Part 1: RC Circuits
(1) Calculate the time constant of the circuit in Figure 1 and theoretically
determine its response to a step function.
The time constant for the RC circuit can be calculated using this equation:
τ = RthC
τ = (1kΩ)*(0.1uF) = (1000Ω)(1x10-7F) = 10-4 seconds
(2) Calculate the time constant of the circuit in Figure 2 and theoretically
determine its response to a step function.
The time constant for the RC circuit can be calculated using this equation:
τ = RthC
τ = (1kΩ)*(0.1uF) = (1000Ω)(1x10-7F) = 10-4 seconds
(3) Calculate the time constant of the circuit in Figure 3.
τ = RthC
Once we open circuit the square voltage source vin(t) and the terminals of the 0.1uF
capacitor, we can calculate Rth for our time constant.
After opening the voltage and capacitor terminals, the 5kΩ and 10kΩ are in parallel,
which then is in series with the 1kΩ resistor… 5kΩ||10kΩ + 1kΩ = Rth = 65/15 kΩ.
Therefore, the time constant
τ = (65/15 kΩ)*(0.1uF) = (4062.5 Ω)*(.1*10-6 F) = 4.0625*10-4 seconds
(4) How can we use square waves to measure the time constant of these circuits?
Let’s say, we have one pulse from t = 1 to t = 2, with a certain amplitude A. We can
model that pulse as an input voltage to charge the capacitor. Since R is a constant value,
we can ignore it in this situation. As the pulse starts from t = 1 and slowly makes it’s
way to t = 2, the capacitor charges itself exponentially to Vo (the amplitude of the pulse
“A”).
When the input voltage drops to 0 (at t = 2), the capacitor will discharge the stored
energy between the electric fields to the network. Then, once a pulse is provided for Vin(t)
with a certain amplitude “A”, the capacitor will charge itself again to the maximum value
it can before the capacitor is triggered to discharge the energy. This process is then
continued until the capacitor is fully discharged permanently (when no more pulses are
provided).
Part 2: Series RLC Circuit
(1) Describe the three types of general behavior you can get as a step response for
an RLC circuit?
The three responses are dependent upon the variable ζ (damping ratio).
Case 1: ζ = 1
This describes the situation when the network response is critically damped. This
is when the system does not oscillate and responds faster than the over-damped situation.
In this system, the natural frequencies are the same as the damping ratio multiplied by the
un-damped natural frequency * -1.
Case 2: ζ > 1
This describes the situation where the network response is over-damped. An overdamped situation entails that the sum of the two decaying exponentials is the natural
response of the network.
Case 3: ζ < 1
This describes the situation where the network response is under-damped. An
under-damped system entails that the natural frequencies are complex numbers. This
system also means that the response is an exponentially damped sinusoid which decays
independent of the damping ratio, ζ.