Transformers and LC Oscillations: Quick Revision

A brief note on Transformers and LC Oscillations. Read it before a test. Try each question at the end before you open its answer.

The transformer

• Principle: mutual induction. The primary makes a changing flux in a soft iron core. The secondary has an emf induced in it. It works on AC only.
• Voltages: $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$. Step-up: $N_s > N_p$. Step-down: $N_s < N_p$.
• Currents (ideal): $V_pI_p = V_sI_s$, so $\dfrac{I_p}{I_s} = \dfrac{N_s}{N_p}$. More voltage means less current.
• Efficiency: $\eta = \dfrac{P_{out}}{P_{in}}$, always below $1$.
• Losses: copper loss $I^2R$ (thick wire), eddy currents (laminated core), hysteresis (soft iron), flux leakage (coils wound close).

Power transmission

• Line current: $I = \dfrac{P}{V}$ at power factor $1$.
• Heat in the line: $I^2r = \dfrac{P^2r}{V^2}$. Ten times the voltage gives a hundred times less loss.
• Voltage lost along the line: $Ir$. It is not the whole line voltage, so $V^2/r$ is wrong.
• The chain: generator, step-up transformer, long line, step-down substation, pole transformer, $230$ V socket.

LC oscillations

• Equation: $L\dfrac{d^2q}{dt^2} + \dfrac{q}{C} = 0$, with $q = q_0\cos\omega_0 t$.
• Frequency: $\omega_0 = \dfrac{1}{\sqrt{LC}}$ and $T = 2\pi\sqrt{LC}$.
• Current: $I_0 = \omega_0q_0$. It is largest when $q = 0$.
• Energy: $\dfrac{q^2}{2C} + \tfrac{1}{2}Li^2 = \dfrac{q_0^2}{2C}$, constant with no resistance.
• Analogy: $q$ and $x$, $i$ and $v$, $L$ and $m$, $1/C$ and $k$. A resistance damps the swing as $e^{-Rt/2L}$.

A worked example

A step-down transformer has $1800$ turns in its primary and $90$ turns in its secondary. It is joined to a $240$ V supply and the secondary gives $5.0$ A. Find the secondary voltage and the primary current. Treat it as ideal.

1. Voltage. $V_s = 240\ \text{V} \times \dfrac{90}{1800} = 12$ V.
2. Power. $P = 12\ \text{V} \times 5.0\ \text{A} = 60$ W.
3. Current. $I_p = \dfrac{60\ \text{W}}{240\ \text{V}} = 0.25$ A.

Practise with new numbers

Check yourself

1. A transformer has $N_p = 300$ and $N_s = 60$. The primary voltage is $150$ V. What is the secondary voltage, and is the transformer step-up or step-down?

2. A line carries $4.0$ A and has a resistance of $25\ \Omega$. How much heat is made in it each second?

3. A coil of $1.0$ mH and a capacitor of $4.0\ \mu\text{F}$ form an LC circuit. What is its period?