**Time Domain**

There are three basic components in Linear Circuits:

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

The behavior of a circuit composed of R, C, and L elements and is modeled by differential equations with constant coefficients.

- An electrical system is said to be in
**steady-state**when the variables describing its behavior (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 a
**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.

In a DC circuit, the electro‐motive forces push the electrons along the circuit and resistors to remove that energy by conversion to heat. In AC circuits, currents vary in time, so we have to consider variations in the energy stored in electric and magnetic fields of capacitors and inductors, respectively.

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

V = Q/C, where Q is the magnitude of the charge distribution on either surface, and V is the potential difference between the surfaces.

Differentiate V = Q/C using I = dQ/dt, We get dV/dt = I/C.

**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:

V = L. dI/dt, where V is the back emf across the inductor, dI/dt is the derivative of the current through the inductor and L is the inductance.

A first-order circuit can only contain one energy storage element (a capacitor or an inductor). The circuit will also contain resistance. So there are two types of first order circuits: RC circuit, and RL circuit.

**Voltage-Current Relationships for Passive Elements** **Note:** V_{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**

The analysis of circuits that contain combinations of resistors with capacitors or inductors follows the same general principles as for networks of resistors alone. Following figure shows a basic combination of a resistor with a capacitor, where a switch connected to a battery or ground allows the creation of voltage pulses at the input to the network.

We have 9 unknowns, using the above equations the following can be derived.

( We may assume the input voltage to be either 0 V or the battery emf E, and that it will be steady while the switch remains in a given position. During such periods, we may therefore treat Vin as a constant.)

**Transient Analysis of R-L Circuit: **

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

The transient current through the inductor L at any time t

where, i_{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, current flow approaches V/R.
- The time constant τ in an RL circuit is defined as τ = L/R.
- As τ grows smaller, transient behavior disappears much faster.

The transient voltage across capacitor C at any time t:

where, V_{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 behavior if the components are chosen properly.
- For many values of R-L-C, there will be no oscillation.
- α is the damping factor, which 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.

Current Source:

- 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.

**Resonance: **The circuit is said to be in resonance if the current is in phase with the applied voltage. 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 reactance of the inductance and the capacitance cancel each other is the resonant frequency (or the unity power factor frequency) of this circuit.

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

**Quality factor:** The 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 an 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**

The 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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