CHAPTER 17 (1): RLC CIRCUITS
Chapter 17: RLC Circuits (Part 1)
Series RLC Circuits
Objectives
At the end of the chapter (Part 1), the student will be able to:
Know the types of RLC circuits.
Determine AC impedance of series RLC circuits.
Analyze a series RLC circuit.
Understand the phasor relationship between applied voltages and circuit current in series RLC circuits.
Draw impedance and phasor diagrams of series RLC circuits.
Topics Covered
Analysis of series RLC circuits.
Analysis of parallel RLC circuits (to be covered in Part 2).
Power and power factor in RLC circuits (to be addressed in Part 2).
Revision of Key Concepts
The impedance of a capacitor is represented as:
Z_C = -jX_C (imaginary), where X_C is the capacitive reactance.
In an AC circuit, the opposition to current flow is called impedance, measured in ohms (Ω).
The impedance of a resistor is represented as:
Z_R = R (real).
Impedance of Series RLC Circuit
The impedance of an inductor is represented as:
Z_L = jX_L, where X_L is the inductive reactance.
Total impedance in series is:
Z_T = Z_1 + Z_2 + ... + Z_n.
Analysis of Series RLC Circuits
Total impedance of series RLC circuit:
Combines the impedance of the resistor, inductor, and capacitor.
Components of Impedance
Real part of Z is contributed by resistors.
Imaginary part is contributed by inductors and capacitors.
In polar form:
Z = R + j(X_L - X_C).
Impact of Reactance on Impedance
The imaginary part of Z can vary:
If X_L > X_C, Z will be positive (net inductive).
If X_L < X_C, Z will be negative (net capacitive).
Phasor Relationships
I (current) is the common electrical quantity in a series circuit, chosen as the reference phasor.
Phase difference between voltages and currents in a series RLC circuit is crucial.
Ohm’s Law connects voltages, current, and impedance.
Kirchhoff’s Voltage Law relates voltages around the circuit.
Diagrams Relationships
Impedance diagrams and phasor diagrams are related through the circuit current.
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Analysis Procedure
Calculate circuit impedance.
Calculate circuit current.
Determine associated voltage drops.
Draw phasor and impedance diagrams if necessary.
Example Calculation (17-1)
a) Circuit Analysis
Given: 75 Ω resistor, 25 Ω inductor reactance, and 60 Ω capacitor reactance in series with a 10 V, 10 kHz AC source.
Z = R + jX - jX_C = 75 + j25 - j60 = 75 - j35.
Convert to polar: Z = 82.8 ∠ -25°.
b) Impedance Diagram
Analysis yields: Z = 75 - j35.
c) Phasor Diagram
Voltage drops: 9.08 V across R, 3.03 V across L, and 7.26 V across C.
Total source voltage: 10 V.
d) Time Domain Expressions
Current: I = 121√2 sin(2π x 10 x 10^3 x t) mA
Voltage: V(t) = 10√2 sin[(2π x 10 x 10^3 x t) - 25°] V.
Summary
The voltage across the resistor is in phase with the current.
Voltage across the inductor leads the current by 90°.
Voltage across the capacitor lags the current by 90° in a series RLC circuit.