Transient Response: MODULE I The Transient Response (also known as the Natural Response) is the way the circuit responds to energies stored in storage elements, such as capacitors and inductors. If a capacitor has energy stored within it, then that energy can be dissipated/absorbed by a resistor. How that energy is dissipated is the Transient Response. A resistor capacitor circuit (RC circuit) or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source A resistor inductor circuit (RL circuit) or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source Transient Response of RC circuit When an increasing DC voltage is applied to a discharged Capacitor, the capacitor draws a charging current and charges up, and when the voltage is reduced, the capacitor discharges
in the opposite direction. Because capacitors are able to store electrical energy they act like small batteries and can store or release the energy as required. The charge on the plates of the capacitor is given as: Q = CV. This charging (storage) and discharging (release) of a capacitors energy is never instant but takes a certain amount of time to occur with the time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value being known as its Time Constant ( τ ). The Transient (step) response of the RC circuit as shown V ( t) V for t for t Applying KCL to the circuit V ( t) Vout( t) dvout C R dt dvout Vout( t) V ( t) dt RC RC WKT V out ( t) A Be () () Substitute Equation () into () A RC A RC B RC V RC V RC B e t B RC B e t
The required output voltage Vout Time-Constant RC V V out out ( t) V Ve ( t) V ( e t RC t RC ) The time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value is known as its Time Constant (τ). Mathematically, τ=rc
From the RC circuit Curve, Notice the second term starts from V and exponentially decay to. It will never reach, but approach to asymptotically. As a result Vout starts from and asymptotically approach to V. Series RC Circuit: If the switch in this circuit was initially open, and then closed at time t=, the current in this circuit is: i t) I ( t exp Where I V = the initial current in thee circuit R RC = the time constant for the circuit Another definition of τ is obtained by setting t=τ in i(t). It gives i(τ)=i*(/e). The time constant of an RC circuit is the time required for the current in the circuit to fall to /e of its initial value. Transient Response of RL circuits A RL Series Circuit consists basically of an inductor of inductance L connected in series with a resistor of resistance R. The resistance R is the DC resistive value of the wire turns or loops that goes into making up the inductors coil. Consider the RL series circuit below.
The above RL series circuit is connected across a constant voltage source, (the battery) and a switch. Assume that the switch, S is open until it is closed at a time t =, and then remains permanently closed producing a step response type voltage input. The current, i begins to flow through the circuit but does not rise rapidly to its maximum value of Imax as determined by the ratio of V / R (Ohms Law). Applying KVL to the circuit, to define the individual voltage drops that exist around the circuit and then hopefully use it to give us an expression for the flow of current. V( t) V R V L The voltage across the resistor R is IR (ohms law) V R I R The Voltage drop across the Inductor is di V L L dt The individual voltage drops around the LR series circuit can be given as: V ( t) I R L di dt
Series RL Circuits: If the switch in this circuit is initially open, and then closed at time t=, the current in this circuit will be described as: i( t) I ( e t ) Where I V = the limiting value of the current in the circuit R L R = the time constant for the circuit Transient response of RLC Circuit Series RLC circuit In theory, there are three cases for the way a series RLC circuit will respond when the switch is closed at time t=. In this lab, only the underdamped case will be dealt with. For this case, the current in the circuit is described by: V i exp( t)sin( dt) L d
Where d and R LC L Current in undamped RLC circuit The current in the circuit oscillates due to the sine component in Equation i, but the maximum value it can reach is decaying due to the negative exponential. The envelope that the current must fall within is described by: V i exp( t) or i L d V exp( t) L d The quantity α is referred to as the time constant of the envelope. It is determined by taking the natural logarithm of both sides of the above equation: ln i V ln t d L Which is a linear equation. Transient Response of LC circuit: If the limit R tents to, the series RLC circuit reduces to the Lossless LC circuit
The equation that describes the response of this circuit is d vc vc dt LC Assuming a solution of the form Ae -st the characteristic equation is s Where LC The roots are s j s j And the solution is a linear combination of Ae st and Ae st vc( t) jot A e A e j t o By using Euler s relation Equation vc t) Bcos( t) Bsin( ) ( t The constants A, A or B, B are determined from the initial conditions of the system. For vc(t=)=v and for dvc(t=)/dt = (no current flowing in the circuit initially ), And A A V j A j A Which give And the solution becomes The current flowing in the circuit is A A V V j ( ) ( t jt vc t e e ) V cos( ) t dvc i C CV sin( t) dt
And the voltage across the inductor is easily determined from KVL or from the element relation of the inductor vl= L di/dt vl vc V cos( t) Step and sinusoidal inputs using Laplace transform method Unit Step Function, t ( t), t
To find the Laplace Transform, we apply the definition ( s) ( t) e. e s s st dt s st dt e st So Sinusoidal Function L ( t) ( s) s As before, start with the definition of the Laplace transform Here it becomes useful to use Euler's identity for the sine
So Let's put this over a common denominator Natural Frequency and Damping Ratio Damping and Natural response of RLC circuit Consider a series RLC circuit (one that has a resistor, an inductor and a capacitor) with a constant driving electro-motive force (emf) E. The current equation for the circuit is
di L Ri idt E dt C This is equivalent: di L Ri q E dt C Differentiating, we have d i dt Auxiliary Equation L R i di dt R L R L i C Lm Rm C withroots : m m R 4L C L R 4L L C R Now is called the damping coefficient of the circuit L is the resonant frequency of the circuit LC m, m are called the natural frequencies of the circuit. The nature of the current will depend on the relationship between R,L and C. The form of the roots m, m depends on the values of α and ω. The following three case are possible.
. α= ω : Critically Damped System. m and m are equal and real numbers: no oscillatory behaviour.. α > ω : Over Damped System:m and m are real numbers but unequal: no oscillatory behaviour. 3. α < ω : Under Damped System. m and m are complex numbers. System exhibits oscillatory behaviour. Logarithmic Decrement The logarithmic decrement represents the rate at which the amplitude of a free damped vibration decreases. It is defined as the natural logarithm of the ratio of any two successive amplitudes. It is found from the time response of underdamped vibration (oscilloscope or realtime analyser). Logarithmic decrement, is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement is the natural log of the ratio of the amplitudes of any two successive peaks: ln n x( t) x( t nt) Where x(t) is the amplitude at time t and x (t+ nt) is the amplitude of the peak n periods away, where n is any integer number of successive, positive peaks. The damping ratio is then found from the logarithmic decrement: Thus logarithmic decrement also permits to evaluate the Q factor of the system:
) ( ) ( ln nt t x t x n Q Q The damping ratio can then be used to find the natural frequency ωn, of vibration of the system from the damped natural frequency ωd: d n d d Where T, the period of the waveform, is the time between two successive amplitude peaks of the underdamped system. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about.5; it does not apply at all for a damping ratio greater than. because the system is overdamped. Simplified Variation The damping ratio can be found for any two adjacent peaks. This method is used when n= and is derived from the general method above: ln x x Where x and x are any two successive peaks And for system << (not too close to the critically damped regime, where =). ln x x Response to non-sinusoidal periodic inputs
Any waveform that differs from the basic description of the sinusoidal waveform is referred to as nonsinusoidal. The most obvious and familiar are the dc, square-wave, triangular, sawtooth, and rectified waveforms. Fourier series refers to a series of terms, developed in 8 by Baron Jean Fourier that can be used to represent a nonsinusoidal periodic waveform. In the analysis of these waveforms, we solve for each term in the Fourier series: The Fourier series representation of a nonsinusoidal input can be applied to a linear network using the principle of superposition. Recall that this theorem allowed us to consider the effects of each source of a circuit independently. If we replace the nonsinusoidal input with the terms of the Fourier series deemed necessary for practical considerations, we can use superposition to find the response of the network to each term.