Time

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Time (German: Zeit) is one of the seven base quantities and describes the duration of processes or the interval between events. In electrical engineering, it is fundamental to all time-dependent processes such as oscillations, switching times and charging and discharging processes.

1 Quantity symbol and unit

The quantity symbol of time is $t$.
Its unit (SI base unit) is the second, with the unit symbol $\mathrm{s}$.

$$[t]=\mathrm{s}$$

For time-dependent quantities, the notation $f(t)$ is often used, e.g. $i(t)$ for a time-dependent current or $u(t)$ for a time-dependent voltage.

Since 1967, the second has been defined by atomic time: 1 second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of caesium-133 atoms.

2 Practical and worked examples

2.1 RC time constant

The time constant $\tau$ of an RC circuit determines the speed of charging and discharging processes:

$$\tau =R\cdot C$$

Example: $R=10\mathrm{k\Omega }$, $C=100\mu \mathrm{F}$

$$\tau =10\cdot 10^3\mathrm{\Omega }\cdot 100\cdot 10^{-6}\mathrm{F}=\underline{1.0\mathrm{s}}$$

After $t=5\tau =5.0\mathrm{s}$, the capacitor is 99.3 % charged.

2.2 Period and frequency

For sinusoidal AC quantities, the period $T$ is the reciprocal of the frequency:

$$T=\frac{1}{f}$$

Example: mains frequency $f=50\mathrm{Hz}$

$$T=\frac{1}{50\mathrm{Hz}}=\underline{0.02\mathrm{s}}=20\mathrm{ms}$$

2.3 Switching time of a relay

A relay has a pull-in time of $t_{on}=15\mathrm{ms}$ and a release time of $t_{off}=8\mathrm{ms}$.
The minimum pulse duration for reliable switching: $t_{min}=t_{on}+t_{off}=15\mathrm{ms}+8\mathrm{ms}=\underline{23\mathrm{ms}}$

2.4 Capacitor discharge

A capacitor with capacitance $C$ discharges exponentially through a resistor $R$ according to:

$$u_C(t)=U_0\cdot e^{-\frac{t}{\tau }}$$

with $\tau =R\cdot C$

Example: $U_0=12\mathrm{V}$, $\tau =2.2\mathrm{s}$, required: voltage after $t=5.0\mathrm{s}$

$$u_C(5.0\mathrm{s})=12\mathrm{V}\cdot e^{-\frac{5.0}{2.2}}=12\mathrm{V}\cdot e^{-2.27}=\underline{1.24\mathrm{V}}$$

2.5 Operating time and energy consumption

A heating device with $P=2.0\mathrm{kW}$ is in operation for $t=3.5\mathrm{h}$:

$$W=P\cdot t=2.0\mathrm{kW}\cdot 3.5\mathrm{h}=\underline{7.0\mathrm{kWh}}$$

3 Typical orders of magnitude

Very short times:

• Lightning discharge: $t\approx 100\mu \mathrm{s}$
• Switching times of semiconductors: $t\approx 1$ to $100\mathrm{ns}$
• Processor clock: $t\approx 0.3$ to $3\mathrm{ns}$ (at 0.3 to 3 GHz)

Short times:

• Relay switching times: $t\approx 1$ to $50\mathrm{ms}$
• RC time constants: $t\approx 1\mathrm{ms}$ to $10\mathrm{s}$
• Mains period: $t=20\mathrm{ms}$ (at 50 Hz)

Longer times:

• Electric motor start-up time: $t\approx 1$ to $10\mathrm{s}$ (until the rated rotational frequency is reached)
• Transformer switch-on transient: $t\approx 0.1$ to $1\mathrm{s}$
• Battery charging time: $t\approx 1$ to $10\mathrm{h}$

4 Example phrasings

• ‘The time constant of the RC circuit is $\tau =2.2\mathrm{s}$.’
• ‘The relay switches reliably after $t=12\mathrm{ms}$.’
• ‘One mains period lasts $T=20\mathrm{ms}$.’
• ‘After $t=5\tau$, the capacitor is practically fully charged.’
• ‘The measurement was taken over a period of $t=24\mathrm{h}$.’

5 Related quantities

• Frequency as the reciprocal of the period:

$$f=\frac{1}{T}$$

• Angular frequency:

$$\omega =\frac{2\pi }{T}=2\pi f$$

• Velocity:

$$v=\frac{s}{t}$$

• Time constants:

$$\tau =R\cdot C\text{ or }\tau =\frac{L}{R}$$

• Energy and power:

$$W=P\cdot t$$

• Exponential functions in charging/discharging processes:

$$f(t)=f_0\cdot e^{-\frac{t}{\tau }}$$

6 Related components

• Capacitor: time constants in charging/discharging processes
• Inductor: switch-on and switch-off processes with L/R time constant
• Relays and contactors: pull-in and release times determine switching behaviour