Series RLC Circuit Step Response & RLC Circuit Response

       Series RLC Circuit Step Response

      The measured response of the series RLC second order circuit is compared with expectations

based on the damping ratio and natural frequency of the circuit.

 Vout(t) = Vc(t) = (1/C) ∫ i(t)*dt ,  Vin = Ri + VL + Vc , Vin(t) = Ri + L (di/dt) + Vc

R(di/dt) + L d[(di/dt)]/dt + dVc/dt = 0 , d²i/dt² + (R/L)* di/dt + (1/LC)*i = 0

i(t) = A*e^(St) , S^2 + (R/L)*S + /LC) = 0 ,  α = R/2L , ω_{0 = 1/ (LC)^(1/2)}

Figure 1: A series RLC Circuit

 A 2v square step input voltage is applied to the series RLC circuit. Oscilloscope channel one is

connected to the input of the circuit, and its second channel is also connected to the output.

Figure 2: A real Series RLC circuit

       This circuit is an under-damped because α ˂ ω_0. Resistance of resistor 0.9 Ω and capacitance 

of capacitor 4.26 μF are measured by a DMM. So damping factor α = 450000 (rad/s) , undamped 

natural frequency ω_{0 =} 484501.6 (rad/s) , ω_{d (Theory)} = 179560.0 (rad/s). 

Figure 3: An Input-Output Graph of Series RLC circuit

According to the above graph, period of first under-damped output is 4 mm, also scale is 

1 ms = 100 mm. So (2π/ω_d ) = (4 mm)*(1 ms/100 mm) = 0.04 ms ,  ω_{d (Measure)} = 157000 (rad/s) ,

 f = 25000 Hz

Percent Error of damped natural frequency =  [(179560 - 157000)/(179560)]*100 = 12.6%

_{RLC Circuit Response}

       The measured response of a RLC second order circuit is compared with expectations

based on the damping ratio and natural frequency of the circuit. This circuit is an under-damped because α ˂ ω_0.

Figure 1: A RLC Second-Order Circuit

 A 2v square step input voltage is applied to the RLC circuit. Oscilloscope channel one is

connected to the output of the circuit, and its second channel is also connected to the input.

<sub>When input voltage is in its maximum (2v), capacitor will charge. When input voltage is equal zero, 

capacitor will parallel with L and R2. Of course, time constant of the capacitor is 

47*(8.13</sub> μF _{) = 382} μs , <sub>so the square input period should be equal or more than capacitor time

 constant. Internal resistance of inductor</sub> is measured 1.9 ohms by a DMM that is near to the R2.

  Ri + VL + Vc = 0 ,  Ri + L (di/dt) + Vc = 0

R(di/dt) + L d[(di/dt)]/dt + dVc/dt = 0 , d²i/dt² + (R/L)* di/dt + (1/LC)*i = 0


i(t) = A*e^(St) , S^2 + (R/L)*S + /LC) = 0 ,  α = R/2L , ω_{0 = 1/ (LC)^(1/2)}

ω_{0 =} [_{8.13E(-9)]^(-1/2) = 11090.6 (rad/s) ,}  ω_{d (Theory)} = 11076.95 (rad/s) , f = 1763.8 Hz 

Figure 2: A RLC Circuit with an Input Frequency equal 400 Hz

According to the above graph, period of the under-damped output is 40 mm, also scale is 

1ms = 60 mm. So (2π/ω_d ) = (40 mm)*(1 ms/60 mm) = (2/3) ms ω_{d (Measure)} = 9420 (rad/s) ,

 f = 1500 Hz. Vout(t) = R2*i(t).

Percent Error of damped natural frequency =  [(11076.95 - 9420)/(11076.95)]*100 = 14.96%

Figure 3: A Real RLC Circuit