PMMC, moving iron and electrodynamometer instruments
Deflecting, controlling and damping torques; torque equations, scales, AC/DC response and errors of PMMC, moving-iron and electrodynamometer instruments, with worked numericals.
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Why it matters
Almost every analogue ammeter, voltmeter and wattmeter on a panel or lab bench is one of three movements: permanent-magnet moving-coil (PMMC), moving-iron (MI) or electrodynamometer. Knowing how each produces torque tells you whether it reads average or RMS, works on AC or DC, has a linear or crowded scale, and which errors to expect.
Key ideas
Three torques act in every indicating instrument.
- Deflecting torque Td moves the pointer and depends on the measured quantity.
- Controlling torque Tc opposes it and gives a definite deflection for each value — usually two phosphor-bronze hairsprings (Tc = Kθ), sometimes gravity (Tc ∝ sin θ). The pointer rests where Td = Tc.
- Damping torque acts only while the pointer moves, so it settles quickly without overshoot. It is slightly under-critical in practice.
- PMMC: eddy-current damping in the aluminium former.
- MI and electrodynamometer: air-friction damping (a vane in a chamber). Eddy damping would need a strong magnet that disturbs their weak field.
PMMC. A light rectangular coil of N turns on an aluminium former sits in the radial field of a permanent magnet (soft-iron core between shaped pole pieces makes B uniform and radial). Force on each side gives Td = NBAI. With spring control, θ = (NBA/K)·I, so:
- the scale is linear (uniform);
- the deflection follows the average current; on AC at 50 Hz the average is zero and the pointer just vibrates about zero, so a bare PMMC is DC only (it reads AC through a rectifier, then calibrated in RMS for a sine wave);
- sensitivity is high, power consumption low, and it is the most accurate analogue movement (class 0.1–0.5 possible);
- errors: weakening of magnet and springs with age, temperature changes in coil resistance (reduced with a manganin swamping resistor), and it is costly and delicate.
Moving-iron (MI). The current flows in a fixed coil; a soft-iron piece moves into the field (attraction type) or two irons magnetised alike repel each other (repulsion type). From energy, Td = ½I²·dL/dθ. With spring control θ = (I²/2K)·dL/dθ:
- deflection ∝ I² → reads RMS, works on AC and DC;
- scale is non-uniform (crowded at the start); shaping the iron so that dL/dθ falls with θ makes it more uniform over the upper part;
- robust and cheap, the usual panel meter;
- errors: hysteresis and eddy currents in the iron, stray fields, temperature; on AC, frequency errors because coil inductance changes the voltmeter impedance (corrected with a capacitor across the series resistor); DC readings show hysteresis error.
Electrodynamometer. A fixed coil (split in two halves) and a moving coil, air-cored. Td = I₁I₂·cos φ·dM/dθ, where φ is the phase angle between the two currents.
- With both coils in series (ammeter or voltmeter), Td ∝ I², so it reads RMS and has the same calibration on AC and DC → used as a transfer instrument to calibrate AC meters against DC standards.
- With one coil carrying load current and the other a current proportional to voltage, the mean torque ∝ VI cos φ — the wattmeter.
- Air core means no hysteresis or eddy errors, but the field is weak (about 0.005–0.006 T), so torque/weight ratio is low, sensitivity low and power consumption high; stray fields must be shielded.
Formulas
Td = N·B·A·I = G·I (PMMC)
N = turns; B = air-gap flux density (T); A = coil area (m²) = l × d; I = current (A); G = NBA = displacement constant (N·m/A). Td in N·m.
Tc = K·θ (spring control) ; θ = G·I / K
K = spring constant (N·m/rad or N·m/degree; keep θ in the same unit).
Td = ½·I²·dL/dθ ; θ = (I² / 2K)·dL/dθ (moving-iron)
L = coil inductance (H); dL/dθ in H/rad. I is the RMS current on AC.
Td = I₁·I₂·cos φ·dM/dθ (electrodynamometer, AC mean torque)
M = mutual inductance between fixed and moving coils (H); I₁, I₂ RMS currents (A); φ = phase angle between them. DC: cos φ = 1. Coils in series: Td = I²·dM/dθ.
Worked examples
Example 1 — PMMC full-scale current. A PMMC coil is 30 mm × 25 mm with 100 turns, in a gap flux density of 0.2 T. The springs give K = 2.5 × 10⁻⁶ N·m/degree. Find the current for a full-scale deflection of 90°.
A = 30 × 10⁻³ × 25 × 10⁻³ = 7.5 × 10⁻⁴ m².Tc = Kθ = 2.5 × 10⁻⁶ × 90 = 2.25 × 10⁻⁴ N·m.G = NBA = 100 × 0.2 × 7.5 × 10⁻⁴ = 0.015 N·m/A.I = Tc / G = 2.25 × 10⁻⁴ / 0.015 = 0.015 A.
Answer: 15 mA full-scale current.
Example 2 — moving-iron deflection (GATE level). The inductance of an MI ammeter is L = (10 + 5θ − θ²) μH, θ in radians. The spring constant is 12 × 10⁻⁶ N·m/rad. Find the deflection for a current of 5 A.
dL/dθ = (5 − 2θ) μH/rad.- Equilibrium:
½·I²·dL/dθ = Kθ→½ × 25 × (5 − 2θ) × 10⁻⁶ = 12 × 10⁻⁶ × θ. 62.5 − 25θ = 12θ→θ = 62.5 / 37 = 1.689 rad.- In degrees:
1.689 × 180/π = 96.8°.
Answer: θ = 1.69 rad ≈ 96.8°. Note that because dL/dθ falls with θ, the scale is more open than a pure square law would give.
Example 3 — square law of the series dynamometer. A dynamometer ammeter (coils in series, dM/dθ constant over the working range) gives 80° deflection at 2 A DC. What is the deflection for 1 A RMS, 50 Hz?
θ ∝ I²and it responds to RMS, soθ = 80 × (1/2)² = 20°.
Answer: 20° — the same as for 1 A DC.
Common mistakes
- Saying a PMMC reads "zero" or "half" on AC: on mains-frequency AC it indicates the average, which is zero for a sine wave.
- Writing MI torque as I·dL/dθ — it is ½I²·dL/dθ.
- Treating dL/dθ or dM/dθ as constant when the question gives L or M as a function of θ; differentiate first.
- Mixing degrees and radians between K and θ.
- Using eddy-current damping for MI instruments: their field is too weak and a damping magnet would distort it.
- Assuming MI meters have no frequency error; inductance of the coil makes voltmeter readings fall at high frequency.
For GATE IN
Expect questions comparing PMMC, MI and dynamometer responses to waveforms (average versus RMS, DC offset plus sine), NAT problems on torque equations (PMMC full-scale current, MI deflection from L(θ), dynamometer torque from dM/dθ), and MCQs on damping methods and errors. Practise finding what each meter reads for a mixed waveform such as i = 5 + 10 sin ωt A.
Quick check
- What does a PMMC ammeter read for
i = 5 + 10 sin ωtA? And an MI ammeter? - Which instrument is used as a transfer instrument between DC and AC?
- Why does an MI instrument use air-friction damping?
- In a PMMC, doubling the current changes the deflection by what factor? In an MI instrument with constant dL/dθ? Answers: 1. PMMC 5 A (average); MI √(25 + 50) = 8.66 A (RMS); 2. electrodynamometer; 3. its weak operating field would be disturbed by a damping magnet; 4. ×2 for PMMC, ×4 for MI.
Interview questions
All Electrical and Electronic Measurements interview questionsTry answering each one aloud before you open it.
1.What is a PMMC instrument and how does it work?Concept
A PMMC (Permanent Magnet Moving Coil) instrument is an analog measuring device used to measure direct current (DC). It operates on the principle that a current-carrying coil placed in a magnetic field experiences a torque. The coil is suspended between the poles of a permanent magnet, and when current flows through the coil, it deflects. The deflection is proportional to the current, allowing for measurement. The pointer attached to the coil moves over a calibrated scale to indicate the measurement.
2.Explain the working principle of a moving iron instrument.Concept
Current in a fixed coil magnetises a soft-iron piece that is either attracted into the coil (attraction type) or repelled by a second iron magnetised in the same sense (repulsion type). The deflecting torque is ½I²·dL/dθ, so it depends on the square of the current and the instrument reads RMS on both AC and DC. With spring control the scale is non-uniform, crowded at the low end, and damping is by air friction because the field is too weak for a damping magnet.
3.Describe the construction and working of an electrodynamometer instrument.Concept
An electrodynamometer instrument consists of a fixed coil and a moving coil, both of which carry current. The interaction between the magnetic fields of these coils produces a torque that causes the moving coil to rotate. The rotation is proportional to the product of the currents in the two coils, making it suitable for measuring power in AC circuits. The moving coil is attached to a pointer that moves over a calibrated scale to indicate the measurement.
4.Why is a PMMC instrument not suitable for AC measurements?Application
In a PMMC the torque NBAI reverses with the current because the magnet's field is fixed. On 50 Hz AC the moving system is too heavy to follow, so it responds to the average torque, which is zero for a sine wave; the pointer just vibrates about zero. To measure AC it needs a rectifier in front, and then it is calibrated to show the RMS of a pure sine wave.
5.What happens if a moving iron instrument is used to measure DC?Application
It works, because the torque ½I²·dL/dθ does not depend on the direction of current, and the reading equals the DC value. However, DC readings suffer hysteresis error in the iron: the reading for increasing current differs slightly from that for decreasing current. Stray steady magnetic fields such as the Earth's field also add a small error on DC that averages out on AC. For accurate DC work a PMMC is preferred.
6.In what applications would you prefer an electrodynamometer over a PMMC instrument?Application
An electrodynamometer is preferred over a PMMC instrument in applications where power measurement in AC circuits is required. This is because electrodynamometers can measure true power by accounting for both voltage and current, including their phase relationship. They are also suitable for both AC and DC measurements, making them versatile for various applications, unlike PMMC instruments which are limited to DC.
7.A PMMC coil of 100 turns, 20 mm × 10 mm, sits in a 0.5 T gap and its springs give 2 × 10⁻⁶ N·m per degree. What is the deflection for 2 mA?Numerical
Deflecting torque Td = NBAI = 100 × 0.5 × (20 × 10⁻³ × 10 × 10⁻³) × 0.002 = 2 × 10⁻⁵ N·m. At equilibrium Kθ = Td, so θ = 2 × 10⁻⁵ / 2 × 10⁻⁶ = 10°. The deflection is directly proportional to current, which is why the PMMC scale is linear.
8.A moving iron instrument shows a reading of 5 A when connected to a DC source. What will be the reading if the same instrument is connected to an AC source with the same RMS value?Numerical
A moving iron instrument measures the RMS value of the current, whether it is AC or DC. Therefore, if the instrument shows a reading of 5 A with a DC source, it will also show a reading of 5 A when connected to an AC source with the same RMS value. This is because the deflection is proportional to the square of the current, which is the same for both AC and DC when the RMS value is equal.
9.What are the advantages of using a moving iron instrument over a PMMC instrument?Application
Moving iron instruments have several advantages over PMMC instruments. They can measure both AC and DC, making them more versatile. They are generally more robust and less sensitive to overloads and mechanical shocks. Additionally, moving iron instruments are typically cheaper to manufacture and maintain. However, they may not be as accurate or sensitive as PMMC instruments for DC measurements.
10.Why are electrodynamometer instruments used for precision and calibration work?Application
With its coils in series the dynamometer's torque is I²·dM/dθ, so it reads RMS and has the same calibration on AC and DC. It can therefore be calibrated on DC against a potentiometer and then used as a transfer instrument to calibrate AC meters. It has no iron, so hysteresis and eddy-current errors are absent. Its weaknesses are a weak field (low sensitivity, high power consumption) and susceptibility to stray fields, which is why it is shielded.
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