PMMC Instrument Current Calculation

A Permanent Magnet Moving Coil (PMMC) instrument is a type of analog measuring instrument used for precise DC current and voltage measurements. Its operation relies on the principle that a current-carrying coil placed in a magnetic field experiences a torque.

Working Principle of PMMC Instrument

The core of a PMMC instrument consists of a rectangular coil, usually wound on a light aluminium former, which is suspended in the uniform radial magnetic field of a permanent magnet. When current flows through the coil, a deflecting torque is produced due to the interaction between the coil's magnetic field and the permanent magnet's field. This torque causes the coil to rotate.

To control this rotation and provide a measurable deflection, control springs (typically spiral springs) are attached. These springs produce a controlling torque that opposes the deflecting torque. The pointer, attached to the coil, moves along a calibrated scale.

At steady state, or equilibrium, the deflecting torque is balanced by the controlling torque, and the pointer comes to rest at a position corresponding to the magnitude of the current.

Given Parameters for PMMC Instrument

Let's list the known values provided in the question:

• Deflection ($\theta$) = 100°
• Number of turns in the coil ($N$) = 60 turns
• Width of the coil former ($w$) = 18 mm = $18 \times 10^{-3}$ m = 0.018 m
• Effective length of conductor ($l$) = 25 mm = $25 \times 10^{-3}$ m = 0.025 m
• Uniform magnetic field flux density ($B$) = 0.5 T (Tesla)
• Control spring constant ($K$) = $1.5 \times 10^{-6}$ Nm/degree

Our objective is to calculate the current ($I$) required to produce the specified deflection.

Torque Balance in PMMC Instrument

The two main torques acting on the coil in a PMMC instrument are the deflecting torque and the controlling torque. At equilibrium, these two torques are equal.

• Deflecting Torque ($\mathbf{T_d}$): This torque is produced by the interaction of the current flowing through the coil with the magnetic field. It is given by the formula:

  $$T_d = NBIA$$

  Where:

  • $N$ is the number of turns in the coil
  • $B$ is the magnetic flux density in Tesla (T)
  • $I$ is the current flowing through the coil in Amperes (A)
  • $A$ is the effective area of the coil in square meters ($m^2$)
• Controlling Torque ($\mathbf{T_c}$): This torque is provided by the control springs and opposes the deflecting torque. It increases linearly with the deflection angle. It is given by the formula:

  $$T_c = K\theta$$

  Where:

  • $K$ is the control spring constant in Nm/degree (or Nm/radian)
  • $\theta$ is the deflection angle in degrees (or radians)

Current Calculation for PMMC

At equilibrium, the deflecting torque equals the controlling torque:

$$T_d = T_c$$

$$NBIA = K\theta$$

We need to find the current ($I$). We can rearrange the equation to solve for $I$:

$$I = \frac{K\theta}{NBA}$$

First, let's calculate the effective area ($A$) of the coil. The area is the product of the width and the effective length of the conductor:

$$A = \text{width} \times \text{effective length}$$

$$A = w \times l$$

Substitute the values for width and length in meters:

$$A = (0.018 \text{ m}) \times (0.025 \text{ m})$$

$$A = 0.00045 \text{ m}^2$$

Now, substitute all the known values into the current formula. Since the spring constant $K$ is given in Nm/degree, we use the deflection $\theta$ directly in degrees:

$$I = \frac{(1.5 \times 10^{-6} \text{ Nm/degree}) \times (100 \text{ degrees})}{(60) \times (0.5 \text{ T}) \times (0.00045 \text{ m}^2)}$$

Calculate the numerator:

$$Numerator = 1.5 \times 10^{-6} \times 100 = 1.5 \times 10^{-4} \text{ Nm}$$

Calculate the denominator:

$$Denominator = 60 \times 0.5 \times 0.00045$$

$$Denominator = 30 \times 0.00045$$

$$Denominator = 0.0135 \text{ Tm}^2$$

Now, perform the division:

$$I = \frac{1.5 \times 10^{-4}}{0.0135}$$

$$I \approx 0.0111111 \text{ A}$$

To convert the current from Amperes (A) to milliamperes (mA), multiply by 1000:

$$I = 0.0111111 \times 1000 \text{ mA}$$

$$I \approx 11.11 \text{ mA}$$

Final Current for Deflection

The current required to produce a deflection of 100° in the given PMMC instrument is approximately 11.11 mA.