Galvanometer

What a galvanometer is

• A galvanometer is a device used to detect weak electric currents in a circuit.
• Its coil is pivoted (or suspended) between the concave pole faces of a strong laminated horseshoe magnet. When current passes through the coil, it deflects, and the deflection is proportional to the current.
• The deflection is read with a pointer attached to the coil; the position of the pointer on the scale indicates the current through the galvanometer or the potential difference across it.
• The coil has a moderate resistance, about 100 ohm, and the instrument has a small current carrying capacity, about 1 mA. With suitable modification it can therefore be used as an ammeter or as a voltmeter.
• Diagram in words (Fig. 9.12): a rectangular coil wound on a soft iron core, pivoted between the curved (concave) pole pieces N and S of a permanent horseshoe magnet, with a hair spring providing the restoring torque and a pointer sweeping a scale.

9.5.1 Galvanometer as an ammeter

What must change

• Let $I_g$ be the full-scale deflection current and $G$ the resistance of the coil of the moving coil galvanometer (MCG). To convert it into an ammeter — a current measuring instrument, always connected in series with the resistance $R$ through which the current is to be measured — three modifications are necessary.
• Its effective current capacity must be increased to the desired higher value.
• Its effective resistance must be decreased. The finite resistance $G$, when connected in series, itself decreases the current through $R$ that is actually to be measured; in the ideal case an ammeter should have zero resistance.
• It must be protected from possible damage due to the passage of an excessive electric current.

The shunt

• In practice all three are achieved by connecting a low resistance in parallel with the galvanometer, which effectively reduces the resistance of the galvanometer. This low resistance connected in parallel is called a shunt, $S$.
• Diagram in words (Fig. 9.13): the galvanometer coil G with the low resistance $S$ connected directly across its two terminals, the pair forming a single two-terminal ammeter placed in series in the main circuit. The main current $I$ enters, splits into $I_g$ through G and $I_s$ through S, and recombines.
• Uses of the shunt: it diverts a large part of the total current by providing an alternative path and thus protects the instrument from damage; it increases the range of the ammeter; it decreases the resistance between the points to which it is connected.

Derivation

• $I$ and $I_g$ are the currents through the circuit and through the galvanometer respectively, so the current through the shunt is

$$I_s = I - I_g$$

• Since $S$ and $G$ are in parallel, the potential difference across them is the same:

$$GI_g = SI_s \qquad \therefore\ GI_g = S(I-I_g)$$

$$\therefore\ S = \left(\dfrac{I_g}{I-I_g}\right)G \qquad \dots (1)$$

• Equation (1) is useful to calculate the range of current that the galvanometer can measure.
• If the current $I$ is $n$ times the current $I_g$, then $I = nI_g$. Using this in equation (1),

$$S = \dfrac{GI_g}{nI_g - I_g} \qquad \therefore\ S = \dfrac{G}{n-1}$$

• This is the required shunt to increase the range $n$ times.
• If $I_s$ is the current through the shunt, the remaining current $(I-I_s)$ flows through the galvanometer, so $G(I-I_s) = SI_s$, i.e. $GI - GI_s = SI_s$, i.e. $SI_s + GI_s = GI$, giving

$$\dfrac{I_s}{I} = \dfrac{G}{S+G}$$

• This gives the fraction of the total current through the shunt resistance. Correspondingly $\dfrac{I_g}{I} = \dfrac{S}{S+G}$.
• Resistance of the resulting ammeter: $R_A = \dfrac{SG}{S+G} = \dfrac{G}{n}$, which is small, as an ammeter's should be.
• If the only available shunt is too large to give the desired range, a resistance $X$ is added in series with the coil, so that the coil branch has resistance $G+X$ and $\dfrac{I_g}{I} = \dfrac{S}{(G+X)+S}$.

9.5.2 Galvanometer as a voltmeter

What must change

• A voltmeter measures the potential difference between two points in an electrical circuit and is always connected in parallel with the component across which the voltage drop is to be measured. To convert an MCG into a voltmeter, three modifications are necessary.
• Its voltage measuring capacity must be increased to the desired higher value.
• Its effective resistance must be increased. If a galvanometer is used directly to measure voltage it draws some current, because of its low resistance, and so the actual potential difference to be measured decreases; a voltmeter should therefore have very high resistance, ideally infinite.
• It must be protected from possible damage due to an excess applied potential difference.

The series resistance

• All these requirements are fulfilled by connecting a resistance of suitable high value $X$ in series with the given MCG.
• Diagram in words (Fig. 9.14): the galvanometer coil G with the high resistance $X$ in series with it, the pair forming a single two-terminal voltmeter connected across the element whose p.d. is wanted.

Derivation

• If $V$ is the voltage to be measured, then

$$V = I_gX + I_gG \qquad \therefore\ I_gX = V - I_gG$$

$$\therefore\ X = \dfrac{V}{I_g} - G \qquad \dots (1)$$

• where $I_g$ is the current flowing through the galvanometer at full-scale deflection.
• If the voltage $V$ is $n_v$ times the voltage $V_g$ across the galvanometer, then $V = n_vV_g = n_v(I_gG)$. Using this in equation (1),

$$X = G(n_v - 1) \qquad \text{where } n_v = \dfrac{V}{V_g} = \dfrac{V}{I_gG}$$

• Resistance of the resulting voltmeter: $R_V = G + X = Gn_v$, which is large, as a voltmeter's should be.

Comparison of an ammeter and a voltmeter

Thermoelectricity — Seebeck effect

• When electric current is passed through a resistor, electrical energy is converted into thermal energy. The reverse process, the direct conversion of thermal energy into electrical energy, was discovered by Seebeck, and the effect is called the thermoelectric effect.
• If two different metals are joined to form a closed circuit and the two junctions are kept at different temperatures, a small emf is produced and a current flows. This emf is called the thermo emf, the effect is the Seebeck effect, and the pair of dissimilar metals forming the junctions is called a thermocouple.
• For an antimony-bismuth thermocouple the current flows from antimony to bismuth at the cold junction (the ABC rule); for a copper-iron couple the current flows from copper to iron at the hot junction.
• The effect is reversible: the direction of the current reverses if the hot and cold junctions are interchanged.
• With the cold junction at $0\,^\circ\text{C}$ and the hot junction at $T\,^\circ\text{C}$,

$$E = \alpha T + \dfrac{1}{2}\beta T^2$$

• where $\alpha$ and $\beta$ are the thermoelectric constants. The graph of thermo emf against temperature is therefore a parabola.

Standard results collected

• Shunt for an ammeter: $S = \left(\dfrac{I_g}{I-I_g}\right)G$; for an $n$-fold range, $S = \dfrac{G}{n-1}$ with $n = \dfrac{I}{I_g}$.
• Current split: $\dfrac{I_s}{I} = \dfrac{G}{S+G}$ and $\dfrac{I_g}{I} = \dfrac{S}{S+G}$.
• Series resistance for a voltmeter: $X = \dfrac{V}{I_g} - G = G(n_v-1)$ with $n_v = \dfrac{V}{I_gG}$.
• Meter resistances: $R_A = \dfrac{SG}{S+G} = \dfrac{G}{n}$ and $R_V = G+X = Gn_v$.
• For an ideal voltmeter the resistance is infinite; for an ideal ammeter it is zero.
• If a shunt is already fitted and the combined resistance $R_{eq}$ is known, recover the shunt from $\dfrac{1}{R_{eq}} = \dfrac{1}{G} + \dfrac{1}{S}$, i.e. $S = \dfrac{R_{eq}G}{G - R_{eq}}$.
• To extend the range of an existing voltmeter of resistance $G$ from $V_g$ to $V$, connect $R = G(n-1)$ in series, where $n = \dfrac{V}{V_g}$.