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Unformatted text preview: General Physics II Lab EM9 AC Circuits General Physics II Lab EM9 AC Circuits Purpose In this experiment, you will learn the frequency response of resistive, capacitive, and inductive impedances in multi-component AC circuits and their applications. Equipment and components Digital LCR meter, function generator, oscilloscope, breadboard, 220 Ω resistor, 0.22 μ F capacitor, and 33 mH inductor. Background In an AC circuit, a sinusoidal driving voltage having amplitude V o may be expressed as , (1) ( ) i ω t o V t = V cos ω t = Re[V e ] o where the complex identity, , is used. In this expression, ω = 2 π f is the angular frequency with f being the frequency in units of Hertz. In a circuit containing reactive components, the applied voltage and current are generally not in phase. With V(t) expressed as in Equation (1), the current I(t) is also sinusoidal having an amplitude I o that depends not only on the magnitude of V(t) but also on the total impedance Z of the circuit. Hence, θ = θ ± θ = θ ± cos ] sin i Re[cos ] e Re[ i ( ) ( ) ] e I Re[ ) t cos( I t cos Z V t I ) t ( i o o o ϕ + ω = ϕ + ω = ϕ + ω = . (2) In this expression, ϕ is the phase difference between the applied voltage and the current in the circuit. In the analysis of electrical circuits to determine the phase and total impedance, the technique is the same if the trigonometric (real) expressions are used or if the complex expressions are used. When using the complex expressions, all that must be done is to take the real part in the end. For instance, given a circuit consisting of an AC voltage source in series with an ideal resistor R as shown in Fig. 1 below, the current can be determined using Kirchhoff’s loop rule and Ohm’s Law. Hence, we have the relationship , (3) IN V =IR where V IN = V(t) is the AC voltage. R Figure 1 The R circuit Revised: 15 November 2010 1/10 General Physics II Lab EM9 AC Circuits Using the real and complex forms, the input voltage is expressed as: Real form Complex form ) t cos( V V o IN ω = t i o IN e V V ω = (4) Solving for the current yields, ) t cos( I ) t cos( R V I o o ω = ω = Taking the real part of these expressions t i o t i o e I e R V I ω ω = = (5) ) t cos( I I o ω = ) t cos( I ] e I Re[ I o t i o ω = = ω (6) In both cases, the end result is the same. It is seen that the current is sinusoidal having amplitude V o /R and angular frequency ω and also that the current is in phase with the applied voltage V IN . The situation is a little more involved mathematically if we were to analyze an AC voltage source in series with a resistor and a capacitor, a series RC circuit as shown in Figure 2. R C Figure 2 The RC circuit For this circuit, Kirchhoff’s loop rule and Ohm’s Law yields , (7) IN C V = IR+IZ where Z C is the impedance of the capacitor and itself is a function of frequency, C i Z C ω = 1 . (8) Solving Eq. (7) for the current yields, ( ) ( ) ( )-1-1 1 π-i tan- i-tan ω RC ω RC 2 IN IN IN IN i 2 2 2 1 1 C ω...
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