Measuring current and voltage
1 Voltage measurement in a direct current circuit
The electrical voltage that drops across a component or across an interconnection of several components can be determined using a voltage measuring device (also called a voltmeter). To do this, the voltmeter must be connected in parallel to this component. It is usually not necessary to interrupt the circuit to do this.
Key point: Internal resistance of ideal voltmeter
An ideal voltmeter (see Figure 1) has an infinitely high internal resistance.
Consequently, no current „flows past the load resistor“ through the measuring device, and the voltage measurement is performed completely without feedback.
A real voltmeter has a finite internal resistance \(R_\mathrm {iV}\), which is connected in parallel to the voltmeter (see Figure 1). The current flowing through this internal resistance distorts the measurement result, which is why it should be as high as possible. In modern, electronic voltage measuring devices, it is usually in the range of \(10^6 \, \Omega \) to \(10^9 \, \Omega \).
The voltage \(U_\mathrm {corr}\), corrected for the measurement error caused by the internal resistance \(R_\mathrm {iV}\), can be calculated as follows, given the internal resistance of the voltage source \(R_\mathrm {i}\):
\begin {equation*} U_{\mathrm {korr}}= U\cdot \left (1+\frac {R_{\mathrm {i}}||R_{\mathrm {a}}}{R_{\mathrm {iV}}}\right ) \end {equation*}
2 Current measurement in a direct current circuit
The current \(I\) flowing through a component can be determined using a current measuring device (also called an ammeter). In contrast to voltage measurement, the ammeter must be connected in series to the component. This requires an interruption of the circuit.
Key point: INominal resistance ideal ammeter
An ideal ammeter (see Figure 2) has no internal resistance.
Consequently, the current can flow through it without any feedback.
A real ammeter has an internal resistance \(R_\mathrm {iA}\) connected in series (Figure 4). Here, too, the measurement result is influenced by this internal resistance. Depending on the measuring range, this typically ranges from a few \(\mu \Omega \) to less than \(\mathrm {m} \Omega \) in electronic ammeters.
Knowing the internal resistances, the measurement error can also be calculated for the ammeter. The corrected current \(I_\mathrm {corr}\), for which the influence of the measuring device is calculated out, can be determined using the following formula:
\begin {equation*} I_{\mathrm {korr}}= I\cdot \left (1+\frac {R_{\mathrm {iA}}}{R_{\mathrm {i}}+R_{\mathrm {a}}}\right ) \end {equation*}
3 Correct measurement of current and voltage
If both the voltage and the current are to be measured in a circuit, the internal resistances of the actual measuring instruments influence each other. Depending on the quality of the available measuring instruments and the prioritisation of the measured variables among themselves, a distinction is made between current-correct and voltage-correct measurement.
3.1 Correct measurement of electricity
In current-correct measurement (Figure 5), also known as voltage error circuitry, greater emphasis is placed on accurately determining the current. In this circuit, the ammeter displays the current flowing through the load resistor, but the voltmeter measures the sum of the voltage drops across the ammeter and the load resistor:
\begin {equation*} U_\mathrm {meas} = U + U_\mathrm {Amperemeter} \end {equation*}
The voltage fault circuit tends to be suitable for circuits with relatively high load resistances. The following inequality is often used as a rule of thumb for the use of this measuring circuit: \begin {equation*} R_{\mathrm {iA}} \ll R_{\mathrm {a}} \end {equation*}
3.2 Correct voltage measurement
In contrast to the current-correct measurement circuit, the voltmeter in the voltage-correct measurement circuit measures the correct voltage drop across the resistor (see Figure 3.2). The ammeter, on the other hand, measures the current flowing through the parallel connection of the voltmeter and load resistor. This measurement arrangement is recommended if greater emphasis is to be placed on the correct measurement of the voltage, or if the following applies to the resistors: \begin {equation*} R_{\mathrm {iV}} \gg R_{\mathrm {a}} \end {equation*}
If it is unclear whether current-correct or voltage-correct measurement should be used, the following rule of thumb can be applied:
Key point: Rule of thumb for correct current/voltage measurement
- Correct current measurement: \(R_{\mathrm {a}} > \sqrt {R_{\mathrm {iA}} \cdot R_{\mathrm {iV}}}\)
- Correct voltage measurement: \(R_{\mathrm {a}} < \sqrt {R_{\mathrm {iA}} \cdot R_{\mathrm {iV}}}\)
