7.6.1. Phasor-diagram solution

From the circuit shown in Fig. 7.10, we see that the resistor, inductor and capacitor are in series. Therefore, the ac current in each element is the same at any time, having the same amplitude and phase. Let it be i = i sin(wt + f) (7.21) where f is the phase difference between the voltage across the source and the current in the circuit.

In a series circuit containing a resistor, inductor, and capacitor, the current flowing through each component is the same, meaning that all three share the same amplitude and phase. We can express this current in a phasor format, which allows us to include a phase difference (denoted as 'f') relative to the source voltage. The use of phasors simplifies the analysis of alternating current circuits by representing sinusoidal functions as rotating vectors.

Think of a group of friends at a party (the current) moving together in the same direction (through the resistive, inductive, and capacitive components) at the same speed (current flow). Occasionally, one friend might move ahead or lag behind slightly (the phase difference), but overall, they are in sync.

On the basis of what we have learnt in the previous sections, we shall construct a phasor diagram for the present case. Let I be the phasor representing the current in the circuit as given by Eq. (7.21). Further, let V, V , V, and V represent the voltage across the inductor, resistor, capacitor and the source, respectively. From previous section, we know that V is parallel to I, V is p/2 behind I and V is p/2 ahead of I.

By constructing a phasor diagram, we represent the voltages across the circuit elements: the resistor (VR), the inductor (VL), and the capacitor (VC), along with the total voltage (V). The current phasor (I) is used as a reference, and the relationships between them include the fact that VL leads the current by 90 degrees (π/2) while VC lags the current by 90 degrees. This visualization helps us see how different voltage and current components interact in a circuit.

Imagine a tug-of-war game with a rope: the team that's pulling the hardest (current) is the reference. The two opposing teams (voltage across the inductor and the capacitor) pull at angles. The team pulling forward (inductor) and the one resting (capacitor) will show how they influence the outcome of the game (the total voltage). This is how we visualize the dynamics of voltage and current in AC circuits with a phasor diagram.

The length of these phasors or the amplitude of V , V and V are: R, C, L v = i R, v = i X ,v =i X (7.22) Rm m Cm m Lm m

In the phasor diagram, each voltage phasor's length is proportional to the current flowing through the circuit multiplied by the respective resistance or reactance of the components. For the resistor, the voltage is simply the current multiplied by the resistance (Ohm's Law). The reactance for the inductor (XL) and the capacitor (XC) defines how much they oppose current change in their respective ways.

Consider a water pipe system: the resistance (R) is like the narrowing of the pipe (the water flow resistance), while the reactances (XL and XC) can be considered as the flexible pipes that can either expand or contract based on how much water is flowing. Each segment directly affects how much water can flow through the system at any given time.

The voltage Equation (7.20) for the circuit can be written as v + v + v = v L R C

This equation states that the total voltage (V) supplied to the circuit is equal to the sum of the voltages across each component. It reflects Kirchhoff’s voltage law, where all voltages in a closed loop must add up to zero. This relationship is crucial for analyzing and solving circuits with multiple components.

Think of it like a team project: the total effort (voltage) is equal to the sum of contributions (individual voltages) from each team member (resistor, inductor, capacitor). Just as each person's input is necessary to complete the project, each voltage adds up to meet the overall supply.