When a series RL circuit is connected across a supply, voltage and current transients occur until the current attains a steady-state condition. Consider the circuit figure 1, where R represents the coil’s resistance or an external resistance. When the switched is closed, current begins to flow into the inductance. The rate of change of current, roc i, will be greatest when the switched is closed. Current and voltage transients will be produced until the current reaches a steady-state level of E/R, at which time the coil effect will be negligible.

Fig.1: Circuit to illustrate RL Time Constant

The time required for the transient current to reach 63.2 % of its maximum value can be calculated by the following equation:

\[\tau =\frac{L}{R}\]

Where

τ= time constant in seconds

L= inductance in Henry

R=resistance in ohms

The time constant τ also represents the time required for the steady-state current to drop 63.2 % when the inductive circuit is opened.

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**Example**

Determine the time constant of the RL Circuit in figure 1 when the switch is closed.

**Solution**

\[\tau =\frac{L}{R}=\frac{1}{5}=0.2s\]

**Universal Time Constant Curve**

Because of the identical nature of the transient response in RL and RC circuits, a common graph may represent both, as in figure 2.

Fig.2: Universal Time Constant Curves for RC and RL Circuits

The X-axis represents time constants, and the Y-axis represents a percentage of full current or voltage. For RL circuits, curves A and B represents i_{L} and V_{L} respectively. For RC circuits, curves A and B represents V_{C} and i_{C} respectively. For both cases, the rise or fall of the curve changes by 63 % in one time constant.

With information obtained from the graph, it is possible to determine the voltage across a capacitor and its charge at any time constant, or fraction thereof, during the charging or discharging cycle. The same statement applies to the inductive circuit.

**Exponential Nature of Time Constant Curve**

The universal time constant graph is based on the following equation, which gives the exponential rise in a capacitive circuit and is derived from the calculus:

\[{{v}_{C}}=E(1-{{e}^{-\frac{t}{RC}}})\]

Where

E=supply voltage

-t/RC=ratio of time to RC time constant

Note: the value of –t/RC is the ratio of actual time of t to the RC time constant.

The following example illustrates the use of this above-mentioned equation when the time elapsed t is jut equal to one time constant. In other words, when t=RC=τ.

**Example**

Determine the rise of voltage across a capacitor in a series RC Circuit in one time constant.

**Solution**

For 1τ, -t/RC=-1. Therefore:

\[{{v}_{C}}=E(1-{{e}^{-\frac{t}{RC}}})=E(1-{{e}^{-1}})=0.6321E\]

The voltage across the capacitor will rise to 0.6321, or 63.21 % of the supply voltage in RC seconds, the value of 1 on the time constant curve. By plotting V_{C} for different time constants, we obtain the universal curve A of figure 2.

The charging current in a series RC Circuit can be calculated for any time constant with the following formula:

\[i={{\operatorname{I}}_{max}}{{e}^{-\frac{t}{RC}}}\]

This equation is the decreasing form of the exponential curve (curve B in figure 2).

**Example**

Calculate the value of capacitive current in a series RC Circuit in one time constant.

**Solution**

For 1τ, -t/RC=-1. Therefore:

\[i\text{=}{{I}_{\max }}{{e}^{-\frac{t}{RC}}}={{I}_{\max }}{{e}^{-1}}\text{=0}\text{.367}{{\text{I}}_{\text{max}}}\]

The charging current in an RC circuit will have dropped to 0.3679, or 36.8 % of its maximum E/R value in one time constant after charging begins. If different time constants plotted, curve B of figure 2 results.

For the series RL circuit, the following formula is used to calculate the inductive current at any instant:

\[i=\operatorname{I}(1-{{e}^{-\frac{tR}{L}}})\]

**Example**

Refer to figure 3, calculate i_{L} at a time 0.2 ms after the circuit is closed.

Fig.3: Series RL Circuit for the above example

** Solution**

First, find tR/L:

\[\frac{tR}{L}=\frac{0.2*{{10}^{-3}}*100}{20*{{10}^{-3}}}=-1\]

Now, calculate i_{L}:

\[i=\frac{10}{100}(1-{{e}^{-1}})=0.0632A\]

This value represents the current in the circuit after one time constant.

The percentages of the steady state values reached after the passage of some of the commonly used multiples of the time constants are tabulated in table 1. For example, for τ=0.5, the percentage is 40 %.

TIME CONSTANT ( τ ) | % ULTIMATE VALUE |
---|---|

0.1 | 10 |

0.5 | 40 |

0.7 | 50 |

0.9 | 60 |

1 | 63.2 |

2 | 86.5 |

3 | 95.1 |

4 | 98.2 |

5 | 99.3 |

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