1. 1 © Unitec New Zealand DE4401 AC PHASORS BASICS

2. AC Resistors 2 © Unitec New Zealand

3. Resistor Current in phase with Voltage 3 © Unitec New Zealand

4. Resistor Power Phase 4 © Unitec New Zealand

5. Phasor Review 5 © Unitec New Zealand

6. Phase Difference 6 © Unitec New Zealand

7. Phasor Lag 7 © Unitec New Zealand

8. Phasor Sum 8 © Unitec New Zealand

9. Phasor Sum Resultant 9 © Unitec New Zealand

10. Phasor Subtraction 10 © Unitec New Zealand

11. AC Capacitor Breakdown voltage Above a particular electric field, known as the dielectric strength E ds, the dielectric in a capacitor becomes conductive. The voltage at which this occurs is called the breakdown voltage V bd of the device, and is given by the product of the dielectric strength and the separation between the conductors, d 11 © Unitec New Zealand Breakdown voltage limits the maximum energy that can be stored safely in a capacitor.

12. Series and parallel circuits Capacitors in a parallel configuration each have the same applied voltage. Their capacitances add up. –Charge is apportioned among them by size. Using the schematic diagram to visualize parallel plates, it is apparent that each capacitor contributes to the total surface area. Connected in series, the schematic diagram reveals that the separation distance, not the plate area, adds up. The total voltage difference from end to end is apportioned to each capacitor according to the inverse of its capacitance. –The entire series acts as a capacitor smaller than any of its components 12 © Unitec New Zealand

13. Steady-state analysis The response of a network has two parts: –the natural response –the forced response It was shown that the natural response (transition) decays rapidly to zero. The forced response for sinusoidal sources persists indefinitely, and therefore is called the steady-state response. Because the natural response quickly decays, the steady-state response is often of more interest. We are going to work with steady-state responses for sinusoidal sources, avoiding the problem of solving the differential equations. 13 © Unitec New Zealand

14. Phasors The steady-state analysis is greatly simplified if the currents and voltages are represented as vectors in the complex-number plane. These vectors are called phasors. Why doing this? Let’s assume we have to work with sine wave voltage: v(t) = V m cos(  t+  ) KVL and KCL may often give us an equations such as: v(t) = 20cos(  t)+ 4cos(  t-45)+ 5sin(  t+90) To solve these, we need to do some tediously slow trigonometry. Using phasors and the techniques of phasor analysis, solving circuits with sinusoidal sources gets much easier. 14 © Unitec New Zealand

15. Definition of a Phasor A phasor is a complex number, represented as a vector in the complex-number plane. Phasor is a complex number whose magnitude is the magnitude of a corresponding sinusoid, and whose phase is the phase of that corresponding sinusoid. 15 © Unitec New Zealand

16. Impedance The impedance is the ratio of the phasor of the voltage to the phasor of the current for that passive element. The unit for impedance is the same as the unit of resistance, since it is a ratio of voltage to current. –The impedance will behave like a resistance behaved in dc circuits. We use the symbol Z for impedance. 16 © Unitec New Zealand

17. Admitance The inverse of the impedance is called the admittance. The admittance is the ratio of the phasor of the current to the phasor of the voltage for that passive element. The ratio of phasor current to phasor voltage will have units of conductance, since it is a ratio of current to voltage. We use the symbol Y for admittance. The admittance will behave like a conductance behaved in dc circuits. 17 © Unitec New Zealand

18. Terminology for Impedance and Admittance The impedance and the admittance for a combination of elements will be complex. Thus, the impedance, or the admittance, can have a real part and an imaginary part. Alternatively, we can think of these values as having magnitude and phase. We have names for the real and imaginary parts. These names are shown below. 18 © Unitec New Zealand

19. Phasor Transforms of Resistors The phasor transform of a resistor is just a resistor. Remember that a resistor is a device with a constant ratio of voltage to current. If you take the ratio of the phasor of the voltage to the phasor of the current for a resistor, you get the resistance. For a resistor, the impedance and admittance are real. 19 © Unitec New Zealand

20. Phasor Transforms of Capacitors: Capacitive Reactance The phasor transform of a capacitor is a capacitor with an admittance of j  C. In other words, the capacitor has an admittance in the phasor domain which increases with frequency. This comes from taking the ratio of phasor current to phasor voltage for a capacitor, and is a direct result of the capacitive current being proportional to the derivative of the voltage. For a capacitor, the impedance and admittance are purely imaginary. The impedance has a negative imaginary part, and the admittance has a positive imaginary part. The impedance is: 20 © Unitec New Zealand

21. Capacitive reactance and frequency 21 © Unitec New Zealand  is frequency in radians per second:  =2  f Capacitive Reactance: Impedance:

22. Example 22 © Unitec New Zealand

23. Phasor Diagram for AC Capacitance 23 © Unitec New Zealand For a pure capacitor, V C “lags” I C by 90 o, or we can say that I C “leads” V C by 90 o The current is always leading the voltage by 1/4 of a cycle or π/2 = 90 o “out-of- phase” with the potential difference across the capacitor because of the charging and discharging process. This effect can also be represented by a phasor diagram where in a purely capacitive circuit the voltage “LAGS” the current by 90 o ( one quarter of a cycle).

24. Q factor The quality factor (or Q) of a capacitor at a given frequency is a measure of its efficiency. The higher the Q factor of the capacitor, the closer it approaches the behaviour of an ideal, lossless, capacitor. 24 © Unitec New Zealand  is frequency in radians per second, C is the capacitance R C is the series resistance of the capacitor