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In a DC motor, series circuit resistance plays a crucial role in controlling the speed of the motor. The total resistance in the motor, which comprises both the armature resistance \( R_A \) and the field winding resistance \( R_F \), affects how much of the supply voltage is actually used to generate the back electromotive force (back EMF), which in turn affects the speed.
To control the speed of the motor, you can add an external resistor \( R_{ext} \) in series. This added resistance will further drop the voltage across the armature, meaning less voltage is available for back EMF, which leads to a reduction in speed.

• Initial total resistance without \( R_{ext} \): \( R_A + R_F = 0.1 \ \Omega \)
• Total resistance with \( R_{ext} \): \( R_A + R_F + R_{ext} \)

Understanding how series circuit resistance affects the speed is key to correctly tuning the motor's performance for various applications.

The torque-speed relationship in a DC motor signifies how these two quantities interplay. When you apply more load (torque) on the motor, it naturally wants to slow down unless additional power is provided.
In this context, the constant torque value \( T_d = 25 \ \text{Nm} \) informs us that regardless of the speed change, the demand for this consistent level of torque persists. However, since DC motors operate under the principle that speed is proportional to the net current supplied after accounting for the resistive voltage drop, altering speed implies shifting the balance through changes in resistance.The formula to understand this is:\[\frac{N_1}{N_2} = \frac{V_T - I_a \cdot R}{V_T - I_a \cdot (R + R_{ext})}\]Here, \( N_1 \) and \( N_2 \) are the initial and target speeds, respectively. The equation underscores that speed changes inversely with the voltage available after accounting for internal and external resistance.
This relationship is pivotal in applications where precise control of motor speed is required while ensuring the torque remains constant.

Armature current \( I_a \) is a significant quantity in understanding motor performance as it directly influences both torque and speed. In a DC motor, torque is generated by the armature current and the torque constant \( k_T \).
To determine \( I_a \) when other quantities like torque remain constant, we use the expression:\[T_d = k_T \cdot I_a\]However, without the torque constant \( k_T \), discovering \( I_a \) often demands empirical data or assumptions based on standard motor behavior. Once \( I_a \) is known, it helps re-calculate effective voltage and consequently the motor speed, especially when making adjustments to control resistance.
Calculating \( I_a \) accurately ensures the torque-speed relationships are correctly leveraged in tuning motor performance through resistance modifications.