KAHIBARO
Table of Contents
Purpose in Circuit Measurements
In direct current circuits, measuring instruments let us determine quantities such as current and voltage without guessing. The two most common basic instruments are the ammeter, which measures current, and the voltmeter, which measures potential difference. Their correct use depends on how they are connected to a circuit and on how much they disturb the circuit being measured.
An ideal measuring instrument would give a perfectly accurate reading and would not change the behavior of the circuit at all. Real instruments always affect the circuit a little, so understanding their internal resistance is essential.
Ammeters
An ammeter measures the current through a part of a circuit. To do this, it must be placed in series with the element whose current is being measured. In a series connection, the same current passes through the element and the ammeter.
If an ammeter had a large resistance, it would reduce the current in the circuit and give a misleading result. For that reason, an ideal ammeter has zero resistance, or as close to zero as possible in practice.
Important rule: An ammeter is connected in series and should have very small internal resistance.
For an ideal ammeter,
$$R_A = 0$$
If the ammeter has internal resistance $R_A$, then it adds extra resistance to the circuit. This changes the current from what it would have been without the meter.
Voltmeters
A voltmeter measures the potential difference between two points. It must be connected in parallel with the component across which the voltage is measured. In a parallel connection, the voltmeter compares the electric potential at the two ends of the component.
If a voltmeter had low resistance, a large current would flow through it, changing the circuit significantly. So an ideal voltmeter has infinite resistance, or in practice a very large resistance.
Important rule: A voltmeter is connected in parallel and should have very large internal resistance.
For an ideal voltmeter,
$$R_V \to \infty$$
A real voltmeter draws a small current, but if its resistance is very large compared with the rest of the circuit, its effect is usually negligible.
Internal Resistance and Measurement Error
Real measuring instruments are not ideal. Their internal resistance causes loading effects.
For an ammeter, the measured circuit current becomes smaller if the meter resistance is not negligible. For a voltmeter, the meter forms a parallel branch with the component being measured, which can reduce the measured voltage across that component if the voltmeter resistance is not much larger than the component resistance.
The effect can be summarized as follows.
| Instrument | Connected | Ideal Internal Resistance | Main Undesired Effect if Not Ideal |
|---|---|---|---|
| Ammeter | Series | $0$ | Reduces circuit current |
| Voltmeter | Parallel | $\infty$ | Draws current, changes voltage distribution |
Example of Voltmeter Loading
Suppose a resistor $R$ is connected in a circuit, and a voltmeter of resistance $R_V$ is placed across it. Then the resistor and voltmeter are in parallel. The equivalent resistance is
$$
R_{\text{eq}} = \frac{R R_V}{R + R_V}
$$
If $R_V$ is very large compared with $R$, then
$$
R_{\text{eq}} \approx R
$$
and the voltmeter has little effect. But if $R_V$ is not much larger than $R$, then the circuit changes noticeably.
To minimize voltmeter loading, the voltmeter resistance should satisfy
$$R_V \gg R$$
where $R$ is the resistance of the part being measured.
Galvanometer as a Basic Meter
Many traditional analog meters are based on a galvanometer, a sensitive device that detects small currents. A galvanometer gives a pointer deflection proportional to the current through it, within its operating range.
By adding suitable resistors, a galvanometer can be turned into either an ammeter or a voltmeter.
To make an ammeter, a small shunt resistor is connected in parallel with the galvanometer so that most of the current bypasses the delicate meter movement.
To make a voltmeter, a large resistor is connected in series with the galvanometer so that only a small current flows through it.
Shunt Resistor for an Ammeter
Suppose a galvanometer has resistance $R_g$ and can safely carry a maximum current $I_g$. To extend it into an ammeter that measures up to total current $I$, a shunt resistor $R_s$ is connected in parallel.
Because the galvanometer and shunt are in parallel, they have the same voltage:
$$
I_g R_g = I_s R_s
$$
where
$$
I_s = I - I_g
$$
So the shunt resistance must be
$$
R_s = \frac{I_g R_g}{I - I_g}
$$
This resistor is usually very small.
Series Resistor for a Voltmeter
To convert a galvanometer into a voltmeter with maximum reading $V$, a series resistor $R$ is added. At full scale, the galvanometer current is $I_g$, so
$$
V = I_g(R_g + R)
$$
Solving for the required series resistance gives
$$
R = \frac{V}{I_g} - R_g
$$
This resistor is usually large.
Galvanometer conversion formulas:
For an ammeter,
$$
R_s = \frac{I_g R_g}{I - I_g}
$$
For a voltmeter,
$$
R = \frac{V}{I_g} - R_g
$$
Analog and Digital Instruments
Analog instruments use a pointer moving across a scale. They are useful for showing continuous changes, but reading them can involve parallax error, which happens when the eye is not directly in front of the pointer.
Digital instruments display numbers directly. They are often easier to read and usually have very high input resistance when used as voltmeters. A digital multimeter can usually measure voltage, current, and resistance in one device.
Multimeters
A multimeter combines several measuring functions in one instrument. In DC circuit work, it is commonly used as an ammeter, voltmeter, and ohmmeter.
When using a multimeter, the correct mode and range must be selected. A wrong setting can produce an incorrect reading or even damage the meter.
As an ammeter, the multimeter is inserted in series.
As a voltmeter, it is connected in parallel.
As an ohmmeter, it measures resistance by using its own internal battery, so the circuit being tested should usually be disconnected from external power.
Safety rule: Never connect an ammeter directly across a battery or power supply, because its very small resistance can cause a very large current.
Reading and Choosing Instruments
A good measuring instrument should have suitable range, sensitivity, and precision for the task. If the range is too small, the meter may overload. If the range is too large, the reading may be less precise.
For voltage measurements, a high-resistance voltmeter is preferred. For current measurements, a low-resistance ammeter is preferred. In many practical cases, the instrument should disturb the circuit much less than the uncertainty you are willing to accept.
Final Ideas
Measuring instruments are part of the circuit when they are connected, so they must be treated as real physical components. The key idea is simple. An ammeter should let current pass easily, and a voltmeter should draw almost no current. Once this is understood, the correct connection methods and the role of internal resistance become clear.