The solution of this equation describes the response of the circuit. ir + 1 C idt Differentiating with respect to time gives Equation t + L di dt = v s(t) (10) L d2 i di + R dt2 dt + 1 C i = dv S dt (11) of 9ģ This is a second-order linear differential equation. The application of KVL around the loop of the RLC circuit shown in Figure 1 gives Equation 10 for zero initial conditions. However, we present here a brief description required to complete the laboratory assignment. Part 2: Time Domain Response of RLC Circuit to Step Excitation The theory of the RLC circuit behavior when excited by a step voltage is beyond the scope of this course. This represents a voltage magnification equal to Q. voltages appearing across C and L are larger than the applied voltage, VS. Q = 1 ω o CR = ω ol R (9) If R is small, Q can be greater than unity, i.e. V C V S = V L = Q (8) V S Where Q is the quality factor of a series RLC circuit. 1 V C = V S = V ω O CR S ω OL R (7) Using Equation 7, the ratio in Equation 8 is found. V C = I MAX 1 jω o C (6) From Equation 4 and 5, Equation 7 is determined. At the resonant frequency, the voltage across the capacitor is described in Equation 6. For frequencies higher or lower than ω0, the current is reduced below its peak value. the voltage across the combined impedance of L and C in series, is zero. I = I MAX = V S R (5) The voltage across the reactance X, i.e. At the resonant frequency ω0, the magnitude of current takes the value of Equation 5. ωl = 1 ωc or ω = ω o = 1 LC (4) This phenomenon is called resonance. I = V S R 2 +X 2 (3) of 9Ģ The current has a maximum value when X = 0, which occurs when Equation 4 is true. X = ωl 1 ωc (2) Using Equation 2, the magnitude of the current phasor is described in Equation 3. Figure 1: Series RLC Circuit I = V s R+jωL+ 1 = V s R+jX jωc (1) Equation 2 describes the impedance of the inductor and the capacitor (capacitors and inductors are examples of simple reactive components, so this impedance or these components are known as reactance). The loop current I is given by Equation 1. Background: Part 1: Frequency Domain Response of RLC Circuit to Sinusoidal Excitation Figure 1 shows a series RLC circuit driven by an ideal voltage source, V S. Equipment: - accessed locally (i) (ii) Your favourite browser. (b) A step excitation under conditions of underdamping, critical damping, and overdamping. 1 LAB 5: Simulation of an RLC Response Using Multisim Live (_/10) Purpose: To use the free online circuit simulator (NI Multisim) to investigate the response of an RLC series circuit to: (a) A sinusoidal excitation.
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