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Mutual Inductance Coupled Coils · Mutual Inductance Coefficient and T-Equivalent

Two coupled coils \(L_1\) and \(L_2\): adjust the coupling coefficient \(k\) (\(0 \to 1\)) to compare the equivalent inductance of series-aiding and series-opposing connections
Mutual inductance \(M = k\sqrt{L_1 L_2}\)  |  Voltage source \(v(t) = V_m \sin(2\pi f t)\), \(V_m = 10\text{V}\)

Coil & Signal Parameters

0.50
0 (No coupling)1 (Full coupling)

\(k=0\) no mutual inductance; \(k=1\) full coupling (ideal case)

10 mH
1 mH100 mH
20 mH
1 mH100 mH
50 Hz
10 Hz500 Hz

Aiding: magnetic fluxes reinforce each other, \(L_{eq}=L_1+L_2+2M\)

Equivalent Parameter Calculation

Mutual Inductance \(M\)
— mH
Current \(L_{eq}\)
— mH
\(L_{eq}\) (Aiding)
— mH
\(L_{eq}\) (Opposing)
— mH
Mutual Inductance Voltage Amplitude \(V_M\)
— V

T-Type Decoupling Equivalent Circuit

Same polarity direction (aiding): middle arm is \(+M\); opposite polarity (opposing): middle arm is \(-M\)

Voltage Waveforms · \(v_1(t)\), \(v_2(t)\), \(v_M(t)\)

Current: Series Aiding
🔵 \(v_1(t)\): voltage across \(L_1\)  |  🟢 \(v_2(t)\): voltage across \(L_2\)  |  🟣 \(v_M(t)\): mutual inductance voltage (\(M \cdot di/dt\))

Equivalent Inductance Comparison · \(L_{eq}\) (Aiding) vs \(L_{eq}\) (Opposing)

🟠 Aiding: equivalent inductance is larger (magnetic field reinforced)  |  🔴 Opposing: equivalent inductance is smaller (magnetic field weakened)

📐 Mutual Inductance Coupling Theory

\(M = k\sqrt{L_1 L_2}\), \(0 \le k \le 1\)
Series aiding: \(L_{eq} = L_1 + L_2 + 2M\)
Series opposing: \(L_{eq} = L_1 + L_2 - 2M\)
Mutual inductance voltage: \(v_M(t) = M \cdot \frac{di}{dt}\)
Dot convention determines the polarity of the mutual inductance voltage.

⚡ Characteristics and Key Observations

• The larger \(k\) is, the greater the difference between aiding and opposing equivalent inductance.
• When \(k=1\), the opposing equivalent inductance can approach zero (when \(L_1=L_2\)).
• In the T-equivalent circuit, the middle arm is \(+M\) for aiding and \(-M\) for opposing.
• Mutual inductance voltage \(V_M\) increases with frequency and coupling coefficient.