15. Circuit Theorems In AC Analysis

• Circuit Theorems in AC analysis

  • Superposition Theorem
  • Thevenin's Theorem
  • Norton's Theorem
  • Maximum power transfer theorem
• Superposition Theorem


  The current in any branch of a multiple-source circuit can be found by determining the currents in that particular branch produced by each source acting alone, with all other sources replaced by their internal impedances. The total current in the given branch is the phasor sum of the individual source currents in that branch

  
• Superposition Theorem Procedure


  1. Leave one source, reduce others to zero

  2. Find current in the branch of interest produced by the one remaining source

  3. Repeat steps (1) and (2) for each source in turn

  4. Add the individual current values as phasors
• Assume ideal sources ie, internal impedance = zero

  
• Find current in R due to VS1

  1. Converting to Polar Form

  2. Use current divider to get current through R
• Find current in R due to VS2

  1. Converting to Polar Form

  2. Use current divider formula to get current in R due to VS2

  3. Converting to rectangular form to perform the addition
• Find the coil current

  1. Current due to IS1(with IS2 open)

  2. Current due to IS2(IS1 open)

  3. Total inductor current IL = IL1 + IL2

  
• Find total current in resistor RL

  1. Total current from VS1

  2. Find the current in Rl due to the d.c. source VS2 (short VS1)

  3. Total current in RL is 1.69 Ð 47.3°mA riding on a d.c. level of 5mA

  
• Thevenin's Theorem


  Any a.c. circuit regardless of complexity can be reduced to Thevenin's equivalent form

  
• Thevenin's equivalent voltage VTh

  VTh is the open circuit voltage between two specified points in a circuit

  
• Thevenin's Equivalent Impedance ZTh

  ZTh is the total impedance appearing between two specified terminals in a given circuit with all sources zeroed and replaced by their internal impedances (voltage = 0 current = ¥)

  
• Determine Vth as seen by RL

  VTh = VAB with RL removed. No current flow through R2 \no voltage drop across it
• Determine Vth for the circuit external to RL

  1. Zth as seen external to RL

  2. Effectively out of circuit with VS zeroed Zth = XC // R

  
• Norton's Theorem


  Parallel (rather than series) equivalent impedance
• Norton Equivalent Circuit


  • Theorem calculate IN ZN
  • IN is defined as the short circuit current between two specified points in a given circuit

  
• Determine In for the circuit 'seen' by the load resistor RL

  1. Short terminals AB to get In

  2. Total current = IS

  3. In = IC2 via current divider

  
• Norton's Equivalent Impedance


  Zn defined same as Zth (see earlier)
• Maximum Power Transfer Theorem


  Max power is transferred when the load impedance is the complex conjugate of the circuits output impedance
• Calculate power delivered to the load at f = 10kHz, 30kHz, 50 kHz, 80 kHz, 100 kHz (plot graph)

  1. Magnitude of total impedance

  2. Current

  3. Load Power

  
• Max power is transferred from the circuit to the load with a power factor of 1
• Determine the frequency at which max power is transferred from the amplifier to the speaker

  Power is maximum when RS - jXC and RW + jXL are complex conjugates