Problem 51 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Assume a voltage source supplies an ac voltage of amplitude \(V_{0}\) between its output terminals. If the output terminals are connected to an external circuit, and an ac current of amplitude \(I_{0}\) flows out of the terminals, then the equivalent resistance of the external circuit is \(R_{\mathrm{eq}}=V_{0} / I_{0}\) . (a) If a resistor \(R\) is connected directly to the output terminals, what is \(R_{\text { eq }} ?(b)\) If a transformer with \(N_{\mathrm{P}}\) and \(N_{\mathrm{S}}\) turns in its primary and secondary, respectively, is placed between the source and the resistor as shown in Fig. 46 what is \(R_{\mathrm{eq}} ?[\) Transformers can be used in ac circuits to alter the apparent resistance of circuit elements, such as loud speakers, in order to maximize transfer of power.

Step-by-Step Solution

Answer

(a) \( R_{\text{eq}} = R \); (b) \( R_{\text{eq}} = R \left( \frac{N_{\mathrm{P}}}{N_{\mathrm{S}}} \right)^2 \)."

Step 1: Understanding part (a) with a direct connection

In part (a), a resistor \( R \) is directly connected to an AC voltage source. The equivalent resistance \( R_{\text{eq}} \) in such a case is simply the resistance of the resistor itself. Therefore, \( R_{\text{eq}} = R \) since there are no additional components in the circuit to alter the resistance.

Step 2: Understanding part (b) with a transformer in series

In part (b), a transformer is connected between the source and the resistor. The transformer has \( N_{\mathrm{P}} \) primary turns and \( N_{\mathrm{S}} \) secondary turns. A transformer's turns ratio impacts the apparent resistance as seen by the source. The relationship for the apparent resistance in terms of the turns ratio is: \[ R_{\text{eq}} = R \left( \frac{N_{\mathrm{P}}}{N_{\mathrm{S}}} \right)^2 \] This means the equivalent resistance as viewed by the source is scaled by the square of the turns ratio.

Key Concepts

AC Circuit

An AC circuit, short for alternating current circuit, involves the flow of electric charge that periodically reverses direction. Unlike direct current (DC), which flows in one direction, AC switches directions, causing the voltage to alternate between positive and negative values. This type of circuit is pervasive in household and industrial applications due to its efficient power distribution over long distances. In an AC circuit, the voltage and currents are typically expressed as sine or cosine waves.

Understanding these waves' properties is crucial, especially their amplitude, which is the maximum height of the wave. In our example, the voltage amplitude is denoted by \( V_0 \), and the current by \( I_0 \). The relationship between voltage and current is paramount and described by the equivalent resistance \( R_{\mathrm{eq}} \).

• Voltage: The potential difference across any two points in an AC circuit, expressed as \( V(t) = V_0 \sin(\omega t) \), where \( \omega \) is the angular frequency.
• Current: The flow of electrons within the circuit, given by \( I(t) = I_0 \sin(\omega t) \).
• Equivalent Resistance: The ratio \( R_{\mathrm{eq}} = V_0 / I_0 \), representing how much the external circuit resists the AC source.

Transformer Turns Ratio

A transformer, used extensively in AC circuits, alters voltage levels using inductively coupled coils called primary and secondary windings. The turns ratio, which is the ratio of the number of turns in the primary coil \( N_{\mathrm{P}} \) to those in the secondary \( N_{\mathrm{S}} \), is crucial in transforming voltages and currents.

Here's how it works: when an AC voltage is applied to the primary coil, it creates a magnetic field that induces a voltage in the secondary coil, depending on the turns ratio. The mathematical expression for this transformation is:

• Primary Voltage to Secondary Voltage: \( V_{\mathrm{S}} = V_{\mathrm{P}} \frac{N_{\mathrm{S}}}{N_{\mathrm{P}}} \)
• Primary Current to Secondary Current: \( I_{\mathrm{S}} = I_{\mathrm{P}} \frac{N_{\mathrm{P}}}{N_{\mathrm{S}}} \)
• Apparent Resistance Transformation: \( R_{\mathrm{eq}} = R \left( \frac{N_{\mathrm{P}}}{N_{\mathrm{S}}} \right)^2 \)

Understanding this ratio allows engineers to design circuits that efficiently transfer power, minimizing losses and maximizing performance. This is integral in electrical distribution across the grid and in devices like chargers and adapters.

Power Transfer Maximization

Maximizing power transfer in an AC circuit is essential, especially in applications involving transformers. The concept involves adjusting circuit elements to ensure maximum energy is transferred from the source to the load, minimizing losses. This is often achieved by impedance matching, where the load's impedance is made equal to the source's internal impedance. However, in circuits involving transformers, you can alter the load's apparent resistance as seen by the source using the turns ratio.

Maximum power transfer comes into play significantly:

• When transmitting electricity over long distances—transformers are used to step up voltage, reducing current and thereby minimizing resistive losses, as power lost is proportional to the square of the current.
• In audio amplifiers and communication systems—adjusting impedance for speakers or antennas ensures that most of the power is delivered to the intended device.

Through transformations, you can ensure that the equivalent resistance matches what the source "expects," allowing for efficient power delivery and optimizing system performance. It's about balancing input and output to ensure that energy is transferred rather than wasted as heat or signal loss.