21.4 DC Voltmeters and Ammeters

Analog Meters: Galvanometers

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, $I_{\text{G}}$, 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 $\text{50 μA}$ has a maximum deflection of its needle when $\text{50 μA}$ flows through it, reads half-scale when $\mathrm{25\; \mu A}$ flows through it, and so on.

If such a galvanometer has a $2\text{5-}\Omega$ resistance, then a voltage of only $V=\text{IR}=\left(\text{50 μA}\right)\left(\text{25 Ω}\right)=1\text{.}\text{25 mV}$ 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 21.32 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 $2\text{5-Ω}$ galvanometer with a $\text{50-μA}$ sensitivity. Then 10 V applied to the meter must produce a current of $\text{50 μA}$. The total resistance must be

$$R_{\text{tot}}=R+r=\frac{V}{I}=\frac{\text{10}\text{V}}{\text{50 μA}}=\text{200}\text{k}\Omega ,\; or$$

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$$R=R_{\text{tot}}-r=\text{200 kΩ}-\text{25}\Omega \approx \text{200}\text{k}\Omega .$$

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($R$ 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 $2\text{5-μA}$ 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.

Figure 21.32 A large resistance $R$ placed in series with a galvanometer G produces a voltmeter, the full-scale deflection of which depends on the choice of $R$. The larger the voltage to be measured, the larger $R$ must be. (Note that $r$ represents the internal resistance of 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 21.33. 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 $2\text{5-}\Omega$ galvanometer with its $\text{50-μA}$ sensitivity. Since $R$ and $r$ are in parallel, the voltage across them is the same.

These $\text{IR}$ drops are $\text{IR}=I_{\text{G}}r$ so that $\text{IR}=\frac{I_{\text{G}}}{I}=\frac{R}{r}$. Solving for $R$, and noting that $I_{\text{G}}$ is $\text{50 μA}$ and $I$ is 0.999950 A, we have

$$R=r\frac{I_{\text{G}}}{I}=(\text{25}\Omega )\frac{\text{50 μA}}{0\text{.}\text{999950 A}}=1\text{.}\text{25}\times {\text{10}}^{-3}\Omega .$$

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Figure 21.33 A small shunt resistance $R$ placed in parallel with a galvanometer G produces an ammeter, the full-scale deflection of which depends on the choice of $R$. The larger the current to be measured, the smaller $R$ must be. Most of the current ($I$) flowing through the meter is shunted through $R$ to protect the galvanometer. (Note that $r$ represents the internal resistance of the galvanometer.) Ammeters may also have multiple scales for greater flexibility in application. The various scales are achieved by switching various shunt resistances in parallel with the galvanometer—the greater the maximum current to be measured, the smaller the shunt resistance must be.

Taking Measurements Alters the Circuit

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 21.34(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 21.34(b).) The voltage across the device is not the same as when the voltmeter is out of the circuit.

Figure 21.34 (a) A voltmeter having a resistance much larger than the device ($R_{\text{Voltmeter}}\text{>>}R$) with which it is in parallel produces a parallel resistance essentially the same as the device and does not appreciably affect the circuit being measured. (b) Here the voltmeter has the same resistance as the device ($R_{\text{Voltmeter}}\cong R$), so that the parallel resistance is half of what it is when the voltmeter is not connected. This is an example of a significant alteration of the circuit and is to be avoided.

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 21.35(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 21.35(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.

Figure 21.35 (a) An ammeter normally has such a small resistance that the total series resistance in the branch being measured is not appreciably increased. The circuit is essentially unaltered compared with when the ammeter is absent. (b) Here the ammeter’s resistance is the same as that of the branch, so that the total resistance is doubled and the current is half what it is without the ammeter. This significant alteration of the circuit is to be avoided.

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.