Well, again
For checking your understanding please do the following exercises:
Every electrical circuit consists of three elements:
These elements will be considered in more detail below.
Given is an electrical conductor („consumer“) at a battery (see Abbildung 1)
Current flow generally requires an energy input first. This energy is at some point extracted from the electric circuit and is usually converted into heat. The reason for this conversion is the resistance e.g. of the conductor or other loads.
A resistor is an electrical component with two connections (or terminals). Components with two terminals are called two-terminal networks or one-port networks (Abbildung 3). Later in the semester, four-terminal networks will also be added.
In general, the cause-and-effect relationship is such that an applied voltage across the resistor produces the current flow. However, the reverse relationship also applies: as soon as an electric current flows across a resistor, a voltage drop is produced over the resistor. In electrical engineering, circuit diagrams use idealized components in a Lumped-element model. The resistance of the wires is either neglected - if it is very small compared to all other resistance values - or drawn as a separate „lumped“ resistor.
The values of the resistors are standardized in such a way, that there is a fixed number of different values between $1~\Omega$ and $10~\Omega$ or between $10~\rm k\Omega$ and $100~\rm k\Omega$. These ranges, which cover values up to the tenfold number, are called decades. In general, the resistors are ordered in the so-called E series of preferred numbers, like 6 values in a decade, which is named E6 (here: $1.0~\rm k\Omega$, $1.5~\rm k\Omega$, $2.2~\rm k\Omega$, $3.3~\rm k\Omega$, $4.7~\rm k\Omega$, $6.8~\rm k\Omega$). As higher the number (e.g. E24) more different values are available in a decade, and as more precise the given value is.
For larger resistors with wires, the value is coded by four to six colored bands (see conversion tool). For smaller resistors without wires, often numbers are printed onto the components (conversion tool)
Abb. 4: examples for a real 15kOhm resistor
Abb. 4: Linear resistors in the U-I diagram
Abb. 6: Non-linear resistors in the U-I diagram
The ideal voltage source generates a defined constant output voltage $U_\rm s$ (in German often $U_\rm q$ for Quellenspannung).
In order to maintain this voltage, it can supply any current.
The current-voltage characteristic also represents this (see Abbildung 7).
The circuit symbol shows a circle with two terminals. In the circuit, the two terminals are short-circuited.
Another circuit symbol shows the negative terminal of the voltage source as a „thick minus symbol“, the positive terminal is drawn wider.
The ideal current source produces a defined constant output current $I_\rm s$ (in German often $I_\rm q$ for Quellenstrom).
For this current to flow, any voltage can be applied to its terminals.
The current-voltage characteristic also represents this (see Abbildung 8).
The circuit symbol shows again a circle with two connections. This time the two connections are left open in the circle and a line is drawn perpendicular to them.
The value of the resistance can also be derived from the geometry of the resistor. For this purpose, an experiment can be carried out with resistors of different shapes. Thereby it can be stated:
Material | $\rho$ in ${{\Omega\cdot {{\rm mm}^2}}\over{{\rm m}}}$ |
---|---|
Silver | $1.59\cdot 10^{-2}$ |
Copper | $1.79\cdot 10^{-2}$ |
Gold | $2.2\cdot 10^{-2}$ |
Aluminium | $2.78\cdot 10^{-2}$ |
Lead | $2.1\cdot 10^{-1}$ |
Graphite | $8\cdot 10^{0}$ |
Battery Acid (Lead-acid Battery) | $1.5\cdot 10^4$ |
Blood | $1.6\cdot 10^{6}$ |
(Tap) Water | $2 \cdot 10^{7}$ |
Paper | $1\cdot 10^{15} ... 1\cdot 10^{17}$ |
Charge carriers are freely movable in the conductor.
Examples:
In semiconductors, charge carriers can be generated by heat and light irradiation. Often a small movement of electrons is already possible at room temperature.
Examples:
In the insulator, charge carriers are firmly bound to the atomic shells.
Examples:
The resistance value is - apart from the influences of geometry and material mentioned so far - also influenced by other effects. These are e.g.:
To summarize these influences in a formula, the mathematical tool of Taylor series is often used. This will be shown here practically for the thermoresistive effect. The thermoresistive effect, or the temperature dependence of the resistivity, is one of the most common influences in components.
The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. For this purpose, the resistance is measured using a voltage source, a voltmeter (voltage measuring device), and an ammeter (current measuring device), and the temperature is changed (Abbildung 9).
The result is a curve of the resistance $R$ versus the temperature $\vartheta$ as shown in Abbildung 10. As a first approximation is a linear progression around an operating point. This results in:
$R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$
Where:
We know that a movement of a charge across a potential difference corresponds to a change in energy. Charge transport therefore automatically means energy expenditure. Often, however, the energy expenditure per unit of time is of interest.
The energy expenditure per time unit represents the power:
$\boxed{P={{\Delta W}\over{\Delta t}}}$ with the unit $[P]={{[W]}\over{[t]}}=1~{\rm {J}\over{s}} = 1~{\rm {Nm}\over{s}} = 1 ~{\rm V\cdot A} = 1~{\rm W}$
For a constant power $P$ and an initial energy $W(t=0~{\rm s})=0$ holds:
$\boxed{W=P \cdot t}$
If the above restrictions do not apply, the generated/needed energy must be calculated via an integral.
Besides the current flow from the source to the consumer (and back), also power flows from the source to the consumer. In the following circuit, the color code shows the incoming and outgoing power.
If we only consider a DC circuit, the following energy is converted between the terminals (see also Abbildung 11 and Abbildung 12):
$W=U_{12}\cdot Q = U_{12} \cdot I \cdot t$
This gives the power (i.e. energy converted per unit time):
$\boxed{P=U_{12} \cdot I}$ with the unit $[P]= 1 ~{\rm V\cdot A} = 1~{\rm W} \quad$ … ${\rm W}$ here stands for the physical unit watts.
For ohmic resistors:
$\boxed{P=R\cdot I^2 = {{U_{12}^2}\over{R}}}$
For loads (passive convention): current enters the $+$ terminal; $P=U\cdot I>0$ → absorption. For sources (active convention): current exits the $+$ terminal; $P=U\cdot I>0$ → delivery.
Example 1 — Copper wire resistance. A copper lead ($\rho=0.0178~{\rm \Omega\,mm^2/m}$) is $l=2.0~{\rm m}$ long and $A=0.50~{\rm mm^2}$. \begin{align*} R &= \rho\,\frac{l}{A}=0.0178~{\rm \Omega\,mm^2/m}\cdot\frac{2.0~{\rm m}}{0.50~{\rm mm^2}}=0.0712~\Omega. \end{align*}
Example 2 — Power forms cross-check. A load takes $I=250~{\rm mA}$ at $U=12.0~{\rm V}$. \begin{align*} P&=U I=12.0~{\rm V}\cdot0.250~{\rm A}=3.00~{\rm W}. \\ R&=\frac{U}{I}=\frac{12.0~{\rm V}}{0.250~{\rm A}}=48.0~\Omega, \quad P=R I^2=48.0~\Omega\cdot(0.250~{\rm A})^2=3.00~{\rm W}. \end{align*}
Example 3 — Temperature coefficient. A metal resistor has $R_{20}=1.00~{\rm k\Omega}$ and $\alpha=3.9\cdot10^{-3}~{\rm K^{-1}}$. Find $R$ at $T=60^\circ{\rm C}$. \begin{align*} R(60^\circ{\rm C})&=1.00~{\rm k\Omega}\,[1+3.9\cdot10^{-3}\cdot(60-20)]\\ &=1.00~{\rm k\Omega}\,[1+0.156]=1.156~{\rm k\Omega}. \end{align*}
Example 4 — Choosing an E-series value. Need $\approx 3.3~{\rm k\Omega}$: nearest E12 is $3.3~{\rm k\Omega}$ (good match). For $\approx 3.0~{\rm k\Omega}$: E12 has no 3.0 → choose $3.0~{\rm k\Omega}$ from E24 or combine series/parallel.
A $R=220~\Omega$ resistor is across $U=9.0~{\rm V}$. Compute $I$ and $P$. Is a $0.25~{\rm W}$ part safe at $25^\circ{\rm C}$?
Assume that a soft pencil lead is $100 ~\%$ graphite. What is the resistance of a $5.0~{\rm cm}$ long and $0.20~{\rm mm}$ wide line if it has a height of $0.20~{\rm µm}$?
The resistivity is given by Tabelle 1.
Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. The cylindrical coil has an inner diameter of $d_{\rm i}=70~{\rm mm}$ and an outer diameter of $d_{\rm a} = 120~{\rm mm}$. The number of turns is $n_{\rm W}=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{\rm Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$.
First, calculate the wound wire length and then the ohmic resistance of the entire coil.
The power supply line to a consumer has to be replaced. Due to the application, the conductor resistance must remain the same.
Which wire cross-section $A_{\rm Cu}$ must be selected?
On the rotor of an asynchronous motor, the windings are designed in copper.
The length of the winding wire is $40~\rm{m}$.
The diameter is $0.4~\rm{mm}$.
When the motor is started, it is uniformly cooled down to the ambient temperature of $20~°\rm{C}$.
During operation the windings on the rotor have a temperature of $90~°\rm{C}$.
$\alpha_{Cu,20~°\rm{C}}=0.0039 ~\frac{1}{\rm{K}}$
$ \beta_{Cu,20~°\rm{C}}=0.6 \cdot 10^{-6} ~\frac{1}{\rm{K}^2}$
$ \rho_{Cu,20~°\rm{C}}=0.0178 ~\frac{\Omega \rm{mm}^2}{\rm{m}}$
Use both the linear and quadratic temperature coefficients! 1. determine the resistance of the wire for $T = 20~°\rm{C}$.
2. what is the increase in resistance $\Delta R$ between $20~\rm °C$ and $90~\rm °C$ for one winding?
You need a lead with $R\le 0.10~\Omega$ (round-trip) over $l=3.0~{\rm m}$ each way (total conductor length $6.0~{\rm m}$). Pick $A$ for copper ($\rho=0.0178~{\rm \Omega\,mm^2/m}$).
A resistor has $R_{20}=4.70~{\rm k\Omega}$, $\alpha=3.7\cdot10^{-3}~{\rm K^{-1}}$. Find $R$ at $0^\circ{\rm C}$ and at $70^\circ{\rm C}$.
The first 5:20 minutes is a recap of the fundamentals of calculating the electric power
An SMD resistor is used on a circuit board for current measurement. The resistance value should be $R=0.20~\Omega$, and the maximum power $P_M=250 ~\rm mW $. What is the maximum current that can be measured?
This can be rearranged into
\begin{align*} I = + \sqrt{ {{P} \over{R} } } \end{align*}
Explanation (video):
Good explanation of resistivity
Video explainer on $R(T)$.
Voltage vs Power vs Energy
Resistor packages
A nice 10-minute intro into power and efficiency (a cutout from 2:40 to 12:15 from a full video of EEVblog)