Lesson 11 AC Circuits AC Ciruits Power Maximum

Lesson 11 AC Circuits ¨AC Ciruits ¨Power ¨Maximum and Instantaneous voltage drops and current ¨Phasor Diagrams ¨Phase angle ¨RLC Circuits ¨Resonance frequency ¨High and Low pass filters ¨Step up and Step down Transformers

AC Generator

AC emf source

Effective (Integrated ) values of I and V i (t ) = I max sin (w t ) w = 2 pf = rms current 2 p ; T is the period of oscillation T Instantaneous power = i (t )v (t ) Heat dissipated =Power used in load = i ( t ) R = I m R sin (w t) Average power over one cycle 2 Pave = 1 T ò T 1 2 w = ( ) I R sin t dt I m R 2 m 2 T 0 ò P ave 2 m I R w = ( ) t dt sin Define 2 2 T 0 = I rms. R Þ I rms = 2 Im 2 2 2

Alternating Current Circuits

ac-R circuit Veff and v(t) Ieff and i(t) t)

Phasor Diagram Phasor diagram for R i (t) R IRmsin( t)= i. R(t) t Current through Load Resistance

Phasor Diagram Phasor diagram for R cont. v (t) R VRmsin( t)= v. R(t) t PD across Load Resistance

Instantaneous current and voltage

v. C(t) t) ac-C circuit i. C(t) Current in Circuit and PD across Capacitor v. C(t) t

v. L(t) t) ac-L circuit i. L(t) Current in Circuit and PD across Inductor v. L(t) t i. L(t)

Summary The phase angle between the current and the voltage: In the resistor is 0 rad In the capacitor is - rad ( Current Ahead) In the inductor is + rad (Current Behind)

series ac-RLC circuit Series RLC circuit

Instantaneous current Current through all elements is the same Thus the instantaneous PD’s must be out of phase

Picture Total Potential Drop across R, L & C.

Phasor Diagram for RLC circuit I v. L(t) v. R(t) v. C(t)

Instantaneous PD

Phasor Diagram for RLC circuit II v. L(t) v. RLC(t) v. R(t) v. C(t)

Instantaneous PD as y-axis projection onto v(t 2) v(t 1) v. RLC(t)

Phase v. RLC(t) Angle Phase Angle V Lm - V Cm tan (f ) = V Rm I m X L - I m X C X L - XC = = Im R R -1 æ X L - XC ö f = tan ç ÷ è R ø

series ac-RLC graph

Impedance The magnitude of the Total Potential Phasor is V m = V + (V Lm - VCm ) 2 2 R = I R + (Im X L - Im XC ) = I m R + (XL - XC ) 2 m 2 2 = Im Z Impedance : Z = R + (XL - XC ) 2 2

Table of definitions

Impedance and reactance

Generalized Ohm's Law. Impedance Z = R + (XL - XC ) 2 2

Phase Angle between total PD across circuit and the current

Power is only used in AC circuit in load resistance 2 ( ) P t =i t R Power Factor (energy is not used in inductor or capacitor) 2 2 = I m sin (w t - f ) R (current is always in phase with PD across total resistance) P ave æe ç rms ç Z è ö ÷ I rms R = ÷ ø 2 m I = = I R R 2 ß e 2 rms I rms R= Z e rms I rms cos( f ) cos (f ) = Power Factor

Power and current Angular frequency depend on angular dependence frequency of circuit

I m (w ) = Vm Z (w ) = e m Z (w ) Max I ; Min Z Z (w ) = R + (X L - X C ) 2 æ 1 ö ÷ = R + ç w. L è w. C ø 2 = 2 Z (w ) is a minimum when 2 2 2 æ ö w LC - 1 2 ÷ R +ç w. C ø è w 2 LC - 1 = 0 which occurs when 1 w = w 0 = LC Û X L = XC

Power as a function of 2 Vm R 2 1 1 Pave (w ) = I m (w ) R = 2 2 2 Z (w ) 1 = 2 2 Vm R 2 1 Vm. R 2 = 2 æ ö L 2 2 2 1 2 ÷ R + ç w. L R + 2 (w - w 0 ) è w w. C ø 2 2 1 V m Rw = 2 2 2 2 R w + L (w - w 0 )

Resonance Circuit uses most power / current when it is in RESONANCE with applied frequency

Imax and Pave versus Pave Im

¨Width of Power curve is a measure of the QUALITY of the circuit ¨Small width - High Quality ¨Sharpness of response to external frequency Quality of circuit

RC Filters I Low Pass Filter Vout Vin

Low Pass Filter

RC Filters II High Pass Filter Vout Vin

High Pass Filter

Step up and Step Transformers I down Transformers

V V Transformers II F d 1 = - N 1 2 = - N 2 B dt d FB dt Fluxes are the same N 2 = V 2 V N 1 1