---
title: RL Circuits
module: Electromagnetic Induction
moduleNumber: 8
lessonNumber: 8
order: 808
summary: >
  Put a resistor and an inductor in series and the current cannot switch on or off at
  will: it climbs to $V_0/R$ and falls away exponentially on a single time scale
  $\tau=L/R$ set by how much flux the coil hoards against how fast the resistor bleeds
  it. We solve the turn-on and turn-off, then confront the practical sting — because the
  coil's current refuses to stop instantly, breaking its path throws up a large voltage,
  which is why real inductive circuits carry freewheel diodes and clamps that trade
  voltage stress against how quickly the current dies.
topics: [Electromagnetic Induction]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 28 — Magnetic Induction; §28-8 RL Circuits; §28-6 Inductance"
---

## Series RL model and sign conventions

An RL transient contains a series resistance and an inductance whose current changes
after a switch event. The resistance includes an intentional resistor, winding
resistance, source resistance, wiring resistance, and any measured load that shares
the same series current. The inductance is represented by a constant $L$ only over a
stated current range, core state, and frequency range. The analysis below uses the
single-loop lumped-element model:

$$
V_s(t)-R_\Sigma I(t)-L\frac{\d I}{\d t}=0,
$$

where $R_\Sigma$ is the total series resistance and $V_s(t)$ is the signed source
voltage in the selected loop direction. The relation follows from the self-induced
emf relation in Tipler and Mosca, §28-6,

$$
\mathcal E_L=-L\frac{\d I}{\d t}.
$$

The sign of the induced emf follows the chosen current and loop traversal. A current
increase in the positive loop direction produces an inductor voltage that opposes
that increase. A current decrease reverses the inductor voltage polarity so that
the existing current direction is supported. One consistent reference direction
removes the apparent contradiction between these two cases.

$$
% caption: Series RL loop with one declared current direction. The source, resistor, and
% coil share a single loop current; the resistor and coil each drop a voltage set by that
% current and its rate of change, and the two drops sum to the source voltage.
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$$

The term $L\,\d I/\d t$ has units of volts,

$$
\left[\mathrm H\frac{\mathrm A}{\mathrm s}\right]
=\left[\frac{\mathrm{V\,s}}{\mathrm A}\frac{\mathrm A}{\mathrm s}\right]
=\mathrm V.
$$

Dimensional consistency distinguishes inductive voltage from the resistive drop
$R_\Sigma I$. A large inductance does not impose a voltage at steady current; it
sets the voltage required for a specified current-rate. A large resistance sets a
large voltage drop for a specified current.

The electrical variables are defined at terminals. A real coil can have winding
resistance in series with its inductance, parasitic capacitance across turns, and
core loss. The simple RL model gathers the winding resistance into $R_\Sigma$ and
neglects capacitance over the time interval of interest. A measured waveform with a
fast oscillation, a current-dependent decay rate, or a voltage limited by a clamp
signals that a wider circuit model is required.

## Current continuity and initial or final conditions

For finite $L$, a finite terminal voltage permits only the finite current-rate

$$
\frac{\d I}{\d t}=\frac{V_s-R_\Sigma I}{L}.
$$

An abrupt current jump would require an impulsive, unbounded inductor voltage in
the ideal model. Current is therefore continuous across an ordinary switch event:

$$
I(0^+)=I(0^-).
$$

The superscripts indicate the values immediately after and immediately before the
event. The statement concerns current through the inductive branch. A source branch
can open while another path maintains the inductor current, and a different branch
current can change discontinuously if it contains no inductance.

At long time after a constant source is connected, the current-rate tends to zero.
The steady current is

$$
I_\infty=\frac{V_0}{R_\Sigma}.
$$

The ideal inductance then has zero terminal voltage, while resistive elements
retain their ordinary $IR$ voltage drops. Calling the ideal inductor a short circuit
describes this steady-state terminal condition only. It does not erase stored energy,
winding resistance, or the transient voltage required to change current.

At a long time after an isolated resistive discharge begins, the current tends to
zero. The initial condition for that decay is the current present immediately before
the source path changes:

$$
I(0)=I_0.
$$

The number $I_0$ might equal a previous steady-state value, a measured current at
an arbitrary switch time, or a prescribed initial state. Substituting an assumed
$V_0/R_\Sigma$ without checking the prior circuit is a common error in multi-switch
problems.

The continuity relation gives a quick circuit reduction method. Determine the
inductor current before the event from the earlier circuit. Copy that value into the
new circuit at $0^+$. Use Kirchhoff's laws and the new connection to find other
branch currents and the instantaneous inductor voltage. Then solve the differential
equation only after the initial condition and final condition have both been checked.

## Step turn-on and the exponential current rise

A constant source $V_0$ connected at $t=0$ with $I(0)=0$ gives

$$
L\frac{\d I}{\d t}+R_\Sigma I=V_0.
$$

Separating variables gives

$$
\frac{\d I}{V_0-R_\Sigma I}=\frac{\d t}{L}.
$$

Integration from $0$ to $t$ produces

$$
I(t)=\frac{V_0}{R_\Sigma}
\left(1-e^{-R_\Sigma t/L}\right).
$$

Define the time constant

$$
\tau=\frac{L}{R_\Sigma}.
$$

The rise law becomes

$$
I(t)=I_\infty\left(1-e^{-t/\tau}\right).
$$

At one time constant, $I=(1-e^{-1})I_\infty\approx0.632I_\infty$. The remaining
current gap $I_\infty-I$ is multiplied by $e^{-1}$ during each additional interval
of length $\tau$. The curve approaches its asymptote continuously; it does not reach
the mathematical steady value at a finite time.

$$
% caption: Current rise after a constant-voltage step. The initial tangent has slope
% $V_0/L$, the curve reaches about 63 percent of its steady value at one time constant
% $\tau=L/R_\Sigma$, and the remaining gap to the steady current decays on that scale.
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$$

At switch closure, the loop equation gives

$$
\left.\frac{\d I}{\d t}\right|_{0^+}=\frac{V_0}{L}.
$$

Resistance has no effect on that slope when the initial current is zero because the
resistive drop is $R_\Sigma I=0$ at that instant. Resistance controls the later
approach through $\tau=L/R_\Sigma$ and the final current $V_0/R_\Sigma$. A measured
initial slope and final current therefore constrain $L$ and $R_\Sigma$ in different
ways.

### Voltage partition during turn-on

For the step response, substitute the current-rise solution into the resistor and
coil voltage expressions:

$$
V_R(t)=R_\Sigma I(t)
=V_0\left(1-e^{-t/\tau}\right),
$$

$$
V_L(t)=L\frac{\d I}{\d t}
=V_0e^{-t/\tau}.
$$

At $t=0^+$, the resistor voltage is zero and the ideal coil carries the full source
voltage. At large $t$, the resistor voltage approaches the source voltage and the
coil voltage approaches zero. These are signed terminal-voltage statements for the
chosen loop direction. Reversing voltage reference polarity reverses both reported
signs without changing the physical transient.

$$
% caption: Complementary resistor and coil voltages after a source step. Their magnitudes
% sum to the constant source voltage at every instant; the crossing occurs when each equals
% half the source voltage and the current has reached half its steady value.
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$$

Multiplication by current gives the instantaneous power balance

$$
V_0 I=I^2R_\Sigma+LI\frac{\d I}{\d t}.
$$

The first term on the right is resistive dissipation. The second term is the rate at
which energy enters the inductive state:

$$
\frac{\d U_L}{\d t}=LI\frac{\d I}{\d t}.
$$

During a monotonic current rise, both terms are nonnegative. Near the initial
switch event, current is small, so source power is also small despite a large coil
voltage. At late time, source power is almost entirely resistive dissipation. Power
and voltage should therefore be interpreted together; a high inductor voltage can
coexist with negligible instantaneous stored-energy rate when current is close to
zero.

$$
% caption: Power balance during current rise. Source power splits into resistor heating
% and the rate of change of stored inductive energy; the energy-rate term grows from zero,
% peaks at a middle time, and returns toward zero as the current settles.
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$$

The time required to reach a selected fraction $f$ of final current follows from

$$
f=1-e^{-t/\tau},
\qquad
t=-\tau\ln(1-f).
$$

For $f=0.90$, $t=2.30\tau$. For $f=0.99$, $t=4.61\tau$. These values describe a
first-order ideal step response. A current threshold measured by a comparator can
therefore estimate the time constant, provided source voltage, resistance, and
threshold calibration remain known over the test.

## Current decay after source removal

Current decay requires a closed path containing the inductance and resistance. With
the source disconnected and a discharge path of total resistance $R_D$,

$$
L\frac{\d I}{\d t}+R_D I=0.
$$

For initial current $I_0$,

$$
I(t)=I_0e^{-t/\tau_D},\qquad \tau_D=\frac{L}{R_D}.
$$

The current direction remains the original reference direction while its magnitude
falls. The inductor terminal voltage reverses polarity relative to the turn-on case
so that the loop current can continue through the resistor. The resistor voltage
also has the sign required to dissipate energy as heat.

$$
% caption: Source-free RL discharge loop. The coil current keeps its pre-switch value and
% circulates through the resistor; the coil terminal polarity reverses from the turn-on
% case so the current continues in the same direction while its magnitude decays.
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$$

The initial decay slope is

$$
\left.\frac{\d I}{\d t}\right|_{0^+}
=-\frac{R_D}{L}I_0.
$$

A larger discharge resistance gives a shorter electrical time constant and a larger
initial coil voltage magnitude

$$
|V_L(0^+)|=R_D I_0.
$$

Clamp voltage and release time form the central switch-protection trade-off. A low-resistance freewheel path reduces
switch voltage but prolongs the current and therefore prolongs magnetic actuator
release. A higher-voltage clamp shortens current decay but absorbs energy at a
higher voltage and needs an appropriate power and energy rating.

$$
% caption: Source-free current decay for two discharge resistances. Both start from the
% same current; the larger resistance gives a steeper fall and a shorter time constant,
% and each curve reaches about 37 percent of its start after one of its own time constants.
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$$

## Turn-off paths, freewheel diodes, and switch protection

At a switch-opening event, the inductor branch retains its pre-event current. The
post-switch circuit must provide a current path. Without an intended path, stray
capacitance, contact separation, semiconductor avalanche, or an air arc can carry
the current after the coil voltage rises to the corresponding conduction threshold.
The resulting terminal voltage follows the current-rate relation and can exceed the
source voltage by a large factor.

A freewheel diode is placed across a dc-driven coil with polarity chosen to be
reverse biased while the normal source current flows. When the controlling switch
opens, the coil reverses its terminal voltage and forward biases the diode. The
current then circulates through the coil, winding resistance, and diode rather than
through the open switch. The switch voltage is limited near the supply plus the
diode forward drop, subject to wiring inductance and layout.

$$
% caption: Low-side switched coil with a freewheel diode. While the switch conducts the
% diode is reverse biased and current flows from the supply down through the coil; when the
% switch opens the coil node rises, the diode conducts, and the coil current recirculates
% through the diode loop instead of the open switch.
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$$

A diode loop with approximately constant forward drop $V_D$ has active interval

$$
L\frac{\d I}{\d t}+R_D I+V_D=0
$$

while $I>0$. The constant diode term gives a shifted exponential rather than the
zero-drop RL curve. With $R_D>0$, the current reaches zero after

$$
t_{\rm off}
=\frac{L}{R_D}
\ln\left(1+\frac{R_DI_0}{V_D}\right).
$$

The limiting result for a negligible resistive drop is $t_{\rm off}=LI_0/V_D$.
An ideal zero-drop diode returns the ordinary exponential decay with $R_D$ equal to
the coil and loop resistance. A practical diode has a finite forward drop, giving a
larger initial decay magnitude and a finite turn-off endpoint.

Protection selection uses an event-level rating record.

- **Peak current:** start with $I_0$, including any residual current from a prior
  pulse. Coil current can greatly exceed the average source current in a pulsed
  actuator.
- **Voltage:** include diode reverse voltage, switch voltage, winding insulation, and
  lead-induced overshoot from $L_{\rm stray}\,\d I/\d t$.
- **Energy:** distribute $U_0=\tfrac12LI_0^2$ among winding resistance, diode, clamp,
  and any arc or parasitic path by integrating each $v_k I$ record.
- **Repetition:** compare the per-event energy with average dissipation
  $f_{\rm rep}E_k$ and the device thermal condition.
- **Layout:** place the intended path close to the coil or switch it protects. Long
  leads add loop inductance and can delay effective clamping at the protected terminals.

Higher-voltage suppression can use a transient-voltage suppressor, a zener-based
clamp, a resistor-capacitor network, or a controlled switch topology. A clamp with
approximately constant positive drop $V_C$ during decay has

$$
L\frac{\d I}{\d t}+R_D I+V_C=0.
$$

For $R_D>0$, the current during the active clamp is

$$
I(t)=\left(I_0+\frac{V_C}{R_D}\right)e^{-R_Dt/L}
-\frac{V_C}{R_D}.
$$

The expression applies until current reaches zero and the clamp stops conducting. The
corresponding active-clamp duration is

$$
t_{\rm clamp}
=\frac{L}{R_D}
\ln\left(1+\frac{R_DI_0}{V_C}\right).
$$

Increasing $V_C$ shortens the release time and raises the clamp voltage and
instantaneous dissipation. Clamp energy, switch rating, electromagnetic emissions, and
actuator-release requirement must be checked as one circuit condition.

Mechanical make-before-break switching establishes a low-resistance loop before
disconnecting the source path. It preserves current continuity without demanding
switch-contact arcing. Semiconductor switches need an explicit freewheel or clamp
network because their off-state path can otherwise experience the full inductive
voltage excursion. Switch timing, diode recovery, and source decoupling become
important when the event duration approaches wiring propagation or device switching
times.

### Stored current energy and resistive dissipation

The work required to establish current in an ideal inductive branch is stored in
the magnetic state associated with that current. With linear inductance, the stored
energy is

$$
U_L=\frac12LI^2.
$$

The formula is used here as the lumped circuit energy associated with an RL
transient. It does not require a magnetic-energy-density calculation. Its derivative
matches the coil power term:

$$
\frac{\d U_L}{\d t}=LI\frac{\d I}{\d t}.
$$

During source-free decay through a resistor $R_D$, the loop equation gives

$$
LI\frac{\d I}{\d t}=-I^2R_D.
$$

Stored energy decreases at the resistor-heating rate. Integration from
$I_0$ to zero yields

$$
\int_0^\infty I^2R_D\,\d t
=\frac12LI_0^2.
$$

The total heat in the discharge resistor equals initial stored energy in the ideal
model. A diode, clamp, winding resistance, arc, or other conducting element shares
that energy according to its instantaneous voltage and current. Protection design
therefore needs an energy budget in addition to a peak-current and peak-voltage
budget.

$$
% caption: Stored inductive energy during a source-free decay. Because energy is
% proportional to current squared, it falls at twice the current rate; after one current
% time constant only about 13.5 percent of the initial energy remains.
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$$

Resistive power during an exponential discharge is

$$
P_R(t)=I_0^2R_D e^{-2t/\tau_D}.
$$

Its integral has a finite value even though the mathematical current trace extends
to arbitrarily large time. Practical calculations use a finite cutoff based on a
current threshold, sensor noise, or required stored-energy fraction. At five time
constants, current is about $0.67\%$ of $I_0$ and stored energy is about
$0.0045\%$ of its initial value.

An inductor initially carrying current can deliver energy at a voltage above the
source voltage that charged it. The voltage follows from the current-rate and
available discharge path; the total available energy remains $\tfrac12LI_0^2$.
High voltage and high energy are separate hazards. A low-inductance coil with large
current can store substantial energy, while a high-inductance sensor coil with small
current can create a damaging high voltage if current interruption is forced into a
very short time.

## Variable sources, resistance changes, and model range

The exponential formulas assume constant $V_s$, constant $R_\Sigma$, and constant
$L$ during the interval. With a prescribed time-varying source and constant
resistance, the solution is

$$
I(t)=I(0)e^{-t/\tau}
+\frac{1}{L}\int_0^t e^{-(t-t')/\tau}V_s(t')\,\d t'.
$$

The integral weights recent source history more strongly than distant history. A
short voltage pulse can raise current only by the amount permitted by its duration
and the inductance. A source waveform that changes slowly compared with $\tau$
produces a current that follows the changing resistive value with a delay set by the
same time scale.

If resistance changes abruptly from $R_1$ to $R_2$ while the inductance remains
linear, current remains continuous and a new time constant applies after the event:

$$
\tau_1=\frac{L}{R_1},
\qquad
\tau_2=\frac{L}{R_2}.
$$

The current at the resistance-change instant becomes the initial condition for the
new interval. A temperature rise in a copper winding usually changes resistance
gradually rather than abruptly. The measured transient then departs slightly from a
single exponential because both $R_\Sigma(t)$ and final current change during the
test.

Current-dependent inductance, core saturation, nonlinear clamp voltage, and
parasitic capacitance change the first-order model. Saturation can reduce effective
inductance at high current, increasing current slope beyond a constant-$L$
prediction. A capacitance across a switching node can exchange energy with the
inductance and create a ring-down rather than a monotonic current trace. A data set
with a changing log-slope requires a model extension. Fit separate intervals only when
the corresponding circuit topology or parameter change has been identified.

## Measuring RL transients and estimating parameters

Current is often measured through a calibrated series shunt resistor. If the shunt
resistance $R_s$ is small enough that it does not materially change the intended
circuit, its voltage gives

$$
I(t)=\frac{V_s^{\rm shunt}(t)}{R_s}.
$$

The shunt contributes to the total series resistance,

$$
R_\Sigma=R_{\rm load}+R_{\rm coil}+R_s+R_{\rm source}+R_{\rm leads}.
$$

Ignoring a current-sense resistor biases both the predicted final current and the
time constant. A shunt with too small a voltage signal suffers from digitizer noise;
a shunt with too large a value changes the transient being measured. Select it from
the expected current range, allowable voltage loss, pulse energy, bandwidth, and
calibration uncertainty.

Measure coil voltage with a known polarity reference. In a low-side switched
circuit, a grounded single-ended instrument may measure the shunt safely while a
differential input is needed across a floating coil. The measurement leads add
capacitance and inductance; long ground leads can show a switching-loop voltage
that differs from the local component terminal voltage. Record the connection
points in the experimental diagram instead of assigning every visible spike to the
coil itself.

The timebase must resolve the shortest relevant interval. A data interval much
larger than $\tau/20$ cannot resolve the initial slope. A bandwidth limit can round a fast
voltage edge and make a clamp appear slower or lower than it is. Triggering on the
switch command gives a repeatable nominal event time, but actual coil current may
begin changing after gate delay, relay contact motion, or source slew. Measure this
offset when comparing a physical trace with an ideal $t=0$ solution.

The time constant can be estimated from a calibrated rise trace. When $I_\infty$ is
known independently,

$$
\ln\left(1-\frac{I(t)}{I_\infty}\right)=-\frac{t}{\tau}.
$$

A plot of the logarithmic remaining fraction versus time has slope $-1/\tau$ in
the first-order range. Near final current, the remaining fraction is small and the
logarithm amplifies voltage noise. A nonlinear least-squares estimate of the
untransformed current trace can use data across the measured range and can include an
unknown timing offset, initial current, or final current.

The threshold method applies when a comparator identifies one current value. If
the measured threshold current is $I_{\rm th}$ at time $t_{\rm th}$ during a
turn-on, then

$$
\tau=-\frac{t_{\rm th}}
{\ln\left(1-I_{\rm th}/I_\infty\right)}.
$$

Threshold uncertainty enters strongly when $I_{\rm th}$ lies too close to
$I_\infty$, because the logarithm denominator grows sensitive to small current
error. A threshold near $0.5I_\infty$ gives a practical compromise between early
timing resolution and late-time sensitivity for many instruments.

Resistance should be measured at the temperature and current condition relevant to
the transient. A two-wire resistance measurement includes lead resistance. A
four-terminal method separates the current-carrying leads from voltage-sensing
leads and reduces that error for a low-resistance coil or shunt. Current can heat a
winding during repeated pulses, so a room-temperature resistance measurement may
predict a time constant that differs from the warmed operating value.

An uncertainty budget identifies the measurement that limits an inferred
inductance. From $L=\tau R_\Sigma$, small independent relative uncertainties combine
approximately as

$$
\left(\frac{u_L}{L}\right)^2
\approx
\left(\frac{u_\tau}{\tau}\right)^2
\;+\;
\left(\frac{u_R}{R_\Sigma}\right)^2.
$$

Timing resolution, current-scale calibration, source drift, resistance temperature
coefficient, and model residuals can contribute to $u_\tau$ or $u_R$. Repeated
transients estimate random scatter. Systematic timing delay or unmeasured coil
resistance requires a model correction rather than more repetitions.

### Selecting current, time scale, and protection requirements

The three quantities $V_0$, $R_\Sigma$, and $L$ determine the basic dc step
response:

$$
I_\infty=\frac{V_0}{R_\Sigma},
\qquad
\tau=\frac{L}{R_\Sigma},
\qquad
\left.\frac{\d I}{\d t}\right|_{0^+}=\frac{V_0}{L}.
$$

Specifying any two response targets constrains the third. A desired final current
sets total resistance for a given source. A desired time constant then sets
inductance. The initial current slope follows automatically. This relation checks
component data: values claimed to give a high final current, long time
constant, and high initial slope from a small source voltage may be mutually
inconsistent.

The final steady resistor dissipation is

$$
P_{\infty}=I_\infty^2R_\Sigma=\frac{V_0^2}{R_\Sigma}.
$$

After electrical current has settled, winding power becomes heat. A
short electrical time constant does not ensure a short thermal time constant. A coil
can reach its electrical steady current in milliseconds while its temperature changes
over seconds or minutes. Repeated pulses add average heating according to duty
cycle, conduction resistance, and cooling conditions. Temperature-dependent winding
resistance then feeds back into final current and electrical time constant.

An isolated rectangular on-pulse of duration $T_{\rm on}$ starting at zero current
has switch-off current

$$
I_{\rm off}=I_\infty\left(1-e^{-T_{\rm on}/\tau}\right).
$$

The pulse current approaches the dc final current only when the on-time spans several time constants.
An actuator with a specified threshold time requires evaluation of $I(T_{\rm on})$.
The dc coil label applies to a different operating interval.

Protection selection begins with the initial current and stored energy at turn-off.

$$
I_0=I(T_{\rm off}^-),
\qquad
U_0=\frac12LI_0^2.
$$

A diode path requires initial current, diode forward current, winding resistance, and
expected release time. A clamp path requires maximum switch voltage, clamp voltage
tolerance, current waveform, energy per event, event
rate, and clamp cooling. A component rated for an isolated pulse may overheat under
many repeated events even when each pulse remains below its single-event energy
limit.

The source also needs a transient current specification. At switch turn-on, source
current may rise slowly because the inductance limits it. During turn-off, a local
freewheel loop can return coil energy without drawing source current. A supply
measurement taken only at the source terminals can therefore miss a large
recirculating coil current. Measure the current in the loop of interest.

The ideal first-order equations do not choose a protection device or establish a
safe switch rating by themselves. Device data must cover maximum repetitive reverse
voltage, forward current, avalanche behavior where applicable, thermal resistance,
and package temperature. Winding insulation and connector spacing impose further
voltage limits. A protection network should be tested at the actual cable length,
load current, supply range, and switching rate because those parameters determine
the stray inductance and energy present at the protected node.

## Worked Examples and Validation

> **Worked example (Constant-voltage turn-on).** A coil circuit has
> $L=40.0\ \mathrm{mH}$, $R_\Sigma=8.00\ \mathrm{\Omega}$, and a $24.0\ \mathrm V$
> source. Convert and identify the quantities before substitution:
>
> $$
> L=4.00\times10^{-2}\ \mathrm H,
> \qquad
> I_\infty=\frac{24.0}{8.00}=3.00\ \mathrm A,
> \qquad
> \tau=\frac{4.00\times10^{-2}}{8.00}
> =5.00\ \mathrm{ms}.
> $$
>
> The initial current slope is
>
> $$
> \left.\frac{\d I}{\d t}\right|_{0^+}
> =\frac{24.0}{4.00\times10^{-2}}
> =600\ \mathrm{A\,s^{-1}}.
> $$
>
> At $t=7.50\ \mathrm{ms}=1.50\tau$,
>
> $$
> I=3.00\left(1-e^{-1.50}\right)
> =2.33\ \mathrm A,
> $$
>
> and the resistor and ideal-coil voltages are
>
> $$
> V_R=R_\Sigma I=(8.00)(2.33)=18.6\ \mathrm V,
> \qquad
> V_L=V_0e^{-1.50}=5.36\ \mathrm V,
> $$
>
> which sum to $24.0\ \mathrm V$ within rounding. The magnetic energy at the steady
> current is
>
> $$
> U_L=\frac12(4.00\times10^{-2})(3.00)^2
> =0.180\ \mathrm J.
> $$
>
> After $7.50\ \mathrm{ms}$ the current is still below its final value even though the
> resistor voltage already exceeds the coil voltage: the two are equal at half the final
> current, so a larger resistor drop means more than half the source voltage has shifted
> into resistive loss.

$$
% caption: Values from the worked turn-on example. The steady current $3.00\ \mathrm A$ is
% set by the total series resistance, the marked sample at $1.5\tau$ gives $2.33\ \mathrm A$,
% and the initial tangent slope is fixed by source voltage and inductance.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.60,0.52)--(6.30,0.52) node[right] {time};
\draw[->,black] (0.60,0.52)--(0.60,3.10) node[above] {current};
\draw[black,dashed] (0.60,2.72)--(5.92,2.72);
\draw[acc,very thick] plot[domain=0.72:5.92,samples=120]
  (\x,{0.60+2.12*(1-exp(-(\x-0.72)/1.10))});
\draw[black,dashed] (2.37,0.52)--(2.37,2.25);
\filldraw[draw=black,fill=black!8] (2.37,2.25) circle (2pt);
\node[above] at (5.20,2.72) {3 ampere steady};
\node[below] at (2.37,0.50) {7.5 millisecond sample};
\node[right] at (2.47,2.20) {2.33 ampere};
\end{tikzpicture}
$$

> **Worked example (Protected decay: diode versus clamp).** The same coil carries
> $I_0=3.00\ \mathrm A$ when its source is removed. A freewheel path with
> $R_D=8.00\ \mathrm\Omega$ gives $\tau_D=L/R_D=5.00\ \mathrm{ms}$, so at
> $t=5.00\ \mathrm{ms}$,
>
> $$
> I=3.00e^{-1}=1.10\ \mathrm A,
> \qquad
> U_L=\frac12(4.00\times10^{-2})(1.10)^2
> =2.44\times10^{-2}\ \mathrm J.
> $$
>
> In the ideal zero-drop-diode approximation the resistor path has received
> $0.180-0.0244=0.156\ \mathrm J$ by that time. The current has fallen to $36.8\%$ of
> its initial value while the energy has fallen to $13.5\%$, because energy depends on
> current squared.
>
> Replace the diode with a $36.0\ \mathrm V$ constant clamp in series with the same
> $8.00\ \mathrm\Omega$ resistance. The current now reaches zero at
>
> $$
> t_{\rm zero}
> =\frac{L}{R_D}\ln\left(1+\frac{R_DI_0}{V_C}\right)
> =(5.00\ \mathrm{ms})\ln\left(1+\frac{24.0}{36.0}\right)
> =2.55\ \mathrm{ms}.
> $$
>
> The faster clamp demands a higher turn-off-loop voltage: at the first instant the
> coil voltage magnitude is $R_DI_0+V_C=60.0\ \mathrm V$. Circuit polarity and switch
> placement set the corresponding device-terminal voltage, which must be read off the
> actual schematic rather than copied from the coil voltage alone.

$$
% caption: Decay comparison for the worked coil. The low-voltage freewheel path holds
% current for one $5\ \mathrm{ms}$ time constant, while the $36\ \mathrm V$ clamp drives the
% current to zero in about $2.55\ \mathrm{ms}$ by accepting a higher loop voltage.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.60,0.52)--(6.30,0.52) node[right] {time};
\draw[->,black] (0.60,0.52)--(0.60,3.10) node[above] {current};
\draw[acc,very thick] plot[domain=0.72:5.92,samples=120]
  (\x,{0.52+2.20*exp(-(\x-0.72)/3.10)});
\draw[black,very thick] plot[domain=0.72:2.36,samples=90]
  (\x,{0.52+2.20*(exp(-0.75*(\x-0.72))-exp(-0.75*1.64))/(1-exp(-0.75*1.64))});
\filldraw[draw=black,fill=black!8] (2.36,0.52) circle (1.6pt);
\node[acc] at (4.44,1.55) {low-voltage path};
\node[black] at (2.75,0.92) {36 volt clamp};
\end{tikzpicture}
$$

> **Worked example (Inductance from a transient measurement).** A calibrated test
> gives $\tau=(5.20\pm0.10)\ \mathrm{ms}$ and
> $R_\Sigma=(7.98\pm0.08)\ \mathrm\Omega$. The central inductance is
>
> $$
> L=\tau R_\Sigma
> =(5.20\times10^{-3})(7.98)
> =4.15\times10^{-2}\ \mathrm H,
> $$
>
> and the approximate relative standard uncertainty is
>
> $$
> \frac{u_L}{L}
> =\sqrt{\left(\frac{0.10}{5.20}\right)^2
> +\left(\frac{0.08}{7.98}\right)^2}
> =0.0216,
> $$
>
> so
>
> $$
> L=(41.5\pm0.9)\ \mathrm{mH}
> $$
>
> under the stated model and calibration. The time-constant uncertainty dominates here;
> improving only the resistance measurement yields little until trigger timing, current
> calibration, and model residuals are reduced.

### Validation, nonideal behavior, and model boundaries

The first-order RL equation requires a specific physical regime.

$$
L\frac{\d I}{\d t}+R_\Sigma I=V_s.
$$

It assumes a single branch current, an approximately constant inductance, a known
series resistance, and negligible capacitance over the observed interval. Experimental
validation compares more than one feature of a trace. The initial slope checks
$V_0/L$. The final current checks $V_0/R_\Sigma$. The curved middle region checks
the time constant $L/R_\Sigma$. Agreement at one endpoint cannot establish the
entire model.

Residuals should be plotted against time after a model parameter estimate. Random
scatter around zero can be consistent with sensor noise. A systematic early-time
deviation can indicate switch delay, finite source slew, or measurement bandwidth.
A systematic late-time deviation can indicate source droop, winding heating, or an
incorrect resistance value. Oscillatory residuals suggest parasitic capacitance or
an external resonant path, which lies outside the monotonic RL approximation.

Source resistance changes the effective source voltage as current rises. A simple
Thevenin source with open-circuit voltage $V_{\rm th}$ and internal resistance
$R_{\rm th}$ gives

$$
V_s=V_{\rm th}-I R_{\rm th}.
$$

The RL equation still has the same form when $R_{\rm th}$ is included in
$R_\Sigma$, provided $V_{\rm th}$ remains constant. A current-limited laboratory
supply or a battery near depletion can change its output law during the transient.
Then the source is neither a constant voltage nor a fixed internal resistance, and
its measured terminal waveform belongs in the model.

Inductance can change with current in a magnetic core. A local differential
inductance is defined by

$$
L_{\rm diff}=\frac{\d \Lambda}{\d I}.
$$

When $L_{\rm diff}$ decreases at high current, the trace rises faster than a
constant-inductance prediction near the upper part of the pulse. A secant
inductance inferred from a full excursion can differ from the small-signal
inductance inferred from the initial slope. State the current interval used in any
reported inductance value.

The winding resistance can also vary with temperature. Copper resistance increases
with temperature, lowering final current and usually shortening the electrical time
constant if inductance remains nearly unchanged. Repeated-current tests should
report pulse spacing, coil temperature or resistance before each run, and whether
the trace is a first pulse or a warmed steady sequence.

Parasitic capacitance becomes visible when switch voltage changes rapidly. It can
temporarily carry current, interact with circuit inductance, and produce ringing.
The initial RL current-continuity rule still applies to the inductor branch, but
the measured switch-node current can contain capacitive current as well. A narrow
voltage ringing trace is not evidence that the coil current itself oscillates by the
same fraction. Separate current and voltage measurements are needed before assigning
the oscillation to a physical branch.

Circuit layout controls a portion of the observed turn-off voltage. The protection
element, switch, and coil form a current loop. Loop area and lead length add stray
inductance, causing a voltage $L_{\rm stray}\,\d I/\d t$ before a remote clamp can
fully act. Place the intended current path close to the switched load, route sense
leads separately from high-current paths, and record where terminal voltages were
measured. These are electrical topology requirements, not cosmetic drawing choices.

### Reporting an RL transient

A reproducible transient report identifies the circuit state before each event, the
source connection after the event, every intentional and measured resistance, the
coil temperature, protection topology, voltage reference polarity, current
reference direction, instrument bandwidth, sample interval, trigger source, and
calibration date. It also reports whether $L$ is an initial-slope estimate, a
time-constant estimate, or a value extracted over a stated current interval.

Use force and energy ratings appropriate to the actual turn-off state. Report
$I_0$, $U_0$, peak switch voltage, protection-element current, and event repetition
rate together. A device can meet the steady current rating while exceeding its
repetitive clamp-energy rating, or meet a voltage rating while producing an actuator
release time outside the requirement. The relevant checks are tied by the same
current trace.

The checklist below connects the model with the measurement.

- **Initial state:** obtain $I(0^-)$ from the preceding circuit rather than from the
  new source schematic.
- **Loop equation:** include every series resistance in the active current path and
  declare voltage polarities before assigning signs.
- **Time scale:** compare sampled rise or decay with $L/R$ using the resistance of
  the active interval.
- **Protection state:** identify whether a diode, clamp, arc, or open circuit carries
  current after switch-off.
- **Model test:** compare initial slope, final current, and intermediate residuals;
  expand the model when one parameter set cannot describe all three.

The record specifies the current path and the conditions under which the exponential
approximation is evaluated.

## Network switching and piecewise initial conditions

The RL branch can sit inside a larger resistive network. Replace the rest of the
network, as seen from the two inductor terminals after a specified switch event, by
its Thevenin equivalent $V_{\rm th}$ in series with $R_{\rm th}$. The branch equation
is then

$$
L\frac{\d I_L}{\d t}+R_{\rm th}I_L=V_{\rm th}.
$$

A constant equivalent source drives current from its initial value $I_0$ toward the
new final value

$$
I_\infty=\frac{V_{\rm th}}{R_{\rm th}}
$$

according to

$$
I_L(t)=I_\infty+(I_0-I_\infty)e^{-tR_{\rm th}/L}.
$$

The usual zero-to-final rise and initial-to-zero decay are special cases. This form
handles a current that rises toward a lower target, decays toward a nonzero target,
or reverses direction after sufficient time. It also makes the active resistance
explicit: use the resistance of the post-switch circuit that carries the inductive
branch current.

The inductor current condition belongs to the branch, while other network currents
follow the instantaneous resistor and source constraints. Immediately after a
switch closes, a parallel resistor current can change from zero to a finite value
while the inductive branch current preserves its earlier value. At long time under
dc conditions, the ideal coil terminal voltage approaches zero, changing how
parallel paths share current. Junction equations must be applied separately at
$0^+$ and at the long-time limit.

An ideal-inductor replacement applies only at a stated endpoint.

- **At $0^+$:** retain the measured or calculated current from $0^-$; a branch
  initially carrying zero current has zero branch current at that instant.
- **During the transient:** retain the differential relation
  $V_L=L\,\d I_L/\d t$ with the declared polarity.
- **At a dc long-time limit:** set $\d I_L/\d t=0$; the ideal inductive terminal
  voltage is zero while any winding resistance remains in series.

Applying the dc long-time replacement at $0^+$ discards the current-continuity
condition. Applying the zero-current turn-on condition after a pre-energized coil
discards the actual stored state. A switching diagram should identify the interval
being simplified before any circuit reduction is made.

The Thevenin reduction assumes linear resistive elements outside the coil. A diode
that changes conduction state, a current-limited source, or a switching clamp yields
different equivalents in different intervals. Rebuild the equivalent circuit after
each topology or conduction-state change. The initial coil current is carried across
the boundary; source and resistor branch currents are recalculated from the new
network.

### Event-by-event verification and safe interpretation

Inductor sign errors often arise from mixing two voltage conventions. Under the
passive sign convention, define $v_L$ as voltage at the terminal where positive
reference current enters. The constitutive relation is

$$
v_L=L\frac{\d I}{\d t}.
$$

The induced emf around a selected loop has the opposite sign,

$$
\mathcal E_L=-L\frac{\d I}{\d t}.
$$

Both statements describe the same physical polarity. The loop equation for a source,
resistor, and inductor written in the current direction is

$$
V_s-R_\Sigma I-v_L=0.
$$

Writing $-L\,\d I/\d t$ for an induced emf and also subtracting a passive inductor
voltage double-counts the sign. Draw terminal polarity marks and current reference
arrows before applying either form.

Power signs provide an independent check. With passive reference directions,

$$
p_R=I^2R_\Sigma\ge0,
\qquad
p_L=v_LI=LI\frac{\d I}{\d t}.
$$

During turn-on, $p_L$ is positive and energy enters the inductive state. During a
source-free decay, $p_L$ is negative and the branch transfers power to resistance
and protection elements. The sign change comes from $\d I/\d t$, not from a reversal of
the current reference direction. A numerical solution that shows resistor power
negative in a passive resistor branch has a polarity or current-reference mistake.

An event record consists of a compact set of physical conditions.

- **Before the event:** state coil current, source connection, coil temperature,
  and which protection components are nonconducting.
- **At $0^+$:** copy inductive-branch current from $0^-$, solve the new network for
  terminal voltages, and verify that all ideal contact and diode assumptions have
  physically valid signs.
- **During the interval:** use the resistance and clamp state of the active current
  loop. The source path may differ from the discharge path.
- **At the next event:** use the current reached at the prior interval endpoint as
  the next initial value, then rebuild the circuit after the topology change.

Contact bounce can insert several short intervals into a nominally single switching
event. A relay contact that opens, closes, and opens again can move current between
a source loop, an arc, and a protection loop. A sampled voltage trace may contain
several switch edges while the coil current evolves continuously through each
permitted path. Event timing should be determined from the measured waveform rather
than from a controller command alone when release voltage or peak device stress is
being assessed.

Wiring resistance and source resistance belong to the active loop, while remote
measurement leads may sample another pair of points. A coil voltage calculated from
$V_s-R_\Sigma I$ refers to the terminals represented by $R_\Sigma$. A voltage
measured at a remote supply connector can include lead drop and ground offset. A
clear schematic distinguishes component terminals, source terminals, switch
terminals, and measurement reference terminals.

The RL model also has a current range. A magnetic core can heat or change state
during a pulse. A protection diode has a current-dependent forward voltage. A
transient suppressor has a voltage-current curve rather than one exact clamp value.
The first-order equation forms a baseline and a way to organize measurements; it
does not replace component characterization at the actual operating conditions.

Safety checks follow directly from the current path. Before opening a coil circuit,
identify the intended post-switch loop and the maximum voltage allowed at the
switch, winding, connector, and measurement input. Estimate stored energy from
$\tfrac12LI_0^2$, then compare it with repeated-pulse capability of the diode or
clamp. Keep conductive tools and measurement leads clear of the switching loop
while high current is present. A visible arc indicates that the intended protection
path did not control the event.

These checks tie the mathematical transient to an actual circuit connection.
They also separate an expected exponential discharge from a voltage-limited,
contact-limited, or measurement-limited waveform.

### Limits of the lumped RL representation

The parameter $L$ is a measured or otherwise supplied branch
property. Its value can be obtained from a separate self-inductance or solenoid
geometry analysis, but that geometry is outside the transient reduction. The RL
equation uses the resulting inductance to predict current-rate. It does not infer
winding turn count, core permeability, or magnetic-energy density from a transient
trace without further material and geometric information.

The one-branch model excludes mutual coupling to another coil. A neighboring
current-carrying circuit can add a voltage term associated with its changing current,
altering the measured branch response. It also excludes intentional capacitance.
When a capacitor, cable capacitance, or semiconductor junction capacitance stores
enough energy to influence the trace, the resulting model contains additional state
variables and cannot be reduced to one RL time constant across the full event.

An ideal zero-resistance inductor would retain current indefinitely in an isolated
loop. Ordinary winding resistance, contacts, core loss, and protection paths provide
real dissipation and give a finite decay time. Superconducting behavior, strongly
nonlinear magnetic media, and high-frequency distributed transmission effects need
their own constitutive and circuit descriptions. The checks above identify
when a measured transient has crossed one of those boundaries.

Within a stated interval, test the model with units and limiting values before using a
numerical curve. The time constant $L/R_\Sigma$ must have seconds as its unit. A
zero initial current under a finite source gives initial slope $V_0/L$. A long
constant-voltage interval gives current $V_0/R_\Sigma$. A source-free resistive
interval decreases stored current energy rather than creating it. These checks
connect the algebraic solution to a physically consistent current path and expose
wrong resistance values, reversed voltage references, and omitted protection
components early in a calculation.

Keep the physical event duration distinct from the formal exponential tail. A
release mechanism may have a current threshold, a data recorder may have a noise
floor, and a protection diode may cease conduction at zero current. Each establishes
a finite endpoint for an engineering calculation. State that endpoint and the
associated current or energy threshold when reporting turn-off time.