DC Motor: Speed, Torque, & Power

Demonstrative Video

Dc Motor Basic Operational Video

Back EMF in DC Motor

\(I\) carrying conductor \(\Rightarrow\) \(B\) \(\Rightarrow\) \(T\) induce \(\Rightarrow\) rotates conductor \(\Rightarrow\) cuts \(\Phi\) \(\Rightarrow\) EMF induce
Direction of induced EMF is opposite to \(V\) \(\Rightarrow\) Counter or Back EMF
\(E_b\) series with \(V\) but opposite in direction, i.e. oppose \(I\) which causes it

Advantage of Back EMF

Electrical to Mechanical energy (\(E_bI_a\)) conversion is possible only because of \(E_b\).
\(E_b\) makes dc motor self-regulating \(\Rightarrow\) \(I=\dfrac{V-E_b}{R_a}\)
- No load \(\Rightarrow\) requires small \(T\) \(\Rightarrow\) for controlling friction and windage loss \(\Rightarrow\) withdraws less \(I\) \(\Rightarrow\) \(E_b \downarrow\)
- load \(\uparrow\) \(\Rightarrow\) \(\left(T_m<T_L\right)\) \(\Rightarrow\) \(N \downarrow\) \(\Rightarrow\) \(E_b \downarrow\) \(\Rightarrow\) \(I \uparrow\) \(\Rightarrow\) \(T \uparrow\)
- load \(\downarrow\) \(\Rightarrow\) \(\left(T_m>T_L\right)\) \(\Rightarrow\) \(N \uparrow\) \(\Rightarrow\) \(E_b \uparrow\) \(\Rightarrow\) \(I \downarrow\) \(\Rightarrow\) \(T \downarrow\)

Condition for Maximum Power, \(P_m\)

\[\begin{aligned} P_{m} & =VI_{a}-I_{a}^{2}R_{a}\nonumber\\ \Rightarrow\dfrac{dP_{m}}{dI_{a}} & =V-2I_{a}R_{a}=0 \Rightarrow I_{a}R_{a} =\dfrac{V}{2} \\ \therefore V & = E_{b}+ I_{a}R_{a} = E_{b}+\dfrac{V}{2} \Rightarrow \boxed{E_{b} =\dfrac{V}{2}}\nonumber \end{aligned}\]

Torque Equation of a DC Motor

\[\begin{aligned} P_{m} & \Rightarrow E_{b}I_{a}=T_{a}\omega\\ & \Rightarrow\left(\dfrac{\Phi PN}{60}\times\dfrac{Z}{A}\right)I_{a}=T_{a}\left(\dfrac{2\pi N}{60}\right)\\ & \Rightarrow T_{a}=\dfrac{1}{2\pi}\times\dfrac{\Phi PZ}{A}\times I_{a}\\ & \boxed{T_{a} =K_{a}\phi I_{a}}\\ T_{sh} & =T_{a}-\mbox{mechanical losses} \end{aligned}\]

For series motor: \(\Phi \propto I_a \Rightarrow T_a \propto I_a^2\)

For shunt motor: \(\Phi =\) constant \(\Rightarrow T_a \propto I_a\)

Speed of a DC Motor

\[\begin{aligned} E_{b} & =V-I_{a}R_{a}\\ \Rightarrow\dfrac{\Phi PN}{60}\times\left(\dfrac{Z}{A}\right) & =V-I_{a}R_{a}\\ \Rightarrow N & =\dfrac{V-I_{a}R_{a}}{\Phi}\times\left(\dfrac{60A}{ZP}\right)\\ \Rightarrow N & =\dfrac{E_{b}}{\Phi}\times\left(\dfrac{60A}{ZP}\right)\\ \Rightarrow & \boxed{N =K\dfrac{E_{b}}{\Phi}} \end{aligned}\] \(\boxed{N\propto E_b}\) and \(\boxed{N \propto 1/\Phi}\) \[\dfrac{N_{2}}{N_{1}}=\dfrac{E_{b2}}{E_{b1}}\times\dfrac{\Phi_{1}}{\Phi_{2}}=\dfrac{E_{b2}}{E_{b1}}\times\dfrac{I_{a1}}{I_{a2}}\]

\[\begin{aligned} & \boxed{N =K\dfrac{E_{b}}{\Phi}}\\ & \boxed{T_{a} \propto\Phi I_{a}} \end{aligned}\]

Relation between Torque and Speed

\(T_a\) is function of \(\Phi\) and \(I_a\) but independent of \(N\)
\(N\) depends on \(T\) and not vice-versa
\(\Phi \uparrow \Rightarrow N \downarrow\) but \(T_a \uparrow\) not possiblebecause \(T\) always tends to produce rotation. If \(T \uparrow\), \(N\) must \(\uparrow\) rather than \(\downarrow\)

Following Sequence happens:

Suppose \(\Phi\) is \(\downarrow\) by \(I_f \downarrow\)
\(\left[E_b \propto N\Phi\right] \downarrow\), \(N\) = constant (inertia of heavy armature)
\(E_b \downarrow \Rightarrow \left[I_a = (V-E_b)/R_a\right] \uparrow\). Moreover, \(\Phi \downarrow \Rightarrow I_a \uparrow \uparrow\)
\(T_a \propto \Phi I_a\), \(\Phi \downarrow\) is counterbalanced by \(I_a \uparrow \uparrow\), result \(T_a \uparrow\)
\(T_a \uparrow \Rightarrow N \uparrow\)

Speed Regulation

Change in speed when the load on the motor is reduced from rated value to zero, expressed as percent of the rated load speed \[\% \mbox{speed regulation} = \dfrac{N.L~speed - F.L~speed}{F.L~speed} \times 100\]