ELECTRICAL TECHNOLOGY EET 103/4 Define and explain sine wave, frequency, amplitude, phase angle, complex number Define, analyze and calculate impedance, inductance, phase shifting Explain and calculate active power, reactive power, power factor Define, explain, and analyze Ohm’s law, KCL, KVL, Source Transformation, Thevenin theorem.
SERIES AND PARALLEL AC CIRCUITS (CHAPTER 15)
15.1 Introduction In this chapter, phasor algebra will be used to develop a quick, direct method for solving both the series and the parallel ac circuits.
15.2 Impedance and the Phasor diagram Resistive Elements v and i are in phase Or; In phasor form; Where;
15.2 Impedance and the Phasor diagram Resistive Elements Applying Ohm’s law; Since v and i are in phase; Hence;
15.2 Impedance and the Phasor diagram The boldface Roman quantity ZR, having both magnitude and an associate angle, is referred to as the impedance of a resistive element. ZR is not a phasor since it does not vary with time. Even though the format R0° is very similar to the phasor notation for sinusoidal current and voltage, R and its associated angle of 0° are fixed, non-varying quantities.
15.2 Impedance and the Phasor diagram Example 15.1 Find i in the figure Solution Applying Ohm’s law;
15.2 Impedance and the Phasor diagram Example 15.1 – solution (cont’d) Inverse-transform;
15.2 Impedance and the Phasor diagram Example 15.2 Find v in the figure Solution
15.2 Impedance and the Phasor diagram Example 15.2 – solution (cont’d) Applying Ohm’s law;
15.2 Impedance and the Phasor diagram Example 15.2 – solution (cont’d) Inverse-transform;
15.2 Impedance and the Phasor diagram Inductive Reactance By phasor transformation; Where;
15.2 Impedance and the Phasor diagram Inductive Reactance By Ohm’s law;
15.2 Impedance and the Phasor diagram Inductive Reactance Since v leads i by 90; Hence; Therefore;
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.3 Find i in the figure Solution
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.3 – solution (cont’d) Applying Ohm’s law; Inverse-transform;
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.3 – solution (cont’d)
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.4 Find v in the figure Solution
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.4 – solution (cont’d) Applying Ohm’s law; Inverse-transform;
15.2 Impedance and the Phasor diagram Inductive Reactance Example 15.4 – solution (cont’d) Inverse-transform;
15.2 Impedance and the Phasor diagram Capacitive Reactance By phasor transformation; Where;
15.2 Impedance and the Phasor diagram Capacitive Reactance By Ohm’s law;
15.2 Impedance and the Phasor diagram Capacitive Reactance Since i leads v by 90; Hence; Therefore;
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.5 Find i in the figure Solution
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.5 – solution Use phasor transformation; By Ohm’s law;
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.5 – Solution (cont’d) Inverse-transform;
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.6 Find v in the figure
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.6 – solution Use phasor transformation; By Ohm’s law;
15.2 Impedance and the Phasor diagram Capacitive Reactance Example 15.6 – solution Inverse- transform;
15.2 Impedance and the Phasor diagram Impedance Diagram For any configuration (series, parallel, series-parallel, etc.), the angle associated with the total impedance is the angle by which the applied voltage leads the source current. For inductive networks, T will be positive, whereas for capacitive networks, T will be negative.
15.2 Impedance and the Phasor diagram
15.3 Series Configuration The overall properties of series ac circuits are the same as those for dc circuits. For instance, the total impedance of a system is the sum of the individual impedances:
15.3 Series Configuration In a series ac configuration having two impedances, the current is the same through each element (as it was for the series dc circuit) and is determined by Ohm’s Law:
15.3 Series Configuration Example 15.7 Draw the impedance diagram and find the total impedance
15.3 Series Configuration Example 15.7 – solution Impedance diagram
15.3 Series Configuration Example 15.8 Draw the impedance diagram and find the total impedance
15.3 Series Configuration Example 15.8 – solution Impedance diagram
15.3 Series Configuration Kirchhoff’s voltage law can be applied in the same manner as it is employed for a dc circuit The power to the circuit can be determined by: Where T is the phase angle between E and I.
15.3 Series Configuration R-L Phasor Notation
15.3 Series Configuration R-L ZT Impedance diagram:
15.3 Series Configuration R-L I
15.3 Series Configuration R-L VR and VL
15.3 Series Configuration R-L Kirchhoff’s voltage law Or;
15.3 Series Configuration R-L Kirchhoff’s voltage law From the above calculation;
15.3 Series Configuration R-L Phasor diagram
15.3 Series Configuration R-L Power Or;
15.3 Series Configuration R-L Power factor
15.3 Series Configuration R-C Phasor Notation
15.3 Series Configuration R-C ZT Impedance diagram:
15.3 Series Configuration R-C E
15.3 Series Configuration R-C VR and VC
15.3 Series Configuration R-C Phasor diagram
15.3 Series Configuration R-C Time domain
15.3 Series Configuration R-C Time domain plot
15.3 Series Configuration R-C Kirchhoff’s voltage law Or;
15.3 Series Configuration R-C Power Or;
15.3 Series Configuration R-C Power factor
15.3 Series Configuration R-L-C TIME DOMAIN PHASOR DOMAIN
15.3 Series Configuration R-L-C
15.3 Series Configuration R-L-C Impedance diagram:
15.3 Series Configuration R-L-C
15.3 Series Configuration R-L-C
15.3 Series Configuration R-L-C
15.3 Series Configuration R-L-C
15.3 Series Configuration R-L-C Phasor diagram
15.3 Series Configuration R-L-C Time-domain plot
15.3 Series Configuration R-L-C Total Power Or; Power factor
15.3 Series Configuration R-L-C Example 15.11(a) I, VR, VL and VC; Calculate I, VR, VL and VC;
15.3 Series Configuration R-L-C Example 15.11(a) – solution Combined the R’s, L’s and C’s;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Transform the circuit into phasor domain;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Find the total impedance;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Apply Ohm’s law;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Apply Ohm’s law;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Apply Ohm’s law;
15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d) Apply Ohm’s law;
15.3 Series Configuration R-L-C Example 15.11(b) Calculate the total power factor.
15.3 Series Configuration R-L-C Example 15.11(b) – solution The total power factor;
15.3 Series Configuration R-L-C Example 15.11(c) Calculate the total average power delivered to the circuit.
15.3 Series Configuration R-L-C Example 15.11(c) – solution The total average power delivered to the circuit.
15.3 Series Configuration R-L-C Example 15.11(d) Draw the phasor diagram;
15.3 Series Configuration R-L-C Example 15.11(d) – solution The phasor diagram;
15.3 Series Configuration R-L-C Example 15.11(e) Obtain the phasor sum of VR, VL and VC and show that it equals the input voltage E.
15.3 Series Configuration R-L-C Example 15.11(e) – solution
15.3 Series Configuration R-L-C Example 15.11(f) Find VR and VC using voltage divider rule.
15.3 Series Configuration R-L-C Example 15.11(f) – solution
15.3 Series Configuration R-L-C Example 15.11(f) – solution
15.6 Summaries of Series ac Circuits For a series ac circuit with reactive elements: The total impedance will be frequency dependent. The impedance of any one element can be greater than the total impedance of the network. The inductive and capacitive reactances are always in direct opposition on an impedance diagram. Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive.
15.6 Summaries of Series ac Circuits For a series ac circuit with reactive elements: At lower frequencies the capacitive elements will usually have the most impact on the total impedance, while at high frequencies the inductive elements will usually have the most impact. The magnitude of the voltage across any one element can be greater than the applied voltage.
15.6 Summaries of Series ac Circuits The magnitude of the voltage across an element as compared to the other elements of the circuit is directly related to the magnitude of its impedance; that is, the larger the impedance of an element , the larger the magnitude of the voltage across the element. The voltages across a coil or conductor are always in the direct opposition on a phasor diagram.
15.6 Summaries of Series AC Circuits The current is always in phase with the voltage across the resistive elements, lags the voltage across all the inductive elements by 90°, and leads the voltage across the capacitive elements by 90°. The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity.