Consider the transformer with the secondary terminated in a load resistance R, as shown. This as a real transformer with a coupling coefficient k<1. Assuming AC excitation at a frequency @, derive 1₂ the frequency-dependent voltage transformation ratio V₂ (@)/(a) in terms of the turns ratio, coupling V coefficient, and mutual inductance L=k√ √LL₂ between the primary and secondary coils. Show that the imperfect coupling gives the circuit a low-pass characteristic with a time constant t = L₂(1-k²)/ R₂. Note the orientation of the windings in the figure, as well as the direction of current 72, which imply the following two-port network equations V₁ = jwL₂1₁-jwLm¹₂ V₂ = jwLm-jwL₂1₂ R₂
Consider the transformer with the secondary terminated in a load resistance R, as shown. This as a real transformer with a coupling coefficient k<1. Assuming AC excitation at a frequency @, derive 1₂ the frequency-dependent voltage transformation ratio V₂ (@)/(a) in terms of the turns ratio, coupling V coefficient, and mutual inductance L=k√ √LL₂ between the primary and secondary coils. Show that the imperfect coupling gives the circuit a low-pass characteristic with a time constant t = L₂(1-k²)/ R₂. Note the orientation of the windings in the figure, as well as the direction of current 72, which imply the following two-port network equations V₁ = jwL₂1₁-jwLm¹₂ V₂ = jwLm-jwL₂1₂ R₂
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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![Consider the transformer with the secondary terminated in a load resistance \( R_L \) as shown. This is a real transformer with a coupling coefficient \( k < 1 \). Assuming AC excitation at a frequency \( \omega \), derive the frequency-dependent voltage transformation ratio \( V_2(\omega)/V_1(\omega) \) in terms of the turns ratio, coupling coefficient, and mutual inductance \( L_m = k\sqrt{L_1L_2} \) between the primary and secondary coils. Show that the imperfect coupling gives the circuit a low-pass characteristic with a time constant \( \tau = L_2(1-k^2)/R_L \). Note the orientation of the windings in the figure, as well as the direction of current \( I_2 \), which imply the following two-port network equations:
\[ V_1 = j\omega L_1 I_1 - j\omega L_m I_2 \]
\[ V_2 = j\omega L_m I_1 - j\omega L_2 I_2 \]
**Diagram Explanation:**
- The diagram depicts a transformer with primary and secondary coils.
- The primary side has an input voltage \( V_1 \) and current \( I_1 \).
- The secondary side shows the output voltage \( V_2 \) and current \( I_2 \), connected to a load resistance \( R_L \).
- The coupling between the coils is less than perfect, represented by the coupling coefficient \( k \).
- Arrows indicate the direction of the currents \( I_1 \) and \( I_2 \) through the respective windings.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2F2cb43441-b811-4cb5-a5a9-38ae731886d6%2F2lktt2_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the transformer with the secondary terminated in a load resistance \( R_L \) as shown. This is a real transformer with a coupling coefficient \( k < 1 \). Assuming AC excitation at a frequency \( \omega \), derive the frequency-dependent voltage transformation ratio \( V_2(\omega)/V_1(\omega) \) in terms of the turns ratio, coupling coefficient, and mutual inductance \( L_m = k\sqrt{L_1L_2} \) between the primary and secondary coils. Show that the imperfect coupling gives the circuit a low-pass characteristic with a time constant \( \tau = L_2(1-k^2)/R_L \). Note the orientation of the windings in the figure, as well as the direction of current \( I_2 \), which imply the following two-port network equations:
\[ V_1 = j\omega L_1 I_1 - j\omega L_m I_2 \]
\[ V_2 = j\omega L_m I_1 - j\omega L_2 I_2 \]
**Diagram Explanation:**
- The diagram depicts a transformer with primary and secondary coils.
- The primary side has an input voltage \( V_1 \) and current \( I_1 \).
- The secondary side shows the output voltage \( V_2 \) and current \( I_2 \), connected to a load resistance \( R_L \).
- The coupling between the coils is less than perfect, represented by the coupling coefficient \( k \).
- Arrows indicate the direction of the currents \( I_1 \) and \( I_2 \) through the respective windings.
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