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I believe that base width modulation is also known as the Early effect. In a transistor which is conducting, but not in saturation, the effective width of the base is modulated by depletion at the collector-base junction. As the collector-base (and collector-emitter) voltage increases, the depletion region gets wider, causing the effective width of the base region (the region between the collector and emitter) to decrease. This in turn causes the current gain, or beta, to be greater at higher collector-base voltages. See the graph below, which I got from a site called **broken link removed**.
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