ELL100: INTRODUCTION TO ELECTRICAL ENGG. Lecture 4 Course Instructors: J.-B. Seo, S. Srirangarajan, S.-D. Roy, and S. Janardhanan Department of Electrical Engineering, IITD

Common Signal Waveforms

Common Signal Waveforms

Common Signal Waveforms Battery, Ideal sources Switch Oscillating Switch AC current, many places Caps

Signal Waveforms (2) Exponential Signals

Signal Waveforms (2) Exponential Signals a : Instantaneous value of function A : Maximum Amplitude t : time variable Tau : time constant

Signal Waveforms (2) Exponential Signals e = 2.718 a : Instantaneous value of function A : Maximum Amplitude t : time variable Tau : time constant

Signal Waveforms (2) Exponential Signals a : Instantaneous value of function A : Maximum Amplitude t : time variable Tau : time constant So, after ‘tau’ time an exponential function reduces by ~ 63 %

Signal Waveforms (2) Exponential Signals a : Instantaneous value of function A : Maximum Amplitude t : time variable Tau : time constant So, after ‘tau’ time an exponential function reduces by ~ 63 %

Signal Waveforms (2) Exponential Signals a : Instantaneous value of function A : Maximum Amplitude t : time variable Tau : time constant So, after ‘tau’ time an exponential function reduces by ~ 63 %

Example 1

Example 1

Example 1 Solution Step 1: Calculate VR Step 2: Calculate VL

Example 1 Solution Step 1: Calculate VR Step 2: Calculate VL

Example 1 Solution Step 1: Calculate VR Step 2: Calculate VL

Example 1 Solution Step 1: Calculate VR Step 2: Calculate VL

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC Solution

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC Solution

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC Solution

Example 1 Solution Step 3: Calculate VRL Step 4: Calculate VC Solution

Signal Waveforms (3) a : Instantaneous value of function A : Maximum Amplitude t : time variable in seconds T : time for 1 full cycle  Period f : frequency  cycles/sec or Hertz ω : frequency in radians/sec  2πf α : phase angle/offset angle (radians)

Signal Waveforms (3) a : Instantaneous value of function A : Maximum Amplitude t : time variable in seconds T : time for 1 full cycle  Period f : frequency  cycles/sec or Hertz ω : frequency in radians/sec  2πf α : phase angle/offset angle (radians)

Signal Waveforms (3) a : Instantaneous value of function A : Maximum Amplitude t : time variable in seconds T : time for 1 full cycle  Period f : frequency  cycles/sec or Hertz ω : frequency in radians/sec  2πf α : phase angle/offset angle (radians)

Example 2 A Sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20 A at t = 2 ms. Write the equation of current as a function of time.

Example 2 A Sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20 A at t = 2 ms. Write the equation of current as a function of time.

Example 2 A Sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20 A at t = 2 ms. Write the equation of current as a function of time. As per Sine wave equation, A = 20 A ω = 2πf = 2π (60) = 377 rad/s The positive maximum is reached when (ωt + α) = 0 or any multiple of 2π. Letting (ωt + α) = 0, α = - ωt = -377 * 2 * 10-3 = (-) 0.754 rad Or α = - 0.754 * (360°/ 2π) = - 43.2 °

Example 2 A Sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20 A at t = 2 ms. Write the equation of current as a function of time. As per Sine wave equation, A = 20 A ω = 2πf = 2π (60) = 377 rad/s The positive maximum is reached when (ωt + α) = 0 or any multiple of 2π. Letting (ωt + α) = 0, α = - ωt = -377 * 2 * 10-3 = (-) 0.754 rad Or α = - 0.754 * (360°/ 2π) = - 43.2 ° i = 20 cos (377t – 43.2°)

Signal Waveforms (4)

Periodic Waveforms

Periodic Waveforms (2)

Periodic Waveforms (2)

Periodic Waveforms (2)

Periodic Waveforms (2)

Periodic Waveforms (2) For sinusoidal wave total Q over full cycle = 0

Periodic Waveforms (3)

Periodic Waveforms (3)

Periodic Waveforms (3)

Periodic Waveforms (3)

Periodic Waveforms (3)

Periodic Waveforms (3)

Periodic Waveforms (4) Delivering equivalent Energy

Periodic Waveforms (4) Delivering equivalent Energy + = Ieff

Periodic Waveforms (4) Delivering equivalent Energy + = Ieff

Periodic Waveforms (4) = Ieff + Delivering equivalent Energy

Periodic Waveforms (4) = Ieff + Delivering equivalent Energy

Periodic Waveforms (5)

Periodic Waveforms (5)

Periodic Waveforms (5)

Periodic Waveforms (5)

Periodic Waveforms (5) Delivering equivalent Energy + = Ieff

Periodic Waveforms (5) = Ieff + Delivering equivalent Energy + = Ieff So for sinosoidal current, the Ieff/DC equivalent = 0.707 Im = Irms