Large AC ammeters use a high-permeability core that can be clamped around the current to be measured, as shown in the top right. Lets use a simple toroidal model for the clamp as shown below, where the toroid has a mean radius , and cross-sectional area A. A secondary winding of N turns is wrapped on this toroidal core. a. Assuming the current passes through the center of the torus, find the B field inside the core at the mean radius r. b. If the AC current to be measured is expressed as I(t)=1 cos(@t), find an expression for the induced secondary voltage V(t). Use the result of (a) and assume that the magnetic field is approximately constant over the cross section of the core. c. How many turns in the secondary winding would be required to insure that the induced AC voltage amplitude is at least 10 mV for 60 Hz currents of >1 A? Assume core dimensions H4, = 5000. V(t) Ntu of r = 2 cm and A=0.25 cm

Introductory Circuit Analysis (13th Edition)
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Publisher:Robert L. Boylestad
Chapter1: Introduction
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Large AC ammeters use a high-permeability core that can be clamped around the current to be measured, as shown in the top right. Let's use a simple toroidal model for the clamp as shown below, where the toroid has a mean radius \( r_0 \) and cross-sectional area \( A \). A secondary winding of \( N \) turns is wrapped on this toroidal core.

**a.** Assuming the current passes through the center of the torus, find the B field inside the core at the mean radius \( r_0 \).

**b.** If the AC current to be measured is expressed as \( I(t) = I_0 \cos(\omega t) \), find an expression for the induced secondary voltage \( V(t) \). Use the result of (a) and assume that the magnetic field is approximately constant over the cross section of the core.

**c.** How many turns in the secondary winding would be required to ensure that the induced AC voltage amplitude is at least 10 mV for 60 Hz currents of \( >1 \, \text{A} \)? Assume core dimensions of \( r_0 = 2 \, \text{cm} \) and \( A = 0.25 \, \text{cm}^2 \), and a relative permeability of \( \mu_r = 5000 \).

---

**Diagram Explanation:**

The diagram at the bottom right shows a toroidal core with a wire carrying the current \( I(t) \) passing through its center. The core has a secondary winding of \( N \) turns, connected to a circuit measuring the induced voltage \( V(t) \). Arrows indicate the direction of the magnetic field generated by the current. 

The photo in the top right shows a handheld digital meter with clamp probes, illustrating a practical application of the described toroidal model for measuring AC current.
Transcribed Image Text:Large AC ammeters use a high-permeability core that can be clamped around the current to be measured, as shown in the top right. Let's use a simple toroidal model for the clamp as shown below, where the toroid has a mean radius \( r_0 \) and cross-sectional area \( A \). A secondary winding of \( N \) turns is wrapped on this toroidal core. **a.** Assuming the current passes through the center of the torus, find the B field inside the core at the mean radius \( r_0 \). **b.** If the AC current to be measured is expressed as \( I(t) = I_0 \cos(\omega t) \), find an expression for the induced secondary voltage \( V(t) \). Use the result of (a) and assume that the magnetic field is approximately constant over the cross section of the core. **c.** How many turns in the secondary winding would be required to ensure that the induced AC voltage amplitude is at least 10 mV for 60 Hz currents of \( >1 \, \text{A} \)? Assume core dimensions of \( r_0 = 2 \, \text{cm} \) and \( A = 0.25 \, \text{cm}^2 \), and a relative permeability of \( \mu_r = 5000 \). --- **Diagram Explanation:** The diagram at the bottom right shows a toroidal core with a wire carrying the current \( I(t) \) passing through its center. The core has a secondary winding of \( N \) turns, connected to a circuit measuring the induced voltage \( V(t) \). Arrows indicate the direction of the magnetic field generated by the current. The photo in the top right shows a handheld digital meter with clamp probes, illustrating a practical application of the described toroidal model for measuring AC current.
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