see also:

• signal generators
• multimeters
• spectrum analyzers
• oscilloscopes
• vector network analyzers (VNAs)
• software defined radio (SDR) receivers

• frequency response analyzers (FRAs) measure how a system responds to varying input frequencies, assessing both magnitude and phase changes across a selected range
• they inject a sinusoidal signal using a signal generators into the device under test (DUT) and simultaneously measure the input and output signals
• by sweeping this signal across a frequency range, it determines the system’s frequency response — the gain and phase shifts at different frequencies
• they use techniques like Discrete Fourier Transform (DFT) to isolate the test signal and reject noise, yielding high accuracy and dynamic range
• results are displayed as Bode, Nyquist, or Nichols plots, visualizing magnitude and phase response as functions of frequency
• these are essential for analyzing filters, amplifiers, system stability, and resonance behavior in electrical, mechanical, audio or optical systems

• impedance = frequency dependent resistance of a circuit element
• impedance of a resistor = R (its resistance)
• impedance of a capacitor = 1/jwC where C = capacitance
• impedance of a resistor and a capacitor in parallel = (R - jwCR²)/(1+w²C²R²)

• if you input a sine wave of frequency w and its equation u = Dsin(wt) and your system outputs a wave of y = DAsin(wt + phi) where phi is the phase delay or lead
• the output wave will have the same frequency as the input wave but it will be altered from the input by an amplitude Gain A(w) and a phase shift phi(w)
• the values for A and phi will change with change of the frequency of the input wave
• the frequency response of a system transfer function G(jw) over a range of frequencies can be calculated from a transfer function using complex numbers ¹⁾
  • A is the modulus of G(jw) and phi = argument of G(jw)
• general analysis features
  • when analyzing a circuit's behavior over a range of frequencies, to describe the circuit's transfer function, we use the complex frequency variable s = sigma+jomega (j = sqrt(-1) )
    • this transfer function, which can represent impedance, is a ratio of polynomials
    • the roots of the denominator polynomial are called poles, and the roots of the numerator polynomial are called zeros
      • a pole is a value of the complex frequency variable s at which the denominator of a transfer function equals zero, causing the system response to theoretically become infinitely large
      • a zero is a value of the complex variable s that makes the numerator of a system's transfer function equal to zero, causing the overall transfer function to be zero at that point
        • when s equals a zero, the output response of the system is zero for that input frequency or value, effectively canceling or blocking that part of the input signal
        • on a pole-zero plot, zeros are typically marked with circles (“o”) in the complex plane
    • plotting these poles and zeros on the complex s-plane is a key part of control systems and circuit analysis
    • the location of poles and zeros in the complex plane, particularly in the RHP, is crucial for determining a circuit's stability and dynamic behavior.
• complex number planes
  • Right-Half Plane (RHP) of the complex s-plane (refers to all points where the real axis is positive ie. quadrants I or IV):
    • the complex plane is divided into a Left-Half Plane (LHP), a Right-Half Plane (RHP), and the imaginary axis (jomega)
    • Poles in the RHP (sigma0):
      • a system with a pole in the RHP is unstable. This means that a disturbance will cause the system's response to grow exponentially over time. For an impedance, this would imply an unstable or non-passive circuit, where energy is generated rather than dissipated.
    • Zeros in the RHP:
      • a zero in the RHP doesn't cause instability, but it is a sign of a non-minimum phase system. These systems are challenging to control because they introduce a phase lag, even while the gain is increasing.

• frequency axis is a log axis with equal space to each decade which makes the frequency response much clearer instead of compressing the changes
• the vertical gain axis also uses a log axis for better visualisation and is denoted as dB while argument vertical axis is usually in degrees and is not logarithmic
  • db = 20Log₁₀modulus(G)
• see https://www.youtube.com/watch?v=uZI_uaHB3wE
• consists of two plots:
  • Magnitude (Gain) vs Frequency:
    • shows how much the system amplifies or attenuates signals at each frequency (represented in dB)
    • points where gain drops by 3 dB (-3 dB point) are important as they indicate bandwidth or cutoff frequency of filters or amplifiers
      • -3dB equates to 50% reduction of the input power and 70.7% reduction of the input voltage
    • you can identify resonant frequencies (peaks) where the system naturally amplifies signals, or dips where signals are attenuated
  • Phase vs Frequency:
    • shows the phase shift introduced by the system at each frequency (in degrees)
    • reflects the lag or lead introduced, relative to the input signal, which affects system stability and timing

• in contrast to the Bode Plot, in Nyquist plots the frequency is not explicit
• it is a plot of the imaginary component (phase shift) on the Y axis vs the real Gain component on the X axis
• see https://www.youtube.com/watch?v=9j7PMgFmAq0
  • in Electrochemical Impedance Spectroscopy (EIS), for a lithium battery on charge, there are usually 3 parts to the Nyquist plot:
    • initial zero imaginary part
    • a semicircular middle part associated with higher frequency electrochemical events such as chemical reactions which starts at the “solution resistance” and returns to the x axis at the “charge transfer resistance”
      • this part equates to a resistor and a capacitor in parallel
    • finally a linear terminal section which is called the “Warberg impedance” and represents lower frequency electrochemical events such as ionic diffusion

• a graphical representation used in electrical engineering, control systems, and signal processing to show the locations of the poles and zeros of a system’s transfer function on the complex frequency plane (often the s-plane)
• the plot displays poles as crosses (“×”) and zeros as circles (“o”) on the complex plane, where the horizontal axis represents the real part and the vertical axis the imaginary part of the complex variable s
• provides a visual insight into the system’s behavior, such as stability, frequency response, and transient characteristics
• the proximity of input frequencies to poles or zeros influences the system output magnitude and phase—close to a pole, the output magnitude tends to increase, while close to a zero, it tends to decrease
• key for assessing system stability: poles in the Right-Half Plane indicate instability, while poles in the Left-Half Plane indicate stability
• helps in filter design and tuning by showing how zeros and poles shape the frequency response
• the pattern and distribution can predict oscillatory behavior, damping, and resonance of electronic and control systems
• FRAs typically do not directly display pole-zero plots as part of their standard output, but the relationship between pole-zero plots and frequency response is very close, and from the frequency response data obtained by FRAs, a pole-zero plot can be derived or computed

• Control System Design: Assessing stability and optimizing feedback loops in power supplies and servo motor systems
• Electronic Component Testing: Characterizing filters, transistors, amplifiers, optocouplers, signal transformers, cables, and measuring resonance in piezo elements.
• Power Electronics: Measuring loop gain and phase margin in voltage regulators and switching power converters
• Electrochemical Impedance Analysis: Studying impedance and admittance of batteries, fuel cells, and other electrochemical devices.
• Audio and RF Design: Testing frequency response of amplifiers, filters, and transmission lines.
  • see https://www.youtube.com/watch?v=f1hqbQlysjA
• Stability Analysis: Verifying system stability by quantitative evaluation of phase and gain margins.

• while both can assess loop gain and phase, FRAs are optimized for stability analysis of control systems, whereas VNAs focus on S-parameter and impedance characterization, especially for high-frequency and RF applications
• VNAs maintain precise port impedances (typically 50Ω), measuring S-parameters (S11, S12, S21, S22) for reflection and transmission
• VNAs excel in very low impedance and high-frequency measurements, performing calibration (Short, Open, Load, Thru) to correct cable and probe artifacts
• VNAs can assess device output impedance, PSRR, and non-invasive phase margin, even when the feedback loop cannot be accessed directly
• VNAs are better for non-invasive stability evaluation of devices where access to the control loop is limited or unavailable
• the S parameter from VNAs can be used to create FRA-like Bode plots but there are important differences in interpretation
  • accurately mapping S-parameters to Bode plots requires injecting the signal into a break in the feedback loop and measuring the ratio of voltages (VA/ VB) across that injection point over frequency, which the VNA’s S21 supports
• FRAs are preferred for control loop stability testing, direct feedback path analysis, and robustness evaluation of power supplies and analog regulators.
• FRAs are generally not great at high frequencies above 10's of MHz and generally have a high port impedance of 1 MOhm