductance bound.
- Infer mutual coupling from series-aiding/opposing measurements.
- Derive voltage and current ratios for an ideal transformer.
- Account for reflected resistance, power loss and finite core limits.
Equations in this section (LaTeX):
\frac{v_s}{v_p}=\frac{N_s}{N_p},\qquad V_{\mathrm{rms}}=\frac{N\omega\Phi_{\max}}{\sqrt2}\approx4.44fN\Phi_{\max}
Meaning: Use the exact 2π/√2 factor when precision matters. The signed instantaneous ratio assumes both voltages are measured from dotted to undotted terminals.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Coupled coils and transformer reasoning
A secondary load changes what the source must supply
When a load draws secondary current, its magnetic effect opposes the change that induced it. The primary draws an additional current so the applied voltage can sustain the required common flux. Ignoring this back-reaction would predict that the secondary can deliver power with no increase in source power.
For an ideal transformer, input real power equals output real power. With a resistive load, both sides are in phase with their respective voltage references when power is described into the transformer at the primary and out of it at the secondary. The current magnitudes change inversely with the turn ratio.
The load resistance as seen from the primary is the secondary resistance multiplied by the square of the primary-to-secondary turn ratio. This is a consequence of both the voltage and current transformations. It is not a change in the physical resistance of the load.
The currents here are RMS magnitudes for a resistive load. For signed winding currents both entering dotted terminals, Np ip + Ns is = 0 in the ideal zero-magnetizing-current limit.
WORKED EXAMPLE
Follow a load through the voltage and current ratios
An ideal transformer has Np = 600, Ns = 150 and primary voltage 48 V RMS. Its secondary drives a 6.0 Ω resistor. Find secondary voltage/current, primary current and the input resistance.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning
Focus: A secondary load changes what the source must supply
Learning goals:
- Distinguish single-turn flux from winding linkage.
- Use dotted terminal references consistently in induced voltages.
- Derive and apply the passive mutual-inductance bound.
- Infer mutual coupling from series-aiding/opposing measurements.
- Derive voltage and current ratios for an ideal transformer.
- Account for reflected resistance, power loss and finite core limits.
Equations in this section (LaTeX):
\frac{I_p}{I_s}=\frac{N_s}{N_p},\qquad R_{\mathrm{in}}=\left(\frac{N_p}{N_s}\right)^2R_L
Meaning: The currents here are RMS magnitudes for a resistive load. For signed winding currents both entering dotted terminals, Np ip + Ns is = 0 in the ideal zero-magnetizing-current limit.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Coupled coils and transformer reasoning
Separate useful ideal relations from loss and core limits
A real transformer has winding resistance, leakage inductance, magnetizing current and core loss. The first two change loaded terminal voltages; finite magnetizing inductance gives a current even when the secondary is open. An open secondary draws no load power, but the real primary can still consume power in copper and the core.
Eddy-current heating and hysteresis loss are different mechanisms. Laminations or a high-resistivity core reduce circulating conduction currents; selecting a material with a small hysteresis-loop area reduces cyclic magnetization loss. Neither improvement makes a real device perfectly ideal.
When efficiency is supplied, define it as useful real output power divided by real input power. Do not convert an efficiency percentage into a voltage ratio or assume it changes the turns ratio. If a problem gives a measured secondary voltage and load resistance, use those observations for the load and then compute the required input power.
WORKED EXAMPLE
Use measured output and efficiency without inventing a new turn ratio
A transformer delivers a measured 10 V RMS to a 5 Ω resistor. Its real-power efficiency at this load is 80%. Find input power and loss.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Coupled coils and transformer reasoning Focus: Separate useful ideal relations from loss and core limits Learning goals: - Distinguish single-turn flux from winding linkage. - Use dotted terminal references consistently in induced voltages. - Derive and apply the passive mutual-inductance bound. - Infer mutual coupling from series-aiding/opposing measurements. - Derive voltage and current ratios for an ideal transformer. - Account for reflected resistance, power loss and finite core limits. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Mutual voltage depends on a changing linked current, not merely on a nonzero current.
- Dots label relative winding signs; all voltage/current references must agree.
- The mutual cross energy may be negative, but passive total storage may not.
- A loaded transformer changes the current demanded from its source.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
For L₁ = L₂ = 0.10 H, M = 0.050 H and i₁ = i₂ = 2 A entering the dots, find stored energy.
Answer in J.
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SELF CHECK 2
An ideal transformer has Np/Ns = 3 and a 4 Ω secondary load. What input resistance does the primary see?
Answer in Ω.
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SELF CHECK 3
Why is the steady sine-wave transformer model unsuitable for a constant nonzero primary voltage?
- A constant voltage would require indefinitely ramping core flux in the ideal equation
- Faraday’s law stops applying to DC
- Turns have no meaning at zero frequency
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SELF CHECK 4
An open secondary has a changing induced voltage. What power can it deliver to an ideal open load?
- Finite voltage guarantees finite delivered power
- Zero, because load current is zero
- Exactly half the input power
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Turn counts set the output voltageIntermediate · Free sample · Hints and guided reasoningHow large can mutual inductance be?Advanced · Hints and guided reasoningMeasure coupling by reversing one coilAdvanced · Hints and guided reasoningTwo loads behind one transformerIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
What do winding dots tell you, and what do they not tell you about the actual currents?
Compare your reasoning
They specify the relative polarity convention for mutual voltage contributions. They do not require both actual currents to enter the dotted terminals. Define references, then allow signed currents.
Revisit the explanation →A few days later: change the situation
An ideal transformer reduces voltage by a factor of four while supplying a resistive load. How do its current magnitudes compare?
Compare your reasoning
The secondary current is four times the primary current under the ideal zero-magnetizing-current model. Equal input and output power requires the inverse current ratio; real losses need additional accounting.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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