(a) Sketch the phasor diagram for an ac circuit with a 105-Ωresistor in series with a 22.5-mH in

(a) Sketch the phasor diagram for an ac circuit with a...

Key Concepts

Phasor Diagrams

Phasor diagrams graphically represent sinusoidally varying AC quantities as rotating vectors in the complex plane. This tool simplifies the analysis of AC circuits by showing the magnitude and phase relationships between voltages and currents across different components, making it easier to understand how resistors, inductors, and capacitors interact.

Impedance in AC Circuits

Impedance is the total opposition that an AC circuit presents to current flow, combining both resistive (real) and reactive (imaginary) components. In series circuits, this complex quantity is calculated by adding the resistance and the net reactance (which is the difference between inductive and capacitive reactances), enabling the determination of overall circuit behavior including phase shifts.

Inductive Reactance

Inductive reactance is the resistance to changes in current offered by an inductor in an AC circuit. It is defined by the relationship X_L = ?L, where ? is the angular frequency and L is the inductance. This reactance contributes a positive imaginary component to the total impedance, causing the current to lag behind the voltage.

Capacitive Reactance

Capacitive reactance is the opposition to changes in voltage offered by a capacitor. It is calculated as X_C = 1/(?C), where ? is the angular frequency and C is the capacitance. In contrast to inductive reactance, capacitive reactance contributes a negative imaginary component to the total impedance, causing the current to lead the voltage.

RMS Values in AC Analysis

Root Mean Square (RMS) values are used to express AC voltages and currents in a form that is equivalent to a DC value in terms of power delivery. RMS values provide a practical measure for power calculations and circuit analysis, making it possible to compare AC and DC systems on an equal basis.

Average Power Consumption in AC Circuits

The average power consumed by an AC circuit is determined by the real power dissipated in its resistive element(s), with reactive components (inductors and capacitors) contributing no net energy consumption over one cycle. Power calculations typically use the RMS values and the phase relationship between voltage and current, with the formula P = I_RMS^2 · R or equivalently P = V_RMS^2 · (R/|Z|^2) for circuits where impedance affects the phase.

Transcript

00:01 To solve the first item, we have to calculate what is the phase, and only then we can draw a phase or diagram.

00:08 And to calculate the phase, we have to do the following.

00:11 Remember that the tangent of the phase is equal to the difference between the inductive and capacitive reactances divided by the resistance.

00:22 So, before calculating the phase, we have to calculate what are the reactances.

00:27 We will do this on the other side.

00:31 Beginning by the reactance of the inductor, we have the following.

00:34 So the reactance of the inductor is equal to 2 pi times the frequency times the inductance.

00:41 Plugging in the values that were given by the problem, we have 2 pi times 60 times 22 .5 times 10 to the minus 3.

00:53 And this is approximately 8 .482 oms.

01:01 And now we calculate the reactance of the capacitor, which is equal to 1 divided by 2 pi times the frequency times the capacitance.

01:10 Plagging in the values given by the problem, we have 1 divided by 2 pi times 60 times 32 .2 times 10 to the minus 6.

01:26 And this is approximately 8 .2 .38 oms.

01:36 And then we can calculate what is the tangent of the phase.

01:40 So the tangent of the phase is equal to 8 .482 minus 8 .82 .388.

01:55 Divided by 105 and then the phase is equal to the inverse tangent of 8 .482 minus 82 .382 divided by 105 which is approximately minus 35 .14 degrees.

02:20 Now we can proceed to draw our phasor diagram.

02:26 Our reference axis.

02:28 We will have three phasers, one for the resistor, one for the inductor, and one for the capacitor, and also another one for the source.

02:36 So there is a total of four phasors.

02:39 Note that the resistance is bigger than both reactances.

02:44 So our biggest phaser will be the resistance phaser.

02:49 So the resistor phaser will be this one.

02:56 And now, now we can draw either the inductor phaser or the capacitor phaser.

03:02 We will choose to draw the inductor phaser first...