An electric circuit transfer function H(s) or H(jw)

𝐻(𝑠) or 𝐻(𝑗𝜔) is the frequency-dependent ratio of a circuit’s output signal (voltage/current) to its input signal, found by transforming the circuit to the s-domain (Laplace) or frequency domain (𝑗𝜔) to analyze how it modifies amplitude and phase, crucial for designing filters and control systems by revealing characteristics like gain and phase shifts. 

Key Concepts 

• Definition: H(s)=Output Signal(s)Input Signal(s)cap H open paren s close paren equals the fraction with numerator Output Signal open paren s close paren and denominator Input Signal open paren s close paren end-fraction𝐻(𝑠)=Output Signal(𝑠)Input Signal(𝑠) (Laplace Domain) or H(jω)=Output PhasorInputPhasorcap H open paren j omega close paren equals the fraction with numerator Output Phasor and denominator cap I n p u t cap P h a s o r end-fraction𝐻(𝑗𝜔)=Output Phasor𝐼𝑛𝑝𝑢𝑡𝑃ℎ𝑎𝑠𝑜𝑟 (Frequency Domain).
• Domain Transformation: Use the Laplace transform to convert time-domain circuits (with resistors, capacitors, inductors) into the s-domain, replacing Vcap V𝑉 with V(s)cap V open paren s close paren𝑉(𝑠), Icap I𝐼 with I(s)cap I open paren s close paren𝐼(𝑠), Rcap R𝑅 with Rcap R𝑅, Lcap L𝐿 with sLs cap L𝑠𝐿, and Ccap C𝐶 with 1/(sC)1 / open paren s cap C close paren1/(𝑠𝐶).
• Linear Time-Invariant Systems (LTI): Transfer functions strictly apply to linear, time-invariant circuits, where components behave consistently over time. 

How It Works (Example) 

1. Identify Input/Output: Pick the input voltage (Vincap V sub i n end-sub𝑉𝑖𝑛) and output voltage (Voutcap V sub o u t end-sub𝑉𝑜𝑢𝑡) for the desired relationship.
2. Transform to s-Domain: Convert the circuit components (R, L, C) and apply the Laplace transform to voltages and currents.
3. Solve the Ratio: Use circuit analysis techniques (like voltage dividers or node equations) to find Vout(s)cap V sub o u t end-sub open paren s close paren𝑉𝑜𝑢𝑡(𝑠) in terms of Vin(s)cap V sub i n end-sub open paren s close paren𝑉𝑖𝑛(𝑠).
4. Form the Function: Divide Vout(s)cap V sub o u t end-sub open paren s close paren𝑉𝑜𝑢𝑡(𝑠) by Vin(s)cap V sub i n end-sub open paren s close paren𝑉𝑖𝑛(𝑠) to get H(s)=Vout(s)Vin(s)cap H open paren s close paren equals the fraction with numerator cap V sub o u t end-sub open paren s close paren and denominator cap V sub i n end-sub open paren s close paren end-fraction𝐻(𝑠)=𝑉𝑜𝑢𝑡(𝑠)𝑉𝑖𝑛(𝑠).
5. Analyze Frequency Response: Substitute s=jωs equals j omega𝑠=𝑗𝜔 (where ωomega𝜔 is frequency) to get H(jω)cap H open paren j omega close paren𝐻(𝑗𝜔), then plot its magnitude |H(jω)|the absolute value of cap H open paren j omega close paren end-absolute-value|𝐻(𝑗𝜔)| (gain) and phase ∠H(jω)angle cap H open paren j omega close paren∠𝐻(𝑗𝜔) versus ωomega𝜔. 

Applications 

• Filtering: Design low-pass, high-pass, band-pass, or notch filters to select or reject specific frequencies.
• Control Systems: Analyze system stability and performance in power supplies, amplifiers, and other dynamic systems.
• System Characterization: Understand how a circuit’s gain and phase change with input frequency.