Output Impedance, Multistage Circuits, and Complex Circuit Analysis

Output Impedance Fundamentals and Multistage Amplifiers

• Output Impedance Definition:

  • Output impedance is calculated by taking the output voltage function and dividing it by the output current function.
  • Both the output voltage and output current are complex expressions defined as functions of the angular frequency $\omega$

• Measurement Protocol for Output Impedance:

  • Output impedance characterizes the internal behavior of the circuit itself.
  • Independent input sources must not be actively driving voltage or current to the output during measurement.
  • To properly measure or calculate output impedance, all independent input voltage sources must be zeroed out ($V_{in} = 0\,\text{V}$).

• Multistage Circuit Applications:

  • Impedance considerations are essential in multi-stage systems, such as power supply networks and multi-stage amplifiers (for example, a three-stage amplifier).
  • When analyzing an intermediate stage (such as the second stage in a three-stage amplifier), the output impedance of the preceding stage directly influences signal behavior.
  • The transfer of signals from one stage to the next depends fundamentally on both input and output impedances across stages.

• Distinction Between Impedance and Transfer Functions:

  • Impedance calculations involve voltage and current measured on the same side or terminal pair of a circuit (e.g., input voltage divided by input current, or output voltage divided by output current).
  • Because both quantities belong to the same side of the circuit, an impedance expression is not classified as a transfer function, though it remains a critical analytical concept.

Complex Number Calculations for Circuit Analysis

• Phase Angle Calculation:

  • The phase angle $\theta$ of a complex number expressed in rectangular components $x$ (real) and $y$ (imaginary) is determined using the inverse tangent function:     $\theta = \arctan\left(\frac{y}{x}\right)$

• Operational Preferences for Complex Numbers:

  • Addition and Subtraction: When adding or subtracting complex numbers, converting them to rectangular form ($x + j\,y$) simplifies the operation.
  • Multiplication and Division: When multiplying or dividing complex numbers, converting them to polar form ($r\angle\theta$) simplifies the operation.

Framework for Circuit Analysis and Transfer Functions

• Primary Objective:

  • The overarching goal of the circuit analysis procedure is to derive the overall system transfer function.

• Systematic Impedance Method:

  • Step 1 ($Z$ - Impedances): The letter $Z$ represents circuit impedances. Convert all circuit elements and components into their frequency-domain impedance equivalents ($Z$).
  • Step 2 ($O$ - Ohm's Law): The letter $O$ represents the application of Ohm's Law ($V = I \times Z$). Combine Ohm's Law with circuit principles to evaluate the impedance network and complete the derivation of the transfer function.