AC Circuits Lecture 2/17/26

AC Circuit Analysis and Controlled Sources

General Overview

  • AC (Alternating Current) supply involves a phase angle, which can be arbitrary, possibly zero.

  • Controlled sources are represented as diamonds in circuit diagrams.

  • Types of controlled sources include:

    • Voltage Controlled Voltage Sources (VCVS)

    • Current Controlled Voltage Sources (CCVS)

    • Voltage Controlled Current Sources (VCCS)

    • Current Controlled Current Sources (CCCS)

DC Current Sources

  • A DC current source is denoted with an arrow, indicating a constant current flow (e.g., 5 Amps) regardless of load.

Voltage and Current Controlled Sources

  • Voltage Controlled Current Source Example:

    • If we denote a voltage as (V1 - V3), the current can be represented as:

    • (I = 5(V1 - V3))

    • Here, the output current is controlled by the voltage difference, multiplied by a scaling factor.

  • Current Controlled Voltage Source Example:

    • Can be represented as:

    • (V = 5I_1)

    • The output voltage is a function of the input current.

Transformer's Operation

  • When dealing with transformers:

    • The secondary winding experiences induced voltage due to current flowing in the primary winding.

    • The relationship can be represented as:

    • (V = M I_1) (where M is mutual inductance) with a 90-degree phase shift.

Steady State in AC Circuits

  • Basic Assumptions for AC Circuit Analysis:

    • In steady state, every current and voltage in a network has the same radian frequency.

    • For instance, a network powered by Hoover Dam will exhibit uniform frequency across currents and voltages.

Transients

  • Transients occur when loads are switched on/off. For example, a washing machine motor does not immediately turn on when powered due to inductance.

  • The time taken to reach a steady state depends on the network's characteristics.

Steady State vs Transients

  • Steady state means all time derivatives are zero:

    • Current through a capacitor: (IC = C \frac{dV}{dt}) implies that if (\frac{dV}{dt} = 0), then (IC = 0).

    • Steady state indicates a system where all derivatives (voltage and current) are at constant values.

Voltage and Current Relationships

  • For AC circuits, the voltage and current can experience phase shifts depending on the components involved:

    • Pure resistors cause zero phase difference between voltage and current.

    • Inductors and capacitors introduce 90-degree phase differences:

    • Inductor: Current lags voltage by 90 degrees.

    • Capacitor: Current leads voltage by 90 degrees.

Power Considerations

  • The power dissipated in a circuit with voltage and current out of phase (e.g., resistive and reactive components) may average out to zero under certain conditions.

  • Voltage across an inductor is given by:

    • (V_L = L \frac{dI}{dt})

  • Power factor can be characterized using:

    • (P = V{rms} I{rms} \cos(\theta)) where (\theta) is the phase angle.

Impedance Calculation

  • Impedance denoted as (Z) generalizes resistance under AC:

    • For an inductor: (Z_L = j \omega L)

    • For a capacitor: (Z_C = \frac{1}{j \omega C})

Mesh Analysis in RLC Circuits

  • When analyzing an RLC series circuit, Kirchhoff’s voltage law yields:

    • (V{s} = I (ZL + Z_C + R))

    • From this, the current can be solved as:

    • (I = \frac{Vs}{ZL + Z_C + R})

Transfer Functions

  • The transfer function (H(\omega)) is defined for the filter as:

    • (H(\omega) = \frac{V{out}}{V{in}})

    • E.g., if the output voltage is related to the input voltage derived from mesh equations.

Bode Plotting in MATLAB

  • A Bode plot visualizes the magnitude and phase response of the transfer function over a range of frequencies:

    • Magnitude in decibels and phase in degrees plotted against log frequency.

    • MATLAB code is used to perform calculations and visually present the response of the circuit.

Resonance in RLC Circuits

  • Resonance occurs when the inductive reactance equals the capacitive reactance, leading to optimal transfer of energy with maximum output.

  • The formula for resonant frequency is given by:

    • (\omega_0 = \frac{1}{\sqrt{LC}})

Conclusion

  • Understanding AC circuit analysis requires an appreciation for steady states, transient responses, and the roles of inductors and capacitors in producing phase shifts. The integration of MATLAB simplifies analysis and enhances understanding with practical applications, such as real-world circuits and filter design.