## 17.4 Phase This section defines the concept of phase in Simple Harmonic Motion (SHM) and Alternating Current (AC) circuits, including the initial phase, and explains its physical significance in determining the state of an oscillator or an electrical quantity at any given time. ### Definition of Phase The phase of a point executing SHM or an alternating quantity is the angle $\theta = \omega t$ which specifies not only the displacement but also the direction of motion of the point. In AC circuits, it specifies the instantaneous value of voltage or current. The instantaneous displacement in SHM can be described by: $$x = x_0 \cos \theta \quad \text{or} \quad x = x_0 \cos \omega t$$ where $x_0$ is the amplitude and $\omega$ is the angular frequency. ### Initial Phase ($\varphi$) In a more general case, the displacement equation for SHM (or the equation for AC voltage/current) is written as: $$x = x_0 \cos(\omega t + \varphi)$$ Here, the term $\varphi$ is called the **initial phase** or **phase constant**. It provides information about the starting position and direction of the oscillation at time $t = 0$. ### Physical Significance of Phase The initial phase $\varphi$ determines the state of the oscillator at the beginning of its motion ($t = 0$). This is illustrated by analyzing the displacement-time graph for different values of $\varphi$. **Case 1: $\varphi = 0^{\circ}$** - The equation becomes $x = x_0 \cos(\omega t)$. - At $t = 0$, the displacement is $x = x_0$, meaning the oscillation starts from the positive extreme position. **Case 2: $\varphi = 90^{\circ}$** - The equation becomes $x = x_0 \cos(\omega t + 90^{\circ})$. - At $t = 0$, the displacement is $x = 0$, meaning the oscillation starts from the mean position, moving towards the negative extreme. **Case 3: $\varphi = 180^{\circ}$** - The equation becomes $x = x_0 \cos(\omega t + 180^{\circ})$. - At $t = 0$, the displacement is $x = -x_0$, meaning the oscillation starts from the negative extreme position. ### Phase Difference The phase difference between two oscillating systems or two AC quantities describes their relative motion or timing. **In Phase:** Two systems are *in phase* if their phase difference is $0^{\circ}$ or $360^{\circ}$ (or any integer multiple of $2\pi$ radians). They reach their maximum and minimum values at the same time. **Out of Phase:** Two systems are *out of phase* if their phase difference is $180^{\circ}$ (or $\pi$ radians). When one is at its positive maximum, the other is at its negative maximum. ### Phasors In AC circuit analysis, alternating quantities are often represented as rotating vectors called **phasors**. - The length of the phasor represents the peak value ($V_0$ or $I_0$). - The angular velocity of the phasor is $\omega$. - The vertical projection of the phasor gives the instantaneous value. ### Phase Relationship in Resistive Circuits In a purely resistive AC circuit, the voltage and current are always in phase. This means they reach their peaks and zeros at the exact same time. | Formula/Concept | Description | | :--- | :--- | | $x = x_0 \cos(\omega t)$ | Displacement in SHM starting from the extreme position ($\varphi = 0$). | | $x = x_0 \cos(\omega t + \varphi)$ | General equation for displacement in SHM with an initial phase $\varphi$. | | In Phase | Phase difference is $0^{\circ}$ or $360^{\circ}$. | | Out of Phase | Phase difference is $180^{\circ}$. |