RCL circuit response to a signal Remember that if you can solve problems (both intuitively and mathematically) for drag/spring/mass, you can also solve problems for resistance/capacitor/inductor. Use this chart for the mechanical/electrical counterparts: Mechanical Spring Electrical Capacitor πΉ = −ππ₯ Drag π= 1 π πΆ Resistance πΉ∝π£ Mass π = π πΌ Inductor ππ£ πΉ=π ππ‘ β = −πΏ ππΌ ππ‘ In this class, we study the response to only two signals: sinusoidal and step functions. Response to a sinusoidal signal How to think about this: Even though, in reality and in most cases, we apply a known voltage to a circuit, and the circuit responds by allowing a current to pass through it, in this case, we begin to assume known the current and then work back to figuring out the voltage. Make sure you review the tutorials on Impedance and Power factor. Those are the points to remember: ο· ο· Resistor o For resistors, the voltage is a potential difference. o Voltage is in phase with the current o π = π πΌ. This equation is true for instantaneous values, peak value, and rms value. Capacitor o For capacitors, the voltage is a potential difference. o The voltage across the capacitor lags ¼ of a cycle behind current. o πππππ = ππΆ πΌππππ . This equation is true for peak value, and rms value, but NOT for instantaneous values. o ο· ο· ππΆ is called capacitive inductance, in units of ohms, and is given by ππΆ = 1 . 2πππΆ o ππΆ is small for high frequencies, this makes capacitors high-pass filters. Inductor o For inductors, the voltage is an e.m.f. o The voltage across the inductor is ¼ of a cycle ahead of current. o πππππ = ππΏ πΌππππ . This equation is true for peak value, and rms value, but NOT for instantaneous values. o ππΏ is called inductive inductance, in units of ohms, and is given by ππΏ = 2πππΏ. o ππΏ is small for low frequencies, this makes inductors low-pass filters. Series RCL o Depending on L, C, R, and f, the phase difference between current and voltage is π π anywhere between − and − 2 2 ππΏ −ππΆ π o The phase difference πΌ is given by tan πΌ = o πππππ = π πΌππππ . This equation is true for peak value, and rms value, but NOT for instantaneous values. o o o π is called impedance, in units of ohms, and is given by π = √π 2 + (ππΏ − ππΆ )2 . The average power dissipated in a RCL circuit is πππ£π = πΌπππ ππππ cos πΌ If ππΏ = ππΆ , the circuit behaves as if only the resistor was there. This happens at the same frequency as the natural oscillations as the RLC oscillators. When this happens, we say we have resonance. Trick and tips: Remember “ELI the ICE man” to help you remember which of I and E leads the other. Use the phasor diagram to have a geometrical representation of Z. Response to a step signal A “step signal” is what happens when a square wave switches. This is also what happens when we close a switch. RC series circuits ο· ο· The voltage across the capacitor follows the voltage across the RC series circuit, following Newton’s Law of cooling, with time constant π = π πΆ. Everything else (current, charge, power, energy, etc.) can be derived from that. RL series circuits ο· The voltage across the resistor follows the voltage across the RL series circuit, following πΏ ο· Newton’s Law of cooling, with time constant π = π . Everything else (current, charge, power, energy, etc.) can be derived from that. RCL series circuits ο· ο· ο· The C and L together make a harmonic oscillator of frequency π = 2π 1 . √πΏπΆ The R adds damping. Review the lab we did in class for underdamped, overdamped, critical damping, critical resistance.
