Inductance Notes Inductance
RL Circuit Analysis & Magnetic Inertia
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Self-Inductance (L)
An inductor resists changes in current by inducing a back-EMF (\( V_L = -L \frac{di}{dt} \)) to oppose changes in magnetic flux.
Steady State
Inductor = Wire
1. The Loop Equation
Apply Kirchhoff's Loop Rule to a circuit with battery (\(\mathcal{E}\)), resistor (R ), and inductor (L ):
\[ \mathcal{E} - iR - L \frac{di}{dt} = 0 \]
Derivation Space: Solve for current \( i(t) \)
2. Charging Phase (Build-Up)
Current grows from zero toward \( I_{max} \).
\[ i(t) = \frac{\mathcal{E}}{R} \left( 1 - e^{-\frac{t}{\tau}} \right) \]
Time Constant: \( \tau = L/R \)
1τ 2τ 3τ 4τ 5τ i(t) Time (t) \( I_{max} \)
3. Discharging Phase (Decay)
Magnetic field collapsing maintains current temporarily.
\[ i(t) = I_0 e^{-\frac{t}{\tau}} \]
Stored Energy: \( \frac{1}{2}LI^2 \)
1τ 2τ 3τ 4τ 5τ \( I_{0} \)
4. Voltage Profiles (up to 5τ)
1τ 2τ 3τ 4τ 5τ Voltage (V) \( \mathcal{E} \) Time (t)
Sketch Inductor Voltage (\( V_L \)) and Resistor Voltage (\( V_R \)) up to 5τ.
Inductance Key Inductance
TEACHER KEY: RL Circuit Analysis
Self-Inductance (L)
Inductors oppose changes in current. Initial behavior (t=0): \( i = 0 \). Steady state behavior (t→∞): \( i = \mathcal{E}/R \).
Steady Current
\( I_{max} = \frac{\mathcal{E}}{R} \)
1. The Loop Equation KEY
\[ \mathcal{E} - iR - L \frac{di}{dt} = 0 \]
Derivation Steps
1. Separate Differentials: \( \frac{di}{\mathcal{E} - iR} = \frac{dt}{L} \)
2. Integrate with limits: \( \int_0^i \frac{di'}{\mathcal{E} - i'R} = \int_0^t \frac{dt'}{L} \)
3. Solve for i: \( -\frac{1}{R} \ln\left(\frac{\mathcal{E} - iR}{\mathcal{E}}\right) = \frac{t}{L} \)
Final Equation: \( i(t) = \frac{\mathcal{E}}{R}(1 - e^{-Rt/L}) \)
Time Constant: \( \tau = L/R \)
2. Charging Phase (Build-Up) KEY
Current reaches 63.2% of \( I_{max} \) after 1 time constant (\( \tau \)).
\[ i(t) = I_{max}(1 - e^{-t/\tau}) \]
1τ 2τ 3τ 4τ 5τ Current (i) Time (t)
3. Discharging Phase (Decay) KEY
Current drops to 36.8% of \( I_0 \) after 1 time constant (\( \tau \)).
\[ i(t) = I_0 e^{-t/\tau} \]
1τ 2τ 3τ 4τ 5τ Current (i)
4. Voltage Profiles KEY
1τ 2τ 3τ 4τ 5τ V_R V_L Voltage (V) Time (t)
Conservation Principle
Kirchhoff's Loop Rule is always satisfied: \( V_R(t) + V_L(t) = \mathcal{E} \) for the charging circuit.
Inductance Guide Inductance
The Complete Guide to Inductance and RL Circuits
I. Introduction
In mechanics, mass is the measure of inertia—resistance to changes in motion. In circuits, the inductor serves the same role for current. An inductor possesses self-inductance (L ) , which causes it to oppose any change in current.
An inductor opposes changes in current by storing energy in a magnetic field . This behavior is rooted in Faraday's Law and Lenz's Law.
Physical Basis
A changing magnetic flux induces an EMF (\(\mathcal{E}\)) in a loop:
\[ \mathcal{E} = -N \frac{d\Phi_B}{dt} \]
Lenz's Law: The induced EMF creates a current whose field opposes the original flux change.
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II. Defining Inductance (L )
When current i flows through a coil, it creates a magnetic field. We define the constant of proportionality as the Inductance (L ) :
\[ L = \frac{N\Phi_B}{i} \]
The SI unit is the Henry (H) . 1 H = 1 Wb/A.
The Solenoid Model
For an ideal solenoid of length l , area A , and turns N :
\[ L = \mu_0 \frac{N^2 A}{l} \]
Inductance depends only on geometry.
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III. The Model RL Circuit
A proper RL circuit model includes a two-way switch for both charging and discharging.
Operational Logic
Position 1: Charge (Build field).
Position 2: Decay (Release field).
\(\mathcal{E}\) S R L
Figure 1.2: Complete RL Model
Immediate (t=0)
Inductor produces a back-EMF opposing the battery. Current is zero.
Steady (t→∞)
Current is constant. Back-EMF is zero. Current is \(\mathcal{E}/R\).
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IV. The Loop Equation
Kirchhoff's Rule
Summing potential differences clockwise through the loop:
\[ \mathcal{E} - iR - L \frac{di}{dt} = 0 \]
Key Insights
As i increases, iR increases.
Consequently, L(di/dt) must decrease.
The rate of current growth slows down over time.
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Inductance Slides Inductance
Dynamics of RL Circuits
Magnetic Inertia
Just as mass resists velocity changes, an inductor resists current changes.
\[ V_L = -L \frac{di}{dt} \]
Lenz's Law
Current Grows → EMF Opposes
Current Drops → EMF Supports
Kirchhoff's Law
\[ \mathcal{E} - iR - L \frac{di}{dt} = 0 \]
Battery
\( \mathcal{E} \)
Resistor
\( iR \)
Inductor
\( L \frac{di}{dt} \)
Build-Up Phase
\[ i(t) = I_{max} ( 1 - e^{-t/\tau} ) \]
Steady State Reach: 5τ
\( \tau = L/R \)
1τ2τ3τ4τ5τ Current (i) Time (t)
Decay Phase
\[ i(t) = I_0 e^{-t/\tau} \]
Stored energy is dissipated as heat in the resistor as the field collapses.
1τ2τ3τ4τ5τ Current (i) Time (t)
The 5τ Threshold
Elapsed Time Build-Up (%) Decay (%) 1τ 63.2% 36.8% 2τ 86.5% 13.5% 3τ 95.0% 5.0% 5τ ~100% ~0%
