E37-23 Assuming no inductors or capacitors in the circuit, then the circuit effectively behaves as a DC circuit. The current through the circuit is i = E/(r + R). The power delivered to R is then P = i∆V = i2 R = E 2 R/(r + R)2 . Evaluate dP/dR and set it equal to zero to find the maximum. Then r−R dP = E 2R , 0= dR (r + R)3 which has the solution r = R. E37-24 (a) Since P av = im 2 R/2 = E m 2 R/2Z 2 , then P av is a maximum √ when Z is a minimum, and vise-versa. Z is a minimum at resonance, when Z = R and f = 1/2π LC. When Z is a minimum C = 1/4π 2 f 2 L = 1/4π 2 (60 Hz)2 (60 mH) = 1.2×10−7 F. (b) Z is a maximum when XC is a maximum, which occurs when C is very small, like zero. (c) When XC is a maximum P = 0. When P is a maximum Z = R so P = (30 V)2 /2(5.0 Ω) = 90 W. (d) The phase angle is zero for resonance; it is 90◦ for infinite XC or XL . (e) The power factor is zero for a system which has no power. The power factor is one for a system in resonance. E37-25
(a) The resistance is R = 15.0 Ω. The inductive reactance is XC =
1 1 = = 61.3 Ω. −1 ωC 2π(550 s )(4.72µF)
The inductive reactance is given by XL = ωL = 2π(550 s−1 )(25.3 mH) = 87.4 Ω. The impedance is then Z=
q
2
(15.0 Ω)2 + ((87.4 Ω) − (61.3 Ω)) = 30.1 Ω.
Finally, the rms current is E rms (75.0 V) = = 2.49 A. Z (30.1 Ω) (b) The rms voltages between any two points is given by irms =
(∆V )rms = irms Z, where Z is not the impedance of the circuit but instead the impedance between the two points in question. When only one device is between the two points the impedance is equal to the reactance (or resistance) of that device. We’re not going to show all of the work here, but we will put together a nice table for you Points ab bc cd bd ac
Impedance Expression Z=R Z = XC Z = XL Z =p |XL − XC | Z = R2 + XC2
Impedance Value Z = 15.0 Ω Z = 61.3 Ω Z = 87.4 Ω Z = 26.1 Ω Z = 63.1 Ω
(∆V )rms , 37.4 V, 153 V, 218 V, 65 V, 157 V,
Note that this last one was ∆Vac , and not ∆Vad , because it is more entertaining. You probably should use ∆Vad for your homework. (c) The average power dissipated from a capacitor or inductor is zero; that of the resistor is PR = [(∆VR )rms ]2 /R = (37.4 V)2 /(15.0Ω) = 93.3 W. 155