Physics 312: Electronics (1) Lecture 7 AC Current I Fundamentals of Electronics Circuits (with CD-ROH) By: Charles Alexander, Hathew Sadika, McGraw Hill (2003)  Introduction of Alternating Current  How do AC circuits work, compared with DC? Advantages? Disadvantages? Westinghouse vs. Edison?  What roles do inductors, capacitors, and resistors play in AC circuits?  How can we mathematically model AC circuits and the complex relationships of voltage and current through all components?

Why use AC? The "War of the Currents" Late 1880's: Westinghouse backed AC, developed by Tesla, Edison backed DC (despite Tesla's advice). Edison killed an elephant (with AC) to prove his point. Turning point when Westinghouse won the contract for the Chicago Worlds fair Westinghouse was right P L =I 2 R L : Lowest transmission loss uses High Voltages and Low Currents With DC, difficult to transform high voltage to more practical low voltage efficiently AC transformers are simple and extremely efficient. Nowadays, distribute electricity at up to 765 kV

Alternating Current pure direct current = DC Direction of charge flow (current) always the same and constant. pulsating DC Direction of charge flow always the same but variable AC = Alternating Current Direction of Charge flow alternates pure DC pulsating DC AC V V V V -V

Alternating currents Voltage (supply) is a sinusoidal function of time V(t) = V max cos  t

We have studied that a loop of wire, spinning in a constant magnetic field will have an induced emf that oscillates with time, That is, it is an AC generator. Alternating Current: AC’s are very easy to generate, they are also easy to amplify and decrease in voltage. This in turn makes them easy to send in distribution grids like the ones that power our homes.

9.2 Sinusoids Current or Alternating Current Consider the sinusoidal voltage V m = the amplitude of the sinusoid ω = the angular frequency in radians/s ωt = the argument of the sinusoid frequency hertz(Hz) Time period Sec

Let us now consider a more general expression for the sinusoid, where (ωt + φ) is the argument and φ is the phase. Both argument and phase can be in radians or degrees. The starting point of v 2 occurs first in time. Therefore, we say that v 2 leads v 1 by φ or that v 1 lags v 2 by φ. If φ ≠ 0, we also say that v 1 and v 2 are out of phase. If φ = 0, then v 1 and v 2 are said to be in phase; they reach their minima and maxima at exactly the same time. Let us examine the two sinusoids and

A sinusoid can be expressed in either sine or cosine form. When comparing two sinusoids, it is expedient to express both as either sine or cosine with positive amplitudes. This is achieved by using the following trigonometric identities:

How can we mathematically model AC circuits and the complex relationships of voltage and current, and power through all components? Phasors! 9.3 PHASORS

Phasors Graphical representation of current/voltage in AC circuits Takes into account relative phases of different voltages Example: current phasor graphs i (t) = I cos  t

Complex Number A complex number z can be written in rectangular form as Where: : x is the real part of z; y is the imaginary part of z. Z can be represented in three ways:

Phasor The idea of phasor representation is based on Euler’s identity. In general, where Re and Im stand for the real part of and the imaginary part of. V is thus the phasor representation of the sinusoid v(t). In other words, a phasor is a complex representation of the magnitude and phase of a sinusoid.

9.4 PHASOR RELATIONSHIPS FOR CIRCUIT ELEMENTS a) Alternating currents across a resistor… Current and Voltage are in phase across resistors

(b) Alternating currents across an Inductor… Thus, and Therefore inductive reactance

Current &Voltage are out of phase across inductors So voltage across the inductor will reach maximum BEFORE the current through it builds to max… CURRENT lags VOLTAGE Voltage leads Current

(c) Alternating currents across a Capacitor… For the capacitor C, assume the voltage across it is v = V m cos(ωt + φ). The current through the capacitor is But and Thus, Therefore Capacitive reactance

Alternating currents across a capacitor… Current and Voltage are out of phase across capacitors Capacitors take time to reach maximum voltage Voltage across capacitor LAGS behind current! VOLTAGE lags CURRENT CURRENT leads Voltage

Comparing ac circuit elements Table 31.1 summarizes the characteristics of a resistor, an inductor, and a capacitor in an ac circuit.

5.We just learned that capacitive reactance is and inductive reactance is. What are the units of reactance? A.Seconds per coulomb. B.Henry-seconds. C.Ohms. D.Volts per Amp. E.The two reactances have different units. Units of Reactance

The Series RLC Circuit where ɸ is some phase angle between the current and the applied voltage. Our aim is to determine ɸ and I m First, we note that because the elements are in series, the current everywhere in the circuit must be the same at any instant. That is, the current at all points in a series AC circuit has the same amplitude and phase. Resistor: Here current and voltage are in phase; so the angle of rotation of voltage phasor V R is the same as that of phasor I. Capacitor: Here current leads voltage by 90°; so the angle of rotation of voltage phasor V C is 90° less than that of phasor I. Inductor: Here current lags voltage by 90°; so the angle of rotation of voltage phasor v L is 90° greater than that of phasor I.

Therefore Maximum current The denominator of the fraction plays the role of resistance and is called the impedance Z of the circuit: where impedance also has units of ohms. Therefore, we can write  When X L > X C (which occurs at high frequencies), the phase angle is positive, signifying that the current lags behind the applied voltage,. We describe this situation by saying that the circuit is more inductive than capacitive.  When X L < X C, the phase angle is negative, signifying that the current leads the applied voltage, and the circuit is more capacitive than inductive.  When X L = X C, the phase angle is zero and the circuit is purely resistive.

Power in an AC Circuit Therefore, we can express the average power P av as Power in an AC circuit is given by But Therefore Average of P over one or more cycles I max, V max, ɸ, and ω are all constants. where the quantity cos ɸ is called the power factor I rms and V rms are called root mean square current and voltage, respectively and given by

Resonance in a Series RLC Circuit A series RLC circuit is said to be in resonance when the current has its maximum value. For I Maximum i.e.  ω 0 is called the resonance frequency of the circuit  At this frequency the current is maximum and in phase with the applied voltage

9.5 IMPEDANCE AND ADMITTANCE

where R = Re Z is the resistance and X = Im Z is the reactance. If X is positive i.e. Z = R + jX impedance is said to be inductive If X is negative i.e. Z = R - jX impedance is said to be capacitive The impedance, resistance, and reactance are all measured in ohms. The impedance may also be expressed in polar form as As a complex quantity, the impedance may be expressed in rectangular form as 9.5 IMPEDANCE AND ADMITTANCE

RMS quantities in AC circuits What's the best way to describe the strength of a varying AC signal? Average = 0; Peak=+/- Sometimes use peak-to-peak Usually use Root-mean-square (RMS)

Root-mean-square values

Current in a personal computer Suppose you have a device that draws 2.7 Amps from a 120V, 60- Hz standard US power plug. What is the: AVERAGE current, 0 Amp Average of the current squared, Current amplitude? Average over 1 period = 0! = 7.3 Amps 2 I rms =.707 I So I = 3.8 Amps