**Notes on Time Domain and Frequency Analysis of Linear Circuits**which will cover the topic as

**Introduction to Time domain and Frequency Domain, Transient Responses and Transient Analysis of Different Circuits, Parallel and Series Resonance.**

**Time Domain**

- Resistance (R)
- Capacitance (C) and
- Inductance (L)

- An electrical system is said to be in a
**steady state**when the variables describing its behaviour (voltages, currents, etc.) are either invariant with time (d.c. circuits) or are periodic functions of time (a.c. circuits). - The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching, are called
**transients**. - An electrical system is said to be in
**transient state**when the variables are changed non-periodically, i.e., when the system is not in steady-state. - The
**transient response**is the fluctuation in current and voltage in a circuit (after the application of a step voltage or current) before it settles down to its steady state.

**Capacitance:**the capacitance C between two oppositely charged surfaces is defined by:

**Inductance:**The usual model for an inductor is a coil (solenoid). By Faraday’s Law of self‐inductance, a changing current in a coil induces a back electromotive force (emf) that opposes the change in current:

**Voltage-Current Relationships for Passive Elements****V**

**Note:**_{R}, V

_{L}and V

_{C}are the voltages across R, L and C respectively while i

_{R}, i

_{L}and i

_{C}are the current through R, L and C respectively.

**Element Transformations**

**Resistor:****Time Domain:**

**s-Domain:**

**Inductor:****Time Domain:**

**s-Domain:**

V(s) = L[sI(s) – i(0)]

**Capacitor (C):****Time Domain:**

**s-Domain:**

i(s) = sCV(s) – Cv

**Steps for Finding Transient Response**- Identify the variable of interest (Inductor current for RL circuit, Capacitor voltage for RC circuit).
- Determine the initial value of the variable.
- Calculate the final value of the variable.
- Calculate the time constant for the circuit.

**Transient Response of RL and RC Circuits****Transient Analysis of R-L Circuit: **

- When the switch is closed, current flows into the capacitor.
- Current flow ceases when the charge collected on the capacitor produces a voltage equal to and opposite to V.
- An equation describing the behaviour is shown; it is both exponential and asymptotic.
- The value RC is called the time constant (τ) in the equation.
- As τ grows smaller, transient behaviour disappears much faster.

_{σ}= Current through L at t → σ i.e., steady state current through L i

_{0}= Current through L at t = 0 R

_{eq}= Thevenin’s equivalent resistance seen across L for t > 0

^{–}

**Transient Analysis of R-C Circuit:**- When the switch is closed, current flow is inhibited as the inductor develops an opposite voltage to the one applied.
- Current slowly begins to flow as the inductor voltage falls toward 0.
- As the transient effect dies, the current flow approaches V/R.
- The time constant τ in an RL circuit is defined as τ = L/R.
- As τ grows smaller, transient behaviour disappears much faster.

The transient voltage across capacitor C at any time t:

_{σ}= Voltage across capacitor at t → σ, i.e., steady state voltage across C V

_{σ}= Voltage across C at t = 0

^{–}R

_{eq}= Thevenin’s equivalent resistance seen across C for t > 0.

**Transient Analysis of RLC Circuit:**- A circuit with R, L, and C can exhibit oscillatory behaviour if the components are chosen properly.
- For many values of R-L-C, there will be no oscillation.
- α is the damping factor that determines the rate at which the oscillation dies out.
- The smaller L and C, the higher frequency of the oscillation.
- If R is too large, the quantity under the square root is negative, which means there is no oscillation.

## Frequency Domain

- A periodic signal can be viewed as being composed of a number of sinusoids.
- Instead of specifying a periodic signal in terms of the time variable t, one can equivalently specify the amplitude and phase density of each sinusoid of frequency contained in the signal.
- It uses the frequency variable ω as an independent variable, and thus it is said to be the frequency domain of the given time domain signal.

**KCL in s-domain:**

- t-domain (time domain): i
_{1}(t)+i_{2}(t)-i_{3}(t)+i_{4}(t)=0

- s-domain (complex frequency domain): I
_{1}(s)+I_{2}(s)-I_{3}(s)+I_{4}(s)=0

**KVL in s-domain:**

- t-domain (time domain): -v
_{1}(t)+v_{2}(t)+v_{3}(t) = 0 - s-domain (complex frequency domain): -V
_{1}(s)+ V_{2}(s)+V_{3}(s) =0

**Signal Sources in s Domain:**Voltage Source:

- t-domain: v(t) = v
_{s}(t), and i(t) depends on circuit.

- s-domain: V(s) = V
_{s}(s), and I(s) depends on circuit.

- t-domain: i(t) = i
_{s}(t), and v(t) depends on circuit. - s-domain: I(s) = I
_{s}(s), and V(s) depends on circuit.

For more information about the time domain analysis and second order system, you can refer to the following video available on the **Byju Exam Prep's **official youtube channel.

**Resonance: **The circuit is said to be in resonance if the current is in phase with the applied voltage. The power factor of the circuit at resonance is unity. At resonance, the circuit behaves like a resistive circuit. The frequency at which the resonance occurs is called the resonant frequency.

There are two types of Resonance circuits: 1. Series Resonance circuit and 2. Parallel Resonance circuit.

**Series Resonance**

The series RLC can be analyzed in the frequency domain using complex impedance relations.

If the voltage source above produces a complex exponential waveform with complex amplitude V(s) and angular frequency s = σ + iω , KVL can be applied:

At resonance

|V_{L}| = |V_{C}| and these are 180^{°} out of phase.

Z_{in} = Input impedance

The frequency at which the inductance and capacitance react cancel each other is this circuit's resonant frequency (or the unity power factor frequency).

**Conditions for ω and ω _{0} in Series Resonance**

**Quality factor:** Quality factor or Q-factor is basically an amplification factor for a resonant circuit.

**Bandwidth:** The bandwidth (ω_{2} – ω_{1}) is called the half-power bandwidth or 3-dB bandwidth.

The bandwidth of the series circuit is defined as the range of frequencies in which the amplitude of the current is equal to or greater than(1/1.414) times its maximum amplitude. This yields the bandwidth B = ω2 - ω1 = R/L.

The frequency at which voltage across the inductor is maximum

The frequency at which voltage across the capacitor is maximum

**Selectivity **It is defined as the ratio of resonant frequency to the bandwidth.

**Key Points**

- Selectivity of series R-L-C circuit with C variable is .
- Selectivity of series R-L-C circuit with L variable is also
- The higher the' selectivity, the higher will be the quality factor.
- The higher the selectivity, the lesser will be the bandwidth.

**Parallel Resonance**

A parallel resonance circuit is also called an anti-resonance circuit. The complex admittance of this circuit is given by adding up the admittances of the components:

At resonance,

- |i
_{L}| = |i_{C}| and these are 180^{°}out of phase

**Conditions for ω and ω _{0} in Series Resonance**

A parallel RLC circuit is an example of a band-stop circuit response that can be used as a filter to block frequencies at the resonance frequency but allow others to pass.

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