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Chapter 5 Steady-State Sinusoidal Analysis Where are we going? A look ahead: HighPass Filter Low Pass Filter Band Pass Filter Three phase Source 5-1 Sinusoidal Currents and Voltages v(t) = Vmsin(wt + q) Vm = peak value w = angular frequency q = phase angle T = period f = Frequency f = 1/T w = 2pf = 2p/T Plot v versus t Label T Plot v versus wt Label q Chapter 5 Steady-State Sinusoidal Analysis Useful Trig Identities sin(z) = cos(z - 90°) Example: v(t) = 10sin(200t + 30°) (a) Write v(t) as a cosine function. (b) Find Vm, w, f, T and q. Definitions Vrms = root mean square voltage Vpp = peak-to-peak voltage Vave = average voltage Pave = average power Vm Vave 0 Vrms 2 V pp 2Vm Pave I rms Vrms R=50W Example: Find v(t) and i(t) Vrms = root mean square voltage Vpp = peak-to-peak voltage Vave = average voltage Pave = average power What are the following for a wall outlet? f Vrms = VDCEquivalent Vm Vave Vpp Notation Summary Sinusoidal v1(t) = V1 cos(wt + q1) Polar Phasor V1 = V1 L q1 Complex Phasor V1 = V1 cos(q1) + j V1 sin(q1) j 1 Math Imaginary C=A + jB q = phase angle Real What is the magnitude of C? C A B 2 2 What is the phase (or direction) of C? B 1 q tan A Example Problems Convert the following voltages to phasors in polar form and complex form. v1(t) = 20 cos(wt - 45°) v2(t) = 20 sin(wt + 60°) Example Problems Convertthe following from complex phasors to polar form. V1 = 30 + j40 V2 = 4 - j20 V1 + V2 Phasor Math in Polar Form (C1q1) (C2q2) = C1 C2 (q1 + q2) (C1q1) /(C2q2) = C1/C2 (q1 - q2) Ohm’s Law for AC Circuits V=IZ Impedance Resistors Suppose that v(t) = Vmcos(wt) What is i(t) ? Hint: v=iR What is the phase relationship between i and v? Capacitors Suppose that v(t) = Vmcos(wt) What is i(t) ? Hint: v = q/C What is the phase relationship between i and v? Inductors Suppose that i(t) = Imcos(wt) What is v(t) ? Hint: v(t) = L di/dt What is the phase relationship between i and v? Reactance, Impedance and Phasors Go to notes… Phasor Diagrams for V and I Impedance Diagrams for R, L and C circuits The R, L, and C Elements Ohm’s Law for Peak Values Resistors: Vp=IpR Capacitors: Vp=IpXC Inductors: Vp=IpXL Go to notes... Example Problems Example 3.15: Convert the following from polar to rectangular form. 1053.13 16-30 25120 Example Problems Example 3.15: Convert the following from rectangular to polar form. 30 + j40 4 - j20 -3 - 4j Impedance Diagrams Resistor ZR = R0 Capacitor ZC = XC-90 Inductor ZL = XL90 RL Circuit Example Connect at AC power supply in series with an inductor and a resistor. How does VR vary with the input frequency? RC Circuit Example Connect at AC power supply in series with an capacitor and a resistor. How does VR vary with the input frequency? RLC Circuit Example Connect at AC power supply in series with an inductor, capacitor and a resistor. How does VR vary with the input frequency? 3.18 Tuned Resonant Networks RLC Series circuits are used in radios. Series RLC networks have a resonant frequency that depends on C and L only. 1 fs 2p LC Whatcapacitance do you need to listen to 107.7 MHz on a radio with a 1mH inductor?

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posted: | 10/30/2011 |

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