The moving coil galvanometer
How a galvanometer works
A moving coil galvanometer detects and measures small currents. Inside it is a coil of many turns, free to turn about a fixed axis, between the poles of a strong magnet.
Look from above. The pole faces are curved, and a soft iron cylinder sits inside the coil. Together they make the field radial: the field lines point straight towards the axis in the gap.
So whichever way the coil turns, its plane always lies along the field. sin θ stays 1, and the torque on the coil is always N I A B.
A spring resists the turning. Its restoring torque is k φ, where φ is the angle turned and k is the torsional constant of the spring.
The coil comes to rest when the two torques balance: k φ equals N I A B. So φ equals N A B over k, times I. The deflection is proportional to the current, so the scale is evenly spaced.
Example: 100 turns, area 4 square centimetres, B equals 0.25 tesla, k equals 2 × 10⁻⁵ newton metre per radian. N A B is 0.01, so φ over I is 500 radians per ampere. A current of 1 milliampere turns the coil by half a radian, about 29°.
φ over I, equal to N A B over k, is the current sensitivity: the deflection per unit current. More turns, a stronger field, or a weaker spring make it more sensitive.
The voltage sensitivity is the deflection per unit voltage: φ over V equals N A B over k R G, where R G is the resistance of the coil.
Doubling the turns doubles the current sensitivity. But the wire is twice as long, so R G doubles too, and the voltage sensitivity stays the same.
Ammeters and voltmeters
A galvanometer cannot measure the current in a circuit directly. It reaches full scale at a few milliamperes, and its coil resistance would change the current being measured.
To make an ammeter, connect a small resistance r s, called a shunt, in parallel with the galvanometer. Most of the current goes through the shunt and only a small part through the coil.
The two are in parallel, so they share the same voltage: I g R G equals, I minus I g, times r s. For a range of I₀, the shunt is r s equals I g R G over, I₀ minus I g.
Example: a galvanometer of resistance 12 ohms gives full-scale deflection at 3 milliamperes. For a 0 to 6 ampere ammeter, r s equals 0.003 times 12, over 6 minus 0.003: about 6 milliohms.
The ammeter’s resistance, R G r s over R G plus r s, is then about 6 milliohms. Tiny, so putting it in series hardly disturbs the circuit.
To make a voltmeter, connect a large resistance R in series with the galvanometer, and put the combination in parallel with the part whose voltage you want.
At full scale, the current through it is I g, so V₀ equals I g, times R G plus R. The series resistance is R equals V₀ over I g, minus R G.
For a 0 to 18 volt voltmeter: 18 over 0.003 is 6000 ohms. Take away the 12 ohms of the coil, and R equals 5988 ohms.
The voltmeter’s resistance is 6000 ohms: very large, so it draws almost no current from the circuit. A good ammeter has a tiny resistance, a good voltmeter a huge one.
Try it: build a meter
- Galvanometer: raise the current and watch the coil turn in the radial field. The deflection grows in proportion to I.
- Double N: the current sensitivity doubles, but the voltage sensitivity hardly changes, because R_G doubles too.
- Ammeter: set the range to 6 A. Almost all of the current flows through the tiny shunt.
- Voltmeter: set the range to 18 V and read the series resistance. Push the applied voltage past the range and the needle goes off the scale.
Galvanometer to voltmeter
A galvanometer coil has a resistance of 12 Ω and gives full-scale deflection for 3 mA. How can it be turned into a voltmeter of range 0–18 V?
- A 6000 Ω resistance in series
- A 5988 Ω resistance in series
- A 5988 Ω resistance in parallel
- A 6 mΩ resistance in parallel
Answer
B. At full scale V₀ = I_g(R_G + R), so R = V₀/I_g − R_G = 18/0.003 − 12 = 6000 − 12 = 5988 Ω in series. 6000 Ω is the total resistance of the voltmeter, which already includes the 12 Ω coil. A resistance in parallel (a shunt) makes an ammeter, not a voltmeter.
Galvanometer to ammeter
The same galvanometer (12 Ω, full scale at 3 mA) is to become an ammeter of range 0–6 A. What should be connected, and how?
- About 6 mΩ in parallel
- About 6 mΩ in series
- 1988 Ω in series
- 1988 Ω in parallel
Answer
A. A shunt goes in parallel: r_s = I_gR_G/(I₀ − I_g) = (0.003 × 12)/(6 − 0.003) = 0.036/5.997 ≈ 6.0 × 10⁻³ Ω. 1988 Ω is 6/0.003 − 12, the voltmeter formula used by mistake.
Summary
- Deflection:
k φ = N I A B ⇒ φ = (NAB/k) I - Sensitivities:
φ/I = NAB/k, φ/V = NAB/(k R_G) - Ammeter (shunt in parallel):
r_s = I_g R_G / (I₀ − I_g) - Voltmeter (resistance in series):
R = V₀/I_g − R_G
- A radial field (curved pole pieces and a soft iron core) keeps the torque at NIAB, so the deflection is proportional to the current.
- More turns raise the current sensitivity, but not necessarily the voltage sensitivity, because R_G grows too.
- An ammeter is a galvanometer with a small shunt in parallel: very low resistance, connected in series.
- A voltmeter is a galvanometer with a large resistance in series: very high resistance, connected in parallel.