Question 2 of 5: Diode Bridge Waveform Sketch

Question 2: Diode Bridge Waveform Sketch (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A resistive bridge driven from $v_{IN}$ (1 kHz triangle, $\pm10$ V peak); the two output nodes $v_1$, $v_2$ are joined by $R_5$ in series with $D_1$ (anode toward $v_1$, cathode at $v_2$; ideal with $0.7$ V drop). Component values:

Find. The time waveforms of $v_1$ and $v_2$ over one period.

Approach. With $D_1$ off the two arms are independent voltage dividers of $v_{IN}$; find each divider ratio, then determine the small window in which the $0.7$ V threshold of $R_5$–$D_1$ is exceeded and re-solve there.

1. Divider ratios (diode off). No current flows through the $R_5$–$D_1$ branch, so $$v_1=v_{IN}\frac{R_3}{R_1+R_3}=0.825\,v_{IN},\qquad v_2=v_{IN}\frac{R_4}{R_2+R_4}=0.902\,v_{IN}.$$ Both are scaled copies of the input triangle, with peak amplitudes $8.25$ V and $9.02$ V.
2. When does $D_1$ conduct? The branch carries current from $v_1$ to $v_2$ only when the anode leads the cathode by a diode drop, i.e. $v_1-v_2\gt 0.7$ V. Using the open-circuit values, $v_1-v_2=(0.825-0.902)v_{IN}=-0.077\,v_{IN}$, which exceeds $0.7$ V only when $$v_{IN}\lt -\frac{0.7}{0.077}=\boxed{-9.08\ \text{V}}.$$ For positive $v_{IN}$, $v_2\gt v_1$ so $D_1$ is reverse-biased; the diode matters only in a thin sliver near the negative peak.
3. Conducting state at the negative peak. At $v_{IN}=-10$ V, solving the two node equations with $v_1-v_2=IR_5+0.7$ gives a small forward current $I\approx 18\ \mu\text{A}$ and $$v_1\approx-8.26\ \text{V},\qquad v_2\approx-9.00\ \text{V},$$ versus the un-clamped $-8.25$ V and $-9.02$ V. The diode pulls the two nodes together by only about $20\ \text{mV}$ — a barely-perceptible flattening.
4. Resulting sketch. Over almost the entire cycle $v_1$ and $v_2$ are triangles of amplitude $8.25$ V and $9.02$ V that track the input; a tiny kink appears on both only in the interval $v_{IN}\in[-10,\,-9.08]$ V near the negative crest.