From one-frequency AC solutions to frequency response

A transfer ratio is meaningless until the input quantity, output quantity and their physical references are fixed. The input may be a voltage or current; the output may also be a voltage or current. Their terminal polarity or positive current direction belongs to the model.

At one declared angular frequency ω, let the real sinusoidal steady-state quantities be represented by AoN peak phasors x and y. If the network is linear and time invariant and the selected output is proportional to the selected input at that frequency, write

The symbol H(jω) is not underlined. It is not one harmonic phasor. It is the complex input/output function evaluated at angular frequency ω.

Suppose the same network and the same input/output references are tested at three angular frequencies. AC-01 gives three independent steady-state ratios:

The network itself has not been changed. We are sampling the same mapping at different ω. In the mathematical model, allowing ω to vary continuously defines the frequency-response function H(jω).

This point is conceptually important: a phasor belongs to one sinusoidal component at one chosen frequency; a frequency response is a function whose argument is frequency.

Consider a constant-coefficient linear differential relation between a chosen input x(t) and output y(t):

For sinusoidal steady state at angular frequency ω, AC-01 established

Therefore the differential equation becomes an algebraic equation in the phasors:

Whenever a bounded sinusoidal steady-state solution exists and A(jω)≠0,

The frequency dependence is therefore not added by hand. It comes from how derivatives act on a sinusoid.

If the input phasor is scaled by any complex constant α while the system remains in its linear regime, linearity gives

Hence

That is why frequency response characterizes the linear network rather than the arbitrary test amplitude. If measured gain changes when input amplitude is changed at fixed frequency and bias, either the system is nonlinear, the operating point changed, a limit was reached, or the measurement itself changed.

Time invariance is equally important: if the system parameters change during the experiment, a single stationary H(jω) may no longer describe it.

The word “gain” is therefore incomplete unless the two quantities are named. A voltage transfer ratio may be dimensionless; a transimpedance is a transfer function too, but it has units of ohms.

AoN keeps the notation separated deliberately:

A general transform such as X(ω) describes frequency content of a signal; H(jω) describes how the system acts on each sinusoidal frequency under the LTI model. They can later be multiplied in transform methods, but they are not the same concept.

Many systems texts reserve the term transfer function for a more general function H(s), often introduced using the Laplace transform, and call H(jω) its frequency response on the imaginary axis when that evaluation is meaningful. AC-02 does not require Laplace theory in this first chapter: it first establishes H(jω) directly from sinusoidal steady state. The broader H(s) language will be introduced only when it adds explanatory value.

Take an ideal series R–C network. Define the input voltage across the complete series pair and the output voltage across the capacitor, with both voltage references using the same return node. At angular frequency ω,

The same current flows through R and C, so the complex voltage-divider relation gives

Multiply numerator and denominator by jωC:

This compact expression contains the complete ideal sinusoidal steady-state input/output behavior of this chosen port definition.

Before plotting anything, perform two physical limits:

At very low frequency the ideal capacitor impedance is large and the capacitor carries nearly the full input voltage. At very high frequency its impedance magnitude becomes small and the output across it tends toward zero.

Choose R=1 kΩ and C=100 nF. Then RC=100 µs. Define the characteristic frequency

Evaluate the exact complex function, not an asymptotic sketch:

If the input is 2∠20° V at the characteristic frequency, the output is simply

The function tells us both amplitude scaling and phase shift. Chapter 02 will formalize those two projections of a complex response.

A practical sinusoidal sweep estimates the same model quantity:

1. Fix the circuit, bias/operating point and the input/output reference definitions.
2. Choose one frequency and wait for the relevant steady state.
3. Measure input and output amplitude and relative phase using instrumentation whose loading is understood.
4. Form the complex ratio output/input.
5. Repeat at other frequencies without silently changing amplitude, termination, probe configuration or reference definition.

A measured sweep is an estimate of the physical system's response under the stated setup. It can disagree with the ideal H(jω) because of component tolerances, parasitics, source/load impedance, probe loading, finite bandwidth, noise or genuine model failure.

1. Explain in one sentence why a phasor and H(jω) are different mathematical objects.
2. A voltage gain is defined with the output polarity reversed. How does H(jω) change?
3. Classify the units of Vout/Vin, Iout/Iin, Vout/Iin and Iout/Vin.
4. From τ dy/dt+y=Kx, derive H(jω)=Y̲/X̲.
5. For the differential equation in exercise 4, find the low-frequency and high-frequency limits.
6. Derive the RC low-pass response from the impedance divider and verify that H(jω₀)=(1−j)/2 at ω₀=1/(RC).
7. Why should a linear frequency-response measurement be repeated at a second input amplitude?
8. A measured RC response differs from the ideal model only above 1 MHz. Give four physically different reasons that do not require an algebra error.

Chapter 02 will take the same complex H(jω) and separate it systematically into magnitude and phase versus normalized frequency. Only after that will poles, zeros and Bode asymptotes be introduced.