33 DC Voltmeters and Ammeters

Analog meters have a needle that swivels to point at numbers on a scale, as opposed to digital meters, which have numerical readouts similar to a hand-held calculator. The heart of most analog meters is a device called a galvanometer, denoted by G. Current flow through a galvanometer, I_G, produces a proportional needle deflection. (This deflection is due to the force of a magnetic field upon a current-carrying wire.)

The two crucial characteristics of a given galvanometer are its resistance and current sensitivity. Current sensitivity is the current that gives a full-scale deflection of the galvanometer’s needle, the maximum current that the instrument can measure. For example, a galvanometer with a current sensitivity of 50 μA has a maximum deflection of its needle when 50 μA flows through it, reads half-scale when 25 μA flows through it, and so on. If such a galvanometer has a 25-Ω resistance, then a voltage of only V = IR = (50 μA)(25 Ω) = 1.25 mV produces a full-scale reading. By connecting resistors to this galvanometer in different ways, you can use it as either a voltmeter or ammeter that can measure a broad range of voltages or currents.

Galvanometer as Voltmeter

Figure 4 shows how a galvanometer can be used as a voltmeter by connecting it in series with a large resistance, R. The value of the resistance R is determined by the maximum voltage to be measured. Suppose you want 10 V to produce a full-scale deflection of a voltmeter containing a 25-Ω galvanometer with a 50-μA sensitivity. Then 10 V applied to the meter must produce a current of 50 μA. The total resistance must be

[latex]{R}_{\text{tot}}=R+r=\frac{V}{I}=\frac{10\text{ V}}{50\text{ }\mu \text{A}}=200\text{ k}\Omega\\[/latex] or

[latex]R={R}_{\text{tot}}-r=200\text{ k}\Omega-25\text{ }\Omega \approx 200\text{ k}\Omega \\[/latex].

(R is so large that the galvanometer resistance, r, is nearly negligible.) Note that 5 V applied to this voltmeter produces a half-scale deflection by producing a 25-μA current through the meter, and so the voltmeter’s reading is proportional to voltage as desired. This voltmeter would not be useful for voltages less than about half a volt, because the meter deflection would be small and difficult to read accurately. For other voltage ranges, other resistances are placed in series with the galvanometer. Many meters have a choice of scales. That choice involves switching an appropriate resistance into series with the galvanometer.

Galvanometer as Ammeter

The same galvanometer can also be made into an ammeter by placing it in parallel with a small resistance R, often called the shunt resistance, as shown in Figure 5. Since the shunt resistance is small, most of the current passes through it, allowing an ammeter to measure currents much greater than those producing a full-scale deflection of the galvanometer. Suppose, for example, an ammeter is needed that gives a full-scale deflection for 1.0 A, and contains the same 25-Ω galvanometer with its 50-μA sensitivity. Since R and r are in parallel, the voltage across them is the same. These IR drops are IR = I_Gr so that [latex]IR=\frac{{I}_{\text{G}}}{I}=\frac{R}{r}\\[/latex]. Solving for R, and noting that I_G is 50 μA and I is 0.999950 A, we have

[latex]R=r\frac{{I}_{\text{G}}}{I}=\left(25\text{ }\Omega\right)\frac{50\text{ }\mu\text{A}}{0.999950\text{ A}}=1.25\times 10^{-3}\text{ }\Omega\\[/latex].

When you use a voltmeter or ammeter, you are connecting another resistor to an existing circuit and, thus, altering the circuit. Ideally, voltmeters and ammeters do not appreciably affect the circuit, but it is instructive to examine the circumstances under which they do or do not interfere. First, consider the voltmeter, which is always placed in parallel with the device being measured. Very little current flows through the voltmeter if its resistance is a few orders of magnitude greater than the device, and so the circuit is not appreciably affected. (See Figure 6(a).) (A large resistance in parallel with a small one has a combined resistance essentially equal to the small one.) If, however, the voltmeter’s resistance is comparable to that of the device being measured, then the two in parallel have a smaller resistance, appreciably affecting the circuit. (See Figure 6(b).) The voltage across the device is not the same as when the voltmeter is out of the circuit.

An ammeter is placed in series in the branch of the circuit being measured, so that its resistance adds to that branch. Normally, the ammeter’s resistance is very small compared with the resistances of the devices in the circuit, and so the extra resistance is negligible. (See Figure 7(a).) However, if very small load resistances are involved, or if the ammeter is not as low in resistance as it should be, then the total series resistance is significantly greater, and the current in the branch being measured is reduced. (See Figure 7(b).) A practical problem can occur if the ammeter is connected incorrectly. If it was put in parallel with the resistor to measure the current in it, you could possibly damage the meter; the low resistance of the ammeter would allow most of the current in the circuit to go through the galvanometer, and this current would be larger since the effective resistance is smaller.

One solution to the problem of voltmeters and ammeters interfering with the circuits being measured is to use galvanometers with greater sensitivity. This allows construction of voltmeters with greater resistance and ammeters with smaller resistance than when less sensitive galvanometers are used. There are practical limits to galvanometer sensitivity, but it is possible to get analog meters that make measurements accurate to a few percent. Note that the inaccuracy comes from altering the circuit, not from a fault in the meter.

Glossary

voltmeter:

an instrument that measures voltage

ammeter:

an instrument that measures current

analog meter:

a measuring instrument that gives a readout in the form of a needle movement over a marked gauge

digital meter:

a measuring instrument that gives a readout in a digital form

galvanometer:

an analog measuring device, denoted by G, that measures current flow using a needle deflection caused by a magnetic field force acting upon a current-carrying wire

current sensitivity:

the maximum current that a galvanometer can read

full-scale deflection:

the maximum deflection of a galvanometer needle, also known as current sensitivity; a galvanometer with a full-scale deflection of 50 μA has a maximum deflection of its needle when 50 μA flows through it

shunt resistance:

a small resistance R placed in parallel with a galvanometer G to produce an ammeter; the larger the current to be measured, the smaller R must be; most of the current flowing through the meter is shunted through R to protect the galvanometer

Selected Solutions to Problems & Exercises

1. 30 μA

3. 1.98 kΩ

5. 1.25 × 10⁻⁴ Ω

7. (a) 3.00 MΩ (b) 2.99 kΩ

9. (a) 1.58 mA (b) 1.5848 V (need four digits to see the difference) (c) 0.99990 (need five digits to see the difference from unity)

11. 15.0 μA

12.

(a)

(b) 10.02 Ω

(c) 0.9980, or a 2.0 × 10^–1 percent decrease

(d) 1.002, or a 2.0 × 10^–1 percent increase

(e) Not significant.

15. (a) −66.7 Ω (b) You can’t have negative resistance. (c) It is unreasonable that I_G is greater than I_tot (see Figure 5). You cannot achieve a full-scale deflection using a current less than the sensitivity of the galvanometer.